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+\section{Sum of 2 Squares in 2 Distinct Ways}
+Tags: Square Numbers, Sums of Squares, Brahmagupta-Fibonacci Identity, Sum of 2 Squares in 2 Distinct Ways
+
+\begin{theorem}
+Let $m, n \in \Z_{>0}$ be [[Definition:Distinct|distinct]] [[Definition:Positive Integer|positive integers]] that can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Distinct|distinct]] [[Definition:Square Number|square numbers]].
+Then $m n$ can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|square numbers]] in at least two [[Definition:Distinct|distinct]] ways.
+\end{theorem}
+
+\begin{proof}
+Let:
+:$m = a^2 + b^2$
+:$n = c^2 + d^2$
+Then:
+{{begin-eqn}}
+{{eqn | l = m n
+ | r = \paren {a^2 + b^2} \paren {c^2 + d^2}
+ | c =
+}}
+{{eqn | r = \paren {a c + b d}^2 + \paren {a d - b c}^2
+ | c = [[Brahmagupta-Fibonacci Identity]]
+}}
+{{eqn | r = \paren {a c - b d}^2 + \paren {a d + b c}^2
+ | c = [[Brahmagupta-Fibonacci Identity/Corollary|Brahmagupta-Fibonacci Identity: Corollary]]
+}}
+{{end-eqn}}
+It remains to be shown that $a \ne b$ and $c \ne d$, then the four numbers:
+:$a c + b d, a d - b c, a c - b d, a d + b c$
+are distinct.
+Because $a, b, c, d > 0$, we have:
+:$a c + b d \ne a c - b d$
+:$a d + b c \ne a d - b c$
+We also have:
+{{begin-eqn}}
+{{eqn | l = a c \pm b d
+ | r = a d \pm b c
+}}
+{{eqn | ll= \leadstoandfrom
+ | l = a c \mp b c - a d \pm b d
+ | r = 0
+}}
+{{eqn | ll= \leadstoandfrom
+ | l = c \paren {a \mp b} - d \paren {a \mp b}
+ | r = 0
+}}
+{{eqn | ll= \leadstoandfrom
+ | l = \paren {a \mp b} \paren {c \mp d}
+ | r = 0
+}}
+{{end-eqn}}
+Thus $a \ne b$ and $c \ne d$ implies $a c \pm b d \ne a d \pm b c$.
+The case for $a c \pm b d \ne a d \mp b c$ is similar.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Bounds for Finite Product of Real Numbers}
+Tags: Inequalities
+
+\begin{theorem}
+Let $a_1, a_2, \ldots, a_n$ be [[Definition:Positive Real Number|positive real numbers]].
+Then:
+:$\displaystyle \sum_{k \mathop = 1}^n a_k \le \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \map \exp {\sum_{k \mathop = 1}^n a_k}$
+\end{theorem}
+
+\begin{proof}
+=== Lower bound ===
+Follows by expanding.
+{{qed|lemma}}
+=== Upper Bound ===
+==== Proof 1 ====
+By [[Exponential of x not less than 1+x]]:
+:$\displaystyle \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \prod_{k \mathop = 1}^n \exp a_k = \map \exp {\sum_{k \mathop = 1}^n a_k}$
+{{qed}}
+==== Proof 2 ====
+By the [[AM-GM Inequality]]:
+:$\displaystyle \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \paren {\frac {n + \sum_{k \mathop = 1}^n a_k} n}^n$
+{{link wanted|link to an article that this is less than $\exp(\sum a_k)$, i.e. that this expression is increasing with $n$}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Bounds for Complex Exponential}
+Tags: Exponential Function
+
+\begin{theorem}
+Let $\exp$ denote the [[Definition:Complex Exponential Function|complex exponential]].
+Let $z \in \C$ with $\cmod z \le \dfrac 1 2$.
+Then
+:$\dfrac 1 2 \cmod z \le \cmod {\exp z - 1} \le \dfrac 3 2 \cmod z$
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Complex Exponential Function|complex exponential]]:
+:$\exp z = \displaystyle \sum_{n \mathop = 1}^\infty \frac {z^n} {n!}$
+Thus
+{{begin-eqn}}
+{{eqn | l = \cmod {\exp z - 1 - z}
+ | r = \cmod {\sum_{n \mathop = 2}^\infty \frac {z^n} {n!} }
+ | c = [[Linear Combination of Convergent Series]]
+}}
+{{eqn | o = \le
+ | r = \sum_{n \mathop = 2}^\infty \cmod {\frac {z^n} {n!} }
+ | c = [[Triangle Inequality for Series]]
+}}
+{{eqn | o = \le
+ | r = \sum_{n \mathop = 2}^\infty \frac {\cmod z^n} 2
+ | c = as $n \ge 2$
+}}
+{{eqn | o = =
+ | r = \frac {\cmod z^2 / 2} {1 - \cmod z}
+ | c = [[Sum of Geometric Sequence]]
+}}
+{{eqn | o = \le
+ | r = \frac 1 2 \cmod z
+ | c = as $\cmod z \le \dfrac 1 2$
+}}
+{{end-eqn}}
+By the [[Triangle Inequality]]:
+:$\dfrac 1 2 \cmod z \le \cmod {\exp z - 1} \le \dfrac 3 2 \cmod z$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum of 2 Squares in 2 Distinct Ways/Sequence}
+Tags: Sum of 2 Squares in 2 Distinct Ways
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] which can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|square numbers]] in two or more different ways begins:
+{{begin-eqn}}
+{{eqn | l = 50
+ | r = 7^2 + 1^2
+ | rr= = 5^2 + 5^2
+}}
+{{eqn | l = 65
+ | r = 8^2 + 1^2
+ | rr= = 7^2 + 4^2
+}}
+{{eqn | l = 85
+ | r = 9^2 + 2^2
+ | rr= = 7^2 + 6^2
+}}
+{{eqn | l = 125
+ | r = 11^2 + 2^2
+ | rr= = 10^2 + 5^2
+}}
+{{eqn | l = 130
+ | r = 11^2 + 3^2
+ | rr= = 9^2 + 7^2
+}}
+{{eqn | l = 145
+ | r = 12^2 + 1^2
+ | rr= = 9^2 + 8^2
+}}
+{{eqn | l = 170
+ | r = 13^2 + 1^2
+ | rr= = 11^2 + 7^2
+}}
+{{end-eqn}}
+\end{theorem}<|endoftext|>
+\section{Sum of 2 Squares in 2 Distinct Ways/Examples/50}
+Tags: Sum of 2 Squares in 2 Distinct Ways, 50
+
+\begin{theorem}
+$50$ is the smallest [[Definition:Positive Integer|positive integer]] which can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|square numbers]] in two [[Definition:Distinct|distinct]] ways:
+{{begin-eqn}}
+{{eqn | l = 50
+ | r = 5^2 + 5^2
+}}
+{{eqn | r = 7^2 + 1^2
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+The smallest two [[Definition:Positive Integer|positive integers]] which can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Distinct|distinct]] [[Definition:Square Number|square numbers]] are:
+{{begin-eqn}}
+{{eqn | l = 5
+ | r = 1^2 + 2^2
+}}
+{{eqn | l = 10
+ | r = 1^2 + 3^2
+}}
+{{end-eqn}}
+We have that:
+:$50 = 5 \times 10$
+Thus:
+{{begin-eqn}}
+{{eqn | r = \paren {1^2 + 2^2} \paren {1^2 + 3^2}
+ | c =
+}}
+{{eqn | r = \paren {1 \times 1 + 2 \times 3}^2 + \paren {1 \times 3 - 2 \times 1}^2
+ | c = [[Brahmagupta-Fibonacci Identity]]
+}}
+{{eqn | r = \paren {1 + 6}^2 + \paren {3 - 2}^2
+ | c =
+}}
+{{eqn | r = 7^2 + 1^2
+ | c =
+}}
+{{eqn | r = 49 + 1
+ | c =
+}}
+{{eqn | r = 50
+ | c =
+}}
+{{end-eqn}}
+and:
+{{begin-eqn}}
+{{eqn | r = \paren {1^2 + 2^2} \paren {1^2 + 3^2}
+ | c =
+}}
+{{eqn | r = \paren {1 \times 1 - 2 \times 3}^2 + \paren {1 \times 3 + 2 \times 1}^2
+ | c = [[Brahmagupta-Fibonacci Identity/Corollary|Brahmagupta-Fibonacci Identity: Corollary]]
+}}
+{{eqn | r = \paren {1 - 6}^2 + \paren {3 + 2}^2
+ | c =
+}}
+{{eqn | r = \paren {6 - 1}^2 + \paren {3 + 2}^2
+ | c =
+}}
+{{eqn | r = 5^2 + 5^2
+ | c =
+}}
+{{eqn | r = 25 + 25
+ | c =
+}}
+{{eqn | r = 50
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum of 2 Squares in 2 Distinct Ways/Examples/65}
+Tags: Sum of 2 Squares in 2 Distinct Ways, 65
+
+\begin{theorem}
+$65$ can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|square numbers]] in two [[Definition:Distinct|distinct]] ways:
+{{begin-eqn}}
+{{eqn | l = 65
+ | r = 8^2 + 1^2
+}}
+{{eqn | r = 7^2 + 4^2
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+We have that:
+:$65 = 5 \times 13$
+Both $5$ and $13$ can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Distinct|distinct]] [[Definition:Square Number|square numbers]]:
+{{begin-eqn}}
+{{eqn | l = 5
+ | r = 1^2 + 2^2
+}}
+{{eqn | l = 13
+ | r = 2^2 + 3^2
+}}
+{{end-eqn}}
+Thus:
+{{begin-eqn}}
+{{eqn | r = \paren {1^2 + 2^2} \paren {2^2 + 3^2}
+ | c =
+}}
+{{eqn | r = \paren {1 \times 2 + 2 \times 3}^2 + \paren {1 \times 3 - 2 \times 2}^2
+ | c = [[Brahmagupta-Fibonacci Identity]]
+}}
+{{eqn | r = \paren {2 + 6}^2 + \paren {3 - 4}^2
+ | c =
+}}
+{{eqn | r = \paren {2 + 6}^2 + \paren {4 - 3}^2
+ | c =
+}}
+{{eqn | r = 8^2 + 1^2
+ | c =
+}}
+{{eqn | r = 64 + 1
+ | c =
+}}
+{{eqn | r = 65
+ | c =
+}}
+{{end-eqn}}
+and:
+{{begin-eqn}}
+{{eqn | r = \paren {1^2 + 2^2} \paren {2^2 + 3^2}
+ | c =
+}}
+{{eqn | r = \paren {1 \times 2 - 2 \times 3}^2 + \paren {1 \times 3 + 2 \times 2}^2
+ | c = [[Brahmagupta-Fibonacci Identity/Corollary|Brahmagupta-Fibonacci Identity: Corollary]]
+}}
+{{eqn | r = \paren {2 - 6}^2 + \paren {3 + 4}^2
+ | c =
+}}
+{{eqn | r = \paren {6 - 2}^2 + \paren {3 + 4}^2
+ | c =
+}}
+{{eqn | r = 4^2 + 7^2
+ | c =
+}}
+{{eqn | r = 16 + 49
+ | c =
+}}
+{{eqn | r = 65
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Continuous Implies Locally Bounded}
+Tags: Boundedness
+
+\begin{theorem}
+Let $X$ be a [[Definition:Topological Space|topological space]].
+Let $M$ be a [[Definition:Metric Space|metric space]].
+Let $f: X \to M$ be [[Definition:Continuous Mapping|continuous]].
+Then $f$ is [[Definition:Locally Bounded Mapping|locally bounded]].
+\end{theorem}
+
+\begin{proof}
+Let $x \in X$.
+Let $U = \map {f^{-1} } {\map B {\map f x, 1} }$.
+{{explain|$\map B {\map f x, 1}$}}
+By [[Definition:Continuous Mapping|continuity]], $U$ is a [[Definition:Neighborhood of Point in Topological Space|neighborhood]] of $x$.
+Because $\map f U \subset \map B {\map f x, 1}$, $f$ is [[Definition:Bounded Mapping|bounded]] on $U$.
+Thus $f$ is [[Definition:Locally Bounded Mapping|locally bounded]].
+{{qed}}
+[[Category:Boundedness]]
+032x9q3dgsc1qdh2h7vr31bk99m0l6f
+\end{proof}<|endoftext|>
+\section{Lower Adjoint at Element is Minimum of Preimage of Singleton of Element implies Composition is Identity}
+Tags: Galois Connections
+
+\begin{theorem}
+Let $L = \struct {S, \preceq}, R = \struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
+Let $g: S \to T, d: T \to S$ be [[Definition:Mapping|mappings]] such that
+:$\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {\set t} }$
+Then $g \circ d = I_T$
+where $I_T$ denotes the [[Definition:Identity Mapping|identity mapping]] of $T$.
+\end{theorem}
+
+\begin{proof}
+Let $t \in T$.
+By assumption:
+:$\map d t = \min \set {g^{-1} \sqbrk {\set t} }$
+By definition of [[Definition:Min Operation|min operation]]:
+:$\map d t = \map \inf {g^{-1} \sqbrk {\set t} }$ and $\map d t \in g^{-1} \sqbrk {\set t}$
+By definition of [[Definition:Image of Subset under Mapping|image of set]]:
+:$\map g {\map d t} \in \set t$
+By definition of [[Definition:Singleton|singleton]]:
+:$\map g {\map d t} = t$
+Thus by definition of [[Definition:Composition of Mappings|composition of mappings]]:
+:$\map {\paren {g \circ d} } t = t$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Bounds for Weierstrass Elementary Factors}
+Tags: Infinite Products
+
+\begin{theorem}
+Let $E_p: \C \to \C$ denote the $p$th [[Definition:Weierstrass Elementary Factor|Weierstrass elementary factor]]:
+:$\map {E_p} z = \begin{cases} 1 - z & : p = 0 \\
+\paren {1 - z} \map \exp {z + \dfrac {z^2} 2 + \cdots + \dfrac {z^p} p} & : \text{otherwise}\end{cases}$
+Let $z \in \C$.
+\end{theorem}
+
+\begin{proof}
+=== Proof of some bound ===
+Let $\cmod z \le \dfrac 1 2$.
+We may assume $p \ge 1$.
+We have:
+:$\map {E_p} z = \map \exp {\map \log {1 - z} + \displaystyle \sum_{k \mathop = 1}^p \frac {z^k} k}$
+Then:
+{{begin-eqn}}
+{{eqn | l = \cmod {\map \log {1 - z} + \sum_{k \mathop = 1}^p \frac {z^k} k}
+ | r = \cmod {-\sum_{k \mathop = p + 1}^\infty \frac{z^k} k}
+ | c = [[Series Expansion of Complex Logarithm]]
+}}
+{{eqn | o = \le
+ | r = \sum_{k \mathop = p + 1}^\infty \frac {\cmod z^k} k
+ | c = [[Triangle Inequality for Series]]
+}}
+{{eqn | o = \le
+ | r = \sum_{k \mathop = p + 1}^\infty \cmod z^k
+ | c = because $k \ge 1$
+}}
+{{eqn | r = \frac {\cmod z^{p + 1} } {1 - \cmod z}
+ | c = [[Sum of Geometric Sequence]]
+}}
+{{eqn | o = \le
+ | r = 2 \cmod z^{p + 1}
+ | c = because $\cmod z \le \dfrac 1 2$
+}}
+{{end-eqn}}
+Because $p \ge 1$:
+:$2 \cmod z^{p + 1} \le \dfrac 1 2$
+By [[Bounds for Complex Exponential]]:
+:$\cmod {\map {E_p} z - 1} \le 3 \cmod z^{p + 1}$
+{{qed}}
+=== Proof of another bound ===
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Order of Product of Entire Functions}
+Tags: Entire Functions
+
+\begin{theorem}
+Let $f, g: \C \to \C$ be [[Definition:Entire Function|entire functions]] of [[Definition:Order of Entire Function|order]] $\alpha$ and $\beta$.
+Then $f g$ has [[Definition:Order of Entire Function|order]] at most $\map \max {\alpha, \beta}$.
+\end{theorem}<|endoftext|>
+\section{Numbers Partitioned into up to 4 Squares in 5 Ways}
+Tags: Sums of Squares
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integers]] can be expressed as the [[Definition:Integer Addition|sum]] of no more than $4$ [[Definition:Square Number|squares]] in $5$ [[Definition:Distinct|distinct]] ways:
+:$50, 52, 54, 58, \ldots$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 50
+ | r = 7^2 + 1^2
+ | c =
+}}
+{{eqn | r = 5^2 + 5^2
+ | c =
+}}
+{{eqn | r = 5^2 + 4^2 + 3^2
+ | c =
+}}
+{{eqn | r = 4^2 + 4^2 + 3^2 + 3^2
+ | c =
+}}
+{{eqn | r = 6^2 + 3^2 + 2^2 + 1^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 52
+ | r = 7^2 + 1^2 + 1^2 + 1^2
+ | c =
+}}
+{{eqn | r = 6^2 + 4^2
+ | c =
+}}
+{{eqn | r = 5^2 + 5^2 + 1^2 + 1^2
+ | c =
+}}
+{{eqn | r = 4^2 + 4^2 + 4^2 + 2^2
+ | c =
+}}
+{{eqn | r = 5^2 + 3^2 + 3^2 + 3^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 54
+ | r = 7^2 + 2^2 + 1^2
+ | c =
+}}
+{{eqn | r = 6^2 + 4^2 + 1^2 + 1^2
+ | c =
+}}
+{{eqn | r = 6^2 + 3^2 + 3^2
+ | c =
+}}
+{{eqn | r = 5^2 + 5^2 + 2^2
+ | c =
+}}
+{{eqn | r = 5^2 + 4^2 + 3^2 + 2^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 58
+ | r = 7^2 + 3^2
+ | c =
+}}
+{{eqn | r = 7^2 + 2^2 + 2^2 + 1^2
+ | c =
+}}
+{{eqn | r = 6^2 + 3^2 + 3^2 + 2^2
+ | c =
+}}
+{{eqn | r = 5^2 + 5^2 + 2^2 + 2^2
+ | c =
+}}
+{{eqn | r = 5^2 + 4^2 + 4^2 + 1^2
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Order of Product of Entire Function with Polynomial}
+Tags: Entire Functions
+
+\begin{theorem}
+Let $f:\C\to\C$ be an [[Definition:Entire Function|entire function]] of [[Definition:Order of Entire Function|order]] $\omega$.
+Let $P:\C\to\C$ be a [[Definition:Zero Polynomial|nonzero]] [[Definition:Complex Polynomial Function|polynomial]].
+Then $f\cdot P$ has [[Definition:Order of Entire Function|order]] $\omega$.
+\end{theorem}
+
+\begin{proof}
+By [[Order of Product of Entire Functions]] and [[Polynomial has Order Zero]], $f\cdot P$ has order at most $\omega$.
+By [[Limit at Infinity of Polynomial]], there exist $r,\delta>0$ such that $|P(z)| \geq \delta$ for $|z|\geq r$.
+Suppose $\displaystyle \log\left(\max_{|z|\leq R}|f(z)P(z)|\right) = O(R^\beta)$ for some $\beta < \omega$.
+By the [[Maximum Modulus Principle]] $\max_{|z|\leq R}|f(z)| \leq \frac1\delta \max_{|z|\leq R}|f(z)P(z)|$ for $R\geq r$.
+Thus $\displaystyle \log\left(\max_{|z|\leq R}|f(z)|\right) = O(R^\beta)$.
+This is a contradiction.
+Thus $f\cdot P$ has [[Definition:Order of Entire Function|order]] $\omega$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Order of Shifted Entire Function}
+Tags: Entire Functions
+
+\begin{theorem}
+Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]] of [[Definition:Order of Entire Function|order]] $\alpha$.
+Let $a \in \C$.
+Then $f \left({z + a}\right)$ has [[Definition:Order of Entire Function|order]] $\alpha$.
+\end{theorem}<|endoftext|>
+\section{Zerofree Entire Function of Finite Order is Exponential of Polynomial}
+Tags: Entire Functions
+
+\begin{theorem}
+Let $f:\C\to\C$ be an [[Definition:Entire Function|entire function]] of [[Definition:Order of Entire Function|finite order]].
+Let $f$ have no zeroes.
+Then $f=\exp P$ for some [[Definition:Polynomial|polynomial]] $P$.
+\end{theorem}
+
+\begin{proof}
+This is an immediate consequence of [[Hadamard Factorization Theorem]].
+{{qed}}
+[[Category:Entire Functions]]
+fv2qke2k3ajg5o0nqogidmbn8so234u
+\end{proof}<|endoftext|>
+\section{Borel-Carathéodory Lemma}
+Tags: Complex Analysis
+
+\begin{theorem}
+Let $D \subset \C$ be an [[Definition:Open Set (Complex Analysis)|open set]] with $0 \in D$.
+Let $R > 0$ be such that $\map B {0, R} \subset D$.
+{{explain|Domain of $R$, definition of $B$}}
+Let $f: D \to \C$ be [[Definition:Analytic Function|analytic]] with $\map f 0 = 0$.
+Let $\map \Re {\map f z} \le M$ for $\cmod z \le R$.
+Let $0 < r < R$.
+Then for $\cmod z \le r$:
+:$(1): \quad \cmod {\map f z} \le \dfrac {2 M r} {R - r}$
+:$(2): \quad \cmod {\map {f^{\paren k} } z} \le \dfrac {2 M R k!} {\paren {R - r}^{k + 1} }$ for all $k \ge 1$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+{{Namedfor|Émile Borel|name2 = Constantin Carathéodory|cat = Borel, Émile|cat2 = Carathéodory}}
+\end{proof}<|endoftext|>
+\section{Order of Sum of Entire Functions}
+Tags: Entire Functions
+
+\begin{theorem}
+Let $f, g: \C \to \C$ be [[Definition:Entire Function|entire functions]] of [[Definition:Order of Entire Function|order]] $\alpha$ and $\beta$.
+Then $f + g$ has [[Definition:Order of Entire Function|order]] at most $\map \max {\alpha, \beta}$, with equality if $\alpha \ne \beta$.
+\end{theorem}
+
+\begin{proof}
+{{MissingLinks|Each step of this proof needs to be backed up with evidence: a link to a result or a definition, preferably.}}
+Let $\map f z = \map \OO {e^{\cmod z^m} }$ and $\map g z = \map \OO {e^{\cmod z^k} }$.
+{{WLOG}}, let $m \ge k$.
+Then it follows that $\map f z + \map g z = \map \OO {e^{\cmod z^m} }$.
+Then the order of $f + g$ is at most $m$.
+Since $\alpha, \beta$ are the infima of $m, k$, it follows that the order of $f + g$ is at most $\alpha$.
+Let $\alpha > \beta$ and $\map f z = \map \OO {e^{\cmod z^m} }$.
+Then:
+:$\map g z = \map {\mathcal o} {e^{\cmod z^m} }$
+So $\dfrac {\map g z} {e^{\cmod z^m} } \to 0$ as $\cmod z \to \infty$.
+In particular:
+:$\dfrac {\map f z + \map g z} {e^{\cmod z^m} } \sim \dfrac {\map f z} {e^{\cmod z^m} } > 0$
+and minimizing $m$ we see that the order of $f + g$ is at least $\alpha$.
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Exponent of Convergence is Less Than Order}
+Tags: Entire Functions
+
+\begin{theorem}
+Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]].
+Let $\omega$ be its [[Definition:Order of Entire Function|order]].
+Let $\tau$ be its [[Definition:Exponent of Convergence|exponent of convergence]].
+Then $\tau \le \omega$.
+\end{theorem}
+
+\begin{proof}
+We may assume $\map f 0 \ne 0$.
+Let $f$ have [[Definition:Finite Set|finitely many]] zeroes.
+Then:
+:$\tau = 0 \le \omega$
+{{qed|lemma}}
+Let $f$ have [[Definition:Infinite Set|infinitely many]] zeroes.
+Let $\sequence {a_n}$ be the [[Definition:Sequence|sequence]] of nonzero zeroes of $f$, repeated according to [[Definition:Multiplicity (Complex Analysis)|multiplicity]] and ordered by increasing modulus.
+Let $r_n = \size {a_n}$ and $R_n = 2 \size {a_n}$.
+By [[Jensen's Inequality (Complex Analysis)|Jensen's Inequality]]:
+:$n \le \dfrac {\map \log {\max_{\size z \le R_n} \size f} - \log \size {\map f 0} } {\log 2}$
+for all $n \in \N$.
+Let $\epsilon > 0$.
+Because $f$ has [[Definition:Order of Entire Function|order]] $\omega$:
+:$n \ll_\epsilon \size {a_n}^{\omega + \epsilon}$
+Thus:
+:$\size {a_n}^{-\paren {\omega + \epsilon} } \ll_\epsilon n^{-1}$
+By [[Convergence of P-Series]]:
+:$\omega + \epsilon \ge \tau$
+Because $\epsilon$ is arbitrary:
+:$\omega \ge \tau$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Class is Proper iff Bijection from Class to Proper Class/Corollary}
+Tags: Gödel-Bernays Class Theory, Class Mappings
+
+\begin{theorem}
+$A$ is [[Definition:Class/Proper Class|proper]] [[Definition:Biconditional|if and only if]] there exists a [[Definition:Class Bijection|bijection]] from $\mathrm P$ to $A$.
+\end{theorem}
+
+\begin{proof}
+{{NotZFC}}
+From [[Biconditional is Transitive]] and [[Class is Proper iff Bijection from Class to Proper Class]], it suffices to show that:
+:There exists a bijection from $A$ to $\mathrm P$ [[Definition:Iff|iff]] there exists a bijection from $\mathrm P$ to $A$.
+The rest follows from [[Inverse of Bijection is Bijection]].
+{{qed}}
+[[Category:Gödel-Bernays Class Theory]] [[Category:Class Mappings]]
+309a4dy28gwmdpy3bfpxv5rnvpwu80i
+\end{proof}<|endoftext|>
+\section{Length of God's Algorithm for Sam Loyd's Fifteen Puzzle}
+Tags: Definitions: Recreational Mathematics, 52
+
+\begin{theorem}
+The [[Definition:Length of Algorithm|length]] of [[God's Algorithm for Sam Loyd's Fifteen Puzzle]] is $52$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|A lot of background material needed first.}}
+\end{proof}<|endoftext|>
+\section{Injection from Proper Class to Class}
+Tags: Gödel-Bernays Class Theory, Class Mappings
+
+\begin{theorem}
+Let $A$ be a [[Definition:Class (Class Theory)|class]].
+Let $\mathrm P$ be a [[Definition:Class/Proper Class|proper class]].
+Let $f: \mathrm P \to A$ be an [[Definition:Class Injection|injection]].
+Then $A$ is [[Definition:Class/Proper Class|proper]].
+\end{theorem}
+
+\begin{proof}
+{{NotZFC}}
+Aiming for [[Definition:Contradiction|contradiction]], suppose that $A$ is not proper.
+Then it must be a [[Definition:Set|set]].
+By [[Injection to Image is Bijection]], it follows that the [[Definition:Restriction of Mapping|restriction]] $f \restriction_{\mathrm P \times f \left({\mathrm P}\right)}$ is a [[Definition:Class Bijection|bijection]].
+By the corollary of [[Class is Proper iff Bijection from Class to Proper Class/Corollary|Class is Proper iff Bijection from Class to Proper Class]], $f \left({\mathrm P}\right)$ is [[Definition:Class/Proper Class|proper]].
+But since $f \left({\mathrm P}\right) \subseteq A$, this contradicts [[Subclass of Set is Set]].
+And so by [[Definition:Contradiction|contradiction]], $A$ cannot be a set.
+Therefore $A$ is [[Definition:Class/Proper Class|proper]].
+{{qed}}
+[[Category:Gödel-Bernays Class Theory]]
+[[Category:Class Mappings]]
+fztqgk4v4mg4qqcuavzhh0sbuzsfhsj
+\end{proof}<|endoftext|>
+\section{Polynomial has Order Zero}
+Tags: Entire Functions
+
+\begin{theorem}
+Let $P:\C\to\C$ be a [[Definition:Complex Polynomial Function|polynomial function]].
+Then $P$ has [[Definition:Order of Entire Function|order]] $0$.
+\end{theorem}
+
+\begin{proof}
+Follows from a slightly stronger result than [[Limit at Infinity of Polynomial over Complex Exponential]].
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Period of Reciprocal of 53 is of Quarter Maximal Length}
+Tags: 53, Examples of Reciprocals
+
+\begin{theorem}
+The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $53$ has $\dfrac 1 4$ the maximum [[Definition:Period of Recurrence|period]], that is: $13$:
+:$\dfrac 1 {53} = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3$
+{{OEIS|A007450}}
+\end{theorem}
+
+\begin{proof}
+Performing the calculation using [[Definition:Long Division|long division]]:
+
+ 0.018867924528301...
+ ---------------------
+53)1.000000000000000000
+ 53 212
+ -- ---
+ 470 280
+ 424 265
+ --- ---
+ 460 150
+ 424 106
+ --- ---
+ 360 440
+ 318 424
+ --- ---
+ 420 160
+ 371 159
+ --- ---
+ 490 100
+ 477 53
+ --- ---
+ 130 ...
+ 106
+ ---
+ 240
+ 212
+ ---
+
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Order is Maximum of Exponent of Convergence and Degree}
+Tags: Entire Functions
+
+\begin{theorem}
+Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]].
+Let $\omega$ be its [[Definition:Order of Entire Function|order]].
+Let $\tau$ be its [[Definition:Exponent of Convergence|exponent of convergence]].
+Let $h$ be the [[Definition:Degree of Polynomial|degree]] of the [[Definition:Polynomial|polynomial]] in its [[Definition:Hadamard's Canonical Factorization|canonical factorization]].
+Then:
+:$\omega = \map \max {\tau, h}$
+\end{theorem}
+
+\begin{proof}
+By [[Exponent of Convergence is Less Than Order]]:
+:$\tau \le \omega$
+By [[Hadamard Factorization Theorem]]:
+:$h \le \omega$
+Thus:
+:$\map \max {\tau, h} \le \omega$
+By [[Order is Less Than Maximum of Exponent of Convergence and Degree]]:
+:$\omega \le \map \max {\tau, h}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Recurring Parts of Multiples of One Fifty-Third}
+Tags: Number Theory, 53
+
+\begin{theorem}
+The [[Definition:Rational Multiplication|multiples]] of $\dfrac 1 {53}$ from $\dfrac 1 {53}$ to $\dfrac {52} {53}$ can be divided into $4$ [[Definition:Set|sets]] of equal [[Definition:Cardinality|size]]:
+:one where the [[Definition:Digit|digits]] of the [[Definition:Recurring Part|recurring part]] consists of a [[Definition:Cyclic Permutation|cyclic permutation]] of $01886 \, 79245 \, 283$
+:one where the [[Definition:Digit|digits]] of the [[Definition:Recurring Part|recurring part]] consists of a [[Definition:Cyclic Permutation|cyclic permutation]] of $03773 \, 58490 \, 566$
+:one where the [[Definition:Digit|digits]] of the [[Definition:Recurring Part|recurring part]] consists of a [[Definition:Cyclic Permutation|cyclic permutation]] of $07547 \, 16981 \, 132$
+:one where the [[Definition:Digit|digits]] of the [[Definition:Recurring Part|recurring part]] consists of a [[Definition:Cyclic Permutation|cyclic permutation]] of $09433 \, 96226 \, 415$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1 / 53
+ | r = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3
+ | c =
+}}
+{{eqn | l = 2 / 53
+ | r = 0 \cdotp \dot 03773 \, 58490 \, 56 \dot 6
+ | c =
+}}
+{{eqn | l = 3 / 53
+ | r = 0 \cdotp \dot 05660 \, 37735 \, 84 \dot 9
+ | c =
+}}
+{{eqn | l = 4 / 53
+ | r = 0 \cdotp \dot 07547 \, 16981 \, 13 \dot 2
+ | c =
+}}
+{{eqn | l = 5 / 53
+ | r = 0 \cdotp \dot 09433 \, 96226 \, 41 \dot 5
+ | c =
+}}
+{{eqn | l = 6 / 53
+ | r = 0 \cdotp \dot 11320 \, 75471 \, 69 \dot 8
+ | c =
+}}
+{{eqn | l = 7 / 53
+ | r = 0 \cdotp \dot 13207 \, 54716 \, 98 \dot 1
+ | c =
+}}
+{{eqn | l = 8 / 53
+ | r = 0 \cdotp \dot 15094 \, 33962 \, 26 \dot 4
+ | c =
+}}
+{{eqn | l = 9 / 53
+ | r = 0 \cdotp \dot 16981 \, 13207 \, 54 \dot 7
+ | c =
+}}
+{{eqn | l = 10 / 53
+ | r = 0 \cdotp \dot 18867 \, 92452 \, 83 \dot 0
+ | c =
+}}
+{{eqn | l = 11 / 53
+ | r = 0 \cdotp \dot 20754 \, 71698 \, 11 \dot 3
+ | c =
+}}
+{{eqn | l = 12 / 53
+ | r = 0 \cdotp \dot 22641 \, 50943 \, 39 \dot 6
+ | c =
+}}
+{{eqn | l = 13 / 53
+ | r = 0 \cdotp \dot 24528 \, 30188 \, 67 \dot 9
+ | c =
+}}
+{{eqn | l = 14 / 53
+ | r = 0 \cdotp \dot 26415 \, 09433 \, 96 \dot 2
+ | c =
+}}
+{{eqn | l = 15 / 53
+ | r = 0 \cdotp \dot 28301 \, 88679 \, 24 \dot 5
+ | c =
+}}
+{{eqn | l = 16 / 53
+ | r = 0 \cdotp \dot 30188 \, 67924 \, 52 \dot 8
+ | c =
+}}
+{{eqn | l = 17 / 53
+ | r = 0 \cdotp \dot 32075 \, 47169 \, 81 \dot 1
+ | c =
+}}
+{{eqn | l = 18 / 53
+ | r = 0 \cdotp \dot 33962 \, 26415 \, 09 \dot 4
+ | c =
+}}
+{{eqn | l = 19 / 53
+ | r = 0 \cdotp \dot 35849 \, 05660 \, 37 \dot 7
+ | c =
+}}
+{{eqn | l = 20 / 53
+ | r = 0 \cdotp \dot 37735 \, 84905 \, 66 \dot 0
+ | c =
+}}
+{{eqn | l = 21 / 53
+ | r = 0 \cdotp \dot 39622 \, 64150 \, 94 \dot 3
+ | c =
+}}
+{{eqn | l = 22 / 53
+ | r = 0 \cdotp \dot 41509 \, 43396 \, 22 \dot 6
+ | c =
+}}
+{{eqn | l = 23 / 53
+ | r = 0 \cdotp \dot 43396 \, 22641 \, 50 \dot 9
+ | c =
+}}
+{{eqn | l = 24 / 53
+ | r = 0 \cdotp \dot 45283 \, 01886 \, 79 \dot 2
+ | c =
+}}
+{{eqn | l = 25 / 53
+ | r = 0 \cdotp \dot 47169 \, 81132 \, 07 \dot 5
+ | c =
+}}
+{{eqn | l = 26 / 53
+ | r = 0 \cdotp \dot 49056 \, 60377 \, 35 \dot 8
+ | c =
+}}
+{{eqn | l = 27 / 53
+ | r = 0 \cdotp \dot 50943 \, 39622 \, 64 \dot 1
+ | c =
+}}
+{{eqn | l = 28 / 53
+ | r = 0 \cdotp \dot 52830 \, 18867 \, 92 \dot 4
+ | c =
+}}
+{{eqn | l = 29 / 53
+ | r = 0 \cdotp \dot 54716 \, 98113 \, 20 \dot 7
+ | c =
+}}
+{{eqn | l = 30 / 53
+ | r = 0 \cdotp \dot 56603 \, 77358 \, 49 \dot 0
+ | c =
+}}
+{{eqn | l = 31 / 53
+ | r = 0 \cdotp \dot 58490 \, 56603 \, 77 \dot 3
+ | c =
+}}
+{{eqn | l = 32 / 53
+ | r = 0 \cdotp \dot 60377 \, 35849 \, 05 \dot 6
+ | c =
+}}
+{{eqn | l = 33 / 53
+ | r = 0 \cdotp \dot 62264 \, 15094 \, 33 \dot 9
+ | c =
+}}
+{{eqn | l = 34 / 53
+ | r = 0 \cdotp \dot 64150 \, 94339 \, 62 \dot 2
+ | c =
+}}
+{{eqn | l = 35 / 53
+ | r = 0 \cdotp \dot 66037 \, 73584 \, 90 \dot 5
+ | c =
+}}
+{{eqn | l = 36 / 53
+ | r = 0 \cdotp \dot 67924 \, 52830 \, 18 \dot 8
+ | c =
+}}
+{{eqn | l = 37 / 53
+ | r = 0 \cdotp \dot 69811 \, 32075 \, 47 \dot 1
+ | c =
+}}
+{{eqn | l = 38 / 53
+ | r = 0 \cdotp \dot 71698 \, 11320 \, 75 \dot 4
+ | c =
+}}
+{{eqn | l = 39 / 53
+ | r = 0 \cdotp \dot 73584 \, 90566 \, 03 \dot 7
+ | c =
+}}
+{{eqn | l = 40 / 53
+ | r = 0 \cdotp \dot 75471 \, 69811 \, 32 \dot 0
+ | c =
+}}
+{{eqn | l = 41 / 53
+ | r = 0 \cdotp \dot 77358 \, 49056 \, 60 \dot 3
+ | c =
+}}
+{{eqn | l = 42 / 53
+ | r = 0 \cdotp \dot 79245 \, 28301 \, 88 \dot 6
+ | c =
+}}
+{{eqn | l = 43 / 53
+ | r = 0 \cdotp \dot 81132 \, 07547 \, 16 \dot 9
+ | c =
+}}
+{{eqn | l = 44 / 53
+ | r = 0 \cdotp \dot 83018 \, 86792 \, 45 \dot 2
+ | c =
+}}
+{{eqn | l = 45 / 53
+ | r = 0 \cdotp \dot 84905 \, 66037 \, 73 \dot 5
+ | c =
+}}
+{{eqn | l = 46 / 53
+ | r = 0 \cdotp \dot 86792 \, 45283 \, 01 \dot 8
+ | c =
+}}
+{{eqn | l = 47 / 53
+ | r = 0 \cdotp \dot 88679 \, 24528 \, 30 \dot 1
+ | c =
+}}
+{{eqn | l = 48 / 53
+ | r = 0 \cdotp \dot 90566 \, 03773 \, 58 \dot 4
+ | c =
+}}
+{{eqn | l = 49 / 53
+ | r = 0 \cdotp \dot 92452 \, 83018 \, 86 \dot 7
+ | c =
+}}
+{{eqn | l = 50 / 53
+ | r = 0 \cdotp \dot 94339 \, 62264 \, 15 \dot 0
+ | c =
+}}
+{{eqn | l = 51 / 53
+ | r = 0 \cdotp \dot 96226 \, 41509 \, 43 \dot 3
+ | c =
+}}
+{{eqn | l = 52 / 53
+ | r = 0 \cdotp \dot 98113 \, 20754 \, 71 \dot 6
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Order of Reciprocal of Entire Function}
+Tags: Entire Functions
+
+\begin{theorem}
+Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]] of [[Definition:Order of Entire Function|order]] $\rho$.
+Let $f$ have no zeroes.
+Then $1/f$ has [[Definition:Order of Entire Function|order]] $\rho$.
+\end{theorem}
+
+\begin{proof}
+By [[Zerofree Analytic Function on Simply Connected Set has Logarithm]], there exists an [[Definition:Entire Function|entire function]] $g$ with $f = \exp g$.
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Jacobi's Necessary Condition/Dependent on N Functions}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $J$ be a [[Definition:Real Functional|functional]], such that:
+:$J \sqbrk {\mathbf y} = \displaystyle \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
+where $\mathbf y = \paren {\sequence {y_i}_{1 \le i \le N} }$ is an N-dimensional real vector.
+Let $\map {\mathbf y} x$ correspond to the [[Definition:Minimum Value of Functional|minimum]] of $J$.
+Let the $N\times N$ matrix $\mathbf P = F_{y_i' y_j'}$ be positive definite along $\map {\mathbf y} x$.
+Then the [[Definition:Real Interval/Open|open interval]] $\openint a b$ contains no [[Definition:Point|points]] [[Definition:Conjugate Point|conjugate]] to $a$.
+\end{theorem}
+
+\begin{proof}
+By [[Necessary Condition for Twice Differentiable N Function dependent Functional to have Minimum]], $J$ is minimised by $y = \map {\mathbf {\hat y} } x$ if:
+:$\delta^2 J \sqbrk {\mathbf {\hat y}; \mathbf h} \ge 0$
+for all admissable [[Definition:Real Function|real functions]] $\mathbf h$.
+By [[Legendre's Condition/Lemma 1/Dependent on N Functions|lemma 1 of Legendre's Condition]]:
+:$\ds \delta^2 J \sqbrk {\mathbf y; \mathbf h} = \int_a^b \paren {\mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$
+where:
+:$\mathbf P = F_{y_i' y_j'}$
+By [[Nonnegative Quadratic N function dependent Functional implies no Interior Conjugate Points]], $\openint a b$ does not contain any [[Definition:Conjugate Point|conjugate points]] {{WRT}} $J$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Probability of no 2 People out of 53 Sharing the Same Birthday}
+Tags: Probability Theory, 53
+
+\begin{theorem}
+Let there be $53$ people in a room.
+The [[Definition:Probability|probability]] that no $2$ of them have the same birthday is approximately $\dfrac 1 {53}$.
+\end{theorem}
+
+\begin{proof}
+{{Refactor|The analyis has already been performed in [[Birthday Paradox]]. Extract that general case and make it a theorem, and then introduce [[Birthday Paradox]] and this page as examples.}}
+Let there be $n$ people in the room.
+Let $\map p n$ be the [[Definition:Probability|probability]] that no two people in the room have the same birthday.
+For simplicity, let us ignore [[Definition:Leap Year|leap years]] and assume there are $365$ days in the [[Definition:Year|year]].
+Let the birthday of person $1$ be established.
+The [[Definition:Probability|probability]] that person $2$ shares person $1$'s birthday is $\dfrac 1 {365}$.
+Thus, the [[Definition:Probability|probability]] that person $2$ does not share person $1$'s birthday is $\dfrac {364} {365}$.
+Similarly, the [[Definition:Probability|probability]] that person $3$ does not share the birthday of either person $1$ or person $2$ is $\dfrac {363} {365}$.
+And further, the [[Definition:Probability|probability]] that person $n$ does not share the birthday of any of the people indexed $1$ to $n - 1$ is $\dfrac {365 - \paren {n - 1} } {365}$.
+Hence the total [[Definition:Probability|probability]] that none of the $n$ people share a birthday is given by:
+:$\map p n = \dfrac {364} {365} \dfrac {363} {365} \dfrac {362} {365} \cdots \dfrac {365 - n + 1} {365}$
+{{begin-eqn}}
+{{eqn | l = \map p n
+ | r = \dfrac {364} {365} \dfrac {363} {365} \dfrac {362} {365} \cdots \dfrac {365 - n + 1} {365}
+ | c =
+}}
+{{eqn | r = \dfrac {365!} {365^n} \binom {365} n
+ | c =
+}}
+{{end-eqn}}
+Setting $n = 53$ and evaluating the above gives:
+:$\map p {53} \approx 0.01887$
+or:
+:$\map p {53} \approx \dfrac 1 {53.01697}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primes not Sum of or Difference between Powers of 2 and 3}
+Tags: Prime Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Prime Number|prime numbers]] which cannot be expressed as either [[Definition:Integer Addition|sum of]] or the [[Definition:Integer Subtraction|difference between]] a [[Definition:Integer Power|power of $2$]] and a [[Definition:Integer Power|power of $3$]] begins:
+:$53, 71, 103, 107, 109, 149, 151, \ldots$
+{{OEIS|A007644}}
+\end{theorem}<|endoftext|>
+\section{Triangular Fibonacci Numbers}
+Tags: Fibonacci Numbers, Triangular Numbers
+
+\begin{theorem}
+The only [[Definition:Fibonacci Number|Fibonacci numbers]] which are also [[Definition:Triangular Number|triangular]] are:
+:$0, 1, 3, 21, 55$
+{{OEIS|A039595}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 0
+ | r = \dfrac {0 \times 1} 2
+}}
+{{eqn | l = 1
+ | r = \dfrac {1 \times 2} 2
+}}
+{{eqn | l = 3
+ | r = \dfrac {2 \times 3} 2
+ | rr= = 1 + 2
+}}
+{{eqn | l = 21
+ | r = \dfrac {6 \times 7} 2
+ | rr= = 8 + 13
+}}
+{{eqn | l = 55
+ | r = \dfrac {10 \times 11} 2
+ | rr= = 21 + 34
+}}
+{{end-eqn}}
+It remains to be shown that these are the only ones.
+Let $F_n$ be the $n$th [[Definition:Fibonacci Number|Fibonacci number]].
+From [[Odd Square is Eight Triangles Plus One]], $F_n$ is [[Definition:Triangular Number|triangular]] {{iff}} $8 F_n + 1$ is [[Definition:Square Number|square]].
+It remains to be demonstrated that $8 F_n + 1$ is [[Definition:Square Number|square]] {{iff}}:
+:$n \in \set{\pm 1, 0, 2, 4, 8, 10}$
+So, let $8 F_n + 1$ be [[Definition:Square Number|square]].
+Then:
+:$n \equiv \begin{cases} \pm 1 \pmod {2^5 \times 5} & : n \text { odd} \\ 0, 2, 4, 8, 10 \pmod {2^5 \times 5^2 \times 11} & : n \text { even} \end{cases}$
+{{ProofWanted|The proof needs to be studied.}}
+\end{proof}<|endoftext|>
+\section{Repdigit Triangular Numbers}
+Tags: Repdigit Numbers, Triangular Numbers
+
+\begin{theorem}
+The only [[Definition:Repdigit Number|repdigit numbers]] which are also [[Definition:Triangular Number|triangular]] are:
+:$55, 66, 666$
+{{OEIS|A045914}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 55
+ | r = \dfrac {10 \times 11} 2
+}}
+{{eqn | l = 66
+ | r = \dfrac {11 \times 12} 2
+}}
+{{eqn | l = 666
+ | r = \dfrac {36 \times 37} 2
+}}
+{{end-eqn}}
+Let $\dfrac {d \paren {10^j - 1}} 9$ be a $j$-[[Definition:Digit|digit]] [[Definition:Repdigit Number|repdigit number]].
+Suppose it is [[Definition:Triangular Number|triangular]].
+For $d = 1$, write:
+{{begin-eqn}}
+{{eqn | l = \frac {k \paren {k + 1} } 2
+ | r = \frac {10^j - 1} 9
+}}
+{{eqn | l = 9 k \paren {k + 1}
+ | r = 2 \times 10^j - 2
+}}
+{{eqn | l = \paren {3 k + 1} \paren {3 k + 2}
+ | r = 2^{j + 1} \times 5^j
+}}
+{{end-eqn}}
+Since $3 k + 1 \perp 3 k + 2$, one of them is equal to $2^{j + 1}$ while the other is equal to $5^j$.
+This gives $2^{j + 1} \pm 1 = 5^j$.
+Since $j \ge 1$, taking modulo $4$:
+:$0 \pm 1 \equiv 1^j \equiv 1 \pmod 4$
+so we have:
+:$2^{j + 1} + 1 = 5^j$
+By [[1 plus Power of 2 is not Perfect Power except 9]], the above equation has no solutions for $j > 1$.
+For $j = 1$ we have equality.
+Hence $1$ is the only [[Definition:Triangular Number|triangular]] [[Definition:Repunit|repunit]].
+{{qed|lemma}}
+By [[Odd Square is Eight Triangles Plus One]]:
+:$x = 1 + \dfrac {8 d \paren {10^j - 1}} 9$ is a [[Definition:Square Number|perfect square]].
+For $d = 2, 4, 7, 9$, $x$ ends in $2$ or $3$, which is not a [[Definition:Square Number|perfect square]] by [[Square Modulo 5/Corollary]].
+{{qed|lemma}}
+For $d = 3, 8$, $x$ ends in $5$.
+Since it is a [[Definition:Square Number|perfect square]], $x$ must end in $25$.
+This is satisfied only when $d = 3, j = 1$.
+For $d = 3, j > 1$ and $d = 8, j = 1$, $x$ ends in $65$.
+For $d = 8, j > 1$, $x$ ends in $05$.
+This corresponds to $\dfrac {x - 1} 8 = 3$.
+{{qed|lemma}}
+For $d = 5$, $x$ is in the form $44 \dots 41$.
+We show that $x$ is [[Definition:Square Number|square]] only when $x = 441$.
+{{finish|see talk}}
+This corresponds to $\dfrac {x - 1} 8 = 55$.
+{{qed|lemma}}
+For $d = 6$, $x$ is either:
+:$49$ for $j = 1$
+:$53 \dots 329$ for $j > 1$, with $j - 1$ $3$'s
+we show that $x$ is [[Definition:Square Number|square]] only when $x = 49, 529, 5329$.
+{{finish|see talk}}
+These correspond to $\dfrac {x - 1} 8 = 6$, $66$ and $666$.
+{{qed|lemma}}
+We have hence demonstrated that the only [[Definition:Triangular Number|triangular]] [[Definition:Repdigit Number|repdigit numbers]] are:
+:$1, 3, 6, 55, 66, 666$
+\end{proof}<|endoftext|>
+\section{Increasing and Ordering on Mappings implies Mapping is Composition}
+Tags: Galois Connections
+
+\begin{theorem}
+Let $L = \left({S, \preceq}\right), R = \left({T, \precsim}\right)$ be [[Definition:Ordered Set|ordered sets]].
+Ley $g:S \to T, d:T \to S$ be [[Definition:Mapping|mappings]] such that
+:$g$ and $d$ are [[Definition:Increasing Mapping|increasing mappings]]
+and
+:$d \circ g \preceq I_S$ and $I_T \precsim g \circ d$
+where $\preceq, \precsim$ denotes the [[Definition:Ordering on Mappings|orderings on mappings]].
+Then $d \circ \left({g \circ d}\right)$ and $g = \left({g \circ d}\right) \circ g$
+\end{theorem}
+
+\begin{proof}
+Let $t \in T$.
+By definition of [[Definition:Ordering on Mappings|ordering on mappings]]:
+:$I_T\left({t}\right) \precsim \left({g \circ d}\right)\left({t}\right)$
+By definition of [[Definition:Identity Mapping|identity mapping]]:
+:$t \precsim \left({g \circ d}\right)\left({t}\right)$
+By definition of [[Definition:Increasing Mapping|increasing mapping]]:
+:$d\left({t}\right) \preceq d\left({\left({g \circ d}\right)\left({t}\right)}\right)$
+By definition of [[Definition:Composition of Mappings|composition of mappings]]:
+:$d\left({t}\right) \preceq \left({d \circ \left({g \circ d}\right)}\right)\left({t}\right)$
+By definition of [[Definition:Ordering on Mappings|ordering on mappings]]:
+:$\left({d \circ g}\right)\left({d\left({t}\right)}\right) \preceq I_S\left({d\left({t}\right)}\right)$
+By definition of [[Definition:Identity Mapping|identity mapping]]:
+:$\left({d \circ g}\right)\left({d\left({t}\right)}\right) \preceq d\left({t}\right)$
+By definition of [[Definition:Composition of Mappings|composition of mappings]]:
+:$\left({\left({d \circ g}\right) \circ d}\right)\left({t}\right) \preceq d\left({t}\right)$
+By [[Composition of Mappings is Associative]]:
+:$\left({d \circ \left({g \circ d}\right)}\right)\left({t}\right) \preceq d\left({t}\right)$
+Thus by definition of [[Definition:Antisymmetric Relation|antisymmetry]]:
+:$d\left({t}\right) = \left({d \circ \left({g \circ d}\right)}\right)\left({t}\right)$
+{{qed|lemma}}
+Thus $g = \left({g \circ d}\right) \circ g$ holds by [[Definition:Mutatis Mutandis|mutatis mutandis]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Closed Form for Square Pyramidal Numbers}
+Tags: Closed Forms, Pyramidal Numbers
+
+\begin{theorem}
+The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Square Pyramidal Number|square pyramidal number]] is:
+:$S_n = \dfrac {n \paren {n + 1} \paren {2 n + 1} } 6$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = S_n
+ | r = \sum_{k \mathop = 1}^n k^2
+ | c = {{Defof|Square Pyramidal Number}}
+}}
+{{eqn | r = \dfrac {n \paren {n + 1} \paren {2 n + 1} } 6
+ | c = [[Sum of Sequence of Squares]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Closed Form for Pentagonal Pyramidal Numbers}
+Tags: Closed Forms, Pyramidal Numbers
+
+\begin{theorem}
+The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Pentagonal Pyramidal Number|pentagonal pyramidal number]] is:
+:$Q_n = \dfrac {n^2 \paren {n + 1} } 2$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = Q_n
+ | r = \sum_{k \mathop = 1}^n P_k
+ | c = {{Defof|Pentagonal Pyramidal Number}}
+}}
+{{eqn | r = \sum_{k \mathop = 1}^n \dfrac {k \paren {3 k - 1} } 2
+ | c = [[Closed Form for Pentagonal Numbers]]
+}}
+{{eqn | r = \dfrac 1 2 \paren {3 \sum_{k \mathop = 1}^n k^2 - \sum_{k \mathop = 1}^n k}
+ | c =
+}}
+{{eqn | r = \dfrac 1 2 \paren {3 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 - \sum_{k \mathop = 1}^n k}
+ | c = [[Sum of Sequence of Squares]]
+}}
+{{eqn | r = \dfrac 1 2 \paren {3 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 - \dfrac {n \paren {n + 1} } 2}
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{eqn | r = \dfrac 1 2 \paren {\frac {n \paren {n + 1} \paren {2 n + 1} - n \paren {n + 1} } 2}
+ | c =
+}}
+{{eqn | r = \dfrac 1 2 \paren {\frac {n \paren {n + 1} \paren {2 n} } 2}
+ | c =
+}}
+{{eqn | r = \frac {n^2 \paren {n + 1} } 2
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Surjection from Class to Proper Class}
+Tags: Gödel-Bernays Class Theory, Class Mappings
+
+\begin{theorem}
+Let $A$ be a [[Definition:Class (Class Theory)|class]].
+Let $\mathrm P$ be a [[Definition:Proper Class|proper class]].
+Let $f: A \to \mathrm P$ be a [[Definition:Class Surjection|surjection]].
+Then $A$ is [[Definition:Proper Class|proper]].
+\end{theorem}
+
+\begin{proof}
+{{NotZFC}}
+{{AimForCont}} $A$ is not [[Definition:Proper Class|proper]].
+Then $A$ must be a [[Definition:Set|set]].
+By the [[Axiom:Axiom of Powers|Axiom of Powers]], $\powerset A$ is also a [[Definition:Set|set]].
+Let $g: f \sqbrk A \to \powerset A$ be defined as:
+:$\map g {\map f a} = f^{-1} \sqbrk {\set {\map f a} }$
+It should be noted that:
+:$\forall a \in A: \map {\paren {g \circ f} } a \ne \O$:
+Suppose that $\map g {\map f a} = \map g {\map f b}$.
+Let $x \in f^{-1} \sqbrk {\set {\map f a} }$.
+Then by the definition of a [[Definition:Singleton|singleton]]:
+:$x \in f^{-1} \sqbrk {\set {\map f a} } \implies \map f x \in \set {\map f a} \implies \map f x = \map f a$
+By the same argument:
+:$x \in f^{-1} \sqbrk {\set {\map f b} } \implies \map f x = \map f b$
+Hhence by the [[Diagonal Relation is Equivalence|properties of equality]]:
+:$\map f a = \map f b$
+It has been shown that $g$ is an [[Definition:Class Injection|injection]].
+Because $f$ is a [[Definition:Class Surjection|surjection]], it follows by definition of [[Definition:Surjection|surjection]] that:
+:$f \sqbrk A = \mathrm P$
+But this contradicts [[Injection from Proper Class to Class]].
+Thus by [[Definition:Contradiction|contradiction]] $A$ must be [[Definition:Proper Class|proper]].
+Hence the result.
+{{qed}}
+[[Category:Gödel-Bernays Class Theory]]
+[[Category:Class Mappings]]
+2uu2rcmfr75ya31wcu0fnuzvv5xm95i
+\end{proof}<|endoftext|>
+\section{Image of Set under Mapping is Set}
+Tags: Gödel-Bernays Class Theory, Class Mappings
+
+\begin{theorem}
+Let $A$ be a [[Definition:Class (Class Theory)|class]].
+Let $\mathrm U$ denote the [[Definition:Universal Class|universal class]].
+Let $f: A \to \mathrm U$ be a [[Definition:Class Mapping|class mapping]].
+Let $S$ be a [[Definition:Subset|subset]] of $A$.
+{{explain|Note that further work is needed on the Subset page to clarify that a "subset" of a "proper class" is indeed a "set". Some words are already on that page, but this needs to be formalised and separated out into its own transcluded "class theoretic" page.}}
+Then the [[Definition:Class Mapping|image]] $f \sqbrk S$ is also a [[Definition:Set|set]].
+If $A$ is a [[Definition:Set|set]], then this result is known as the [[Axiom:Axiom of Replacement|Axiom of Replacement]] in [[Definition:Zermelo-Fraenkel Set Theory|Zermelo-Fraenkel set theory]].
+\end{theorem}
+
+\begin{proof}
+{{NotZFC}}
+Aiming for [[Definition:Contradiction|contradiction]], suppose that $f \sqbrk S$ is not a set.
+Then $f \sqbrk S$ must be [[Definition:Class/Proper Class|proper]].
+By [[Restriction of Mapping to Image is Surjection]], the [[Definition:Restriction of Mapping|restriction]] $f \restriction_{S \times f \sqbrk S}$ is a [[Definition:Class Surjection|surjection]].
+But this contradicts [[Surjection from Class to Proper Class]].
+Thus by [[Definition:Contradiction|contradiction]], $f \sqbrk S$ is a [[Definition:Set|set]].
+Hence the result.
+{{qed}}
+[[Category:Gödel-Bernays Class Theory]]
+[[Category:Class Mappings]]
+fdsyztfl8n7ytacauy3trwyj6gk3y16
+\end{proof}<|endoftext|>
+\section{Square Pyramidal and Triangular Numbers}
+Tags: Triangular Numbers, Pyramidal Numbers
+
+\begin{theorem}
+The only [[Definition:Positive Integer|positive integers]] which are simultaneously [[Definition:Square Pyramidal Number|square pyramidal]] and [[Definition:Triangular Number|triangular]] are:
+:$1, 55, 91, 208 \, 335$
+{{OEIS|A039596}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1
+ | r = \dfrac {1 \paren {1 + 1} \paren {2 \times 1 + 1} } 6
+ | c = [[Closed Form for Square Pyramidal Numbers]]
+}}
+{{eqn | r = \dfrac {1 \times \paren {1 + 1} } 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 55
+ | r = \dfrac {5 \paren {5 + 1} \paren {2 \times 5 + 1} } 6
+ | c = [[Closed Form for Square Pyramidal Numbers]]
+}}
+{{eqn | r = \dfrac {10 \times \paren {10 + 1} } 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 91
+ | r = \dfrac {6 \paren {6 + 1} \paren {2 \times 6 + 1} } 6
+ | c = [[Closed Form for Square Pyramidal Numbers]]
+}}
+{{eqn | r = \dfrac {13 \times \paren {13 + 1} } 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 208 \, 335
+ | r = \dfrac {85 \paren {85 + 1} \paren {2 \times 85 + 1} } 6
+ | c = [[Closed Form for Square Pyramidal Numbers]]
+}}
+{{eqn | r = \dfrac {645 \times \paren {645 + 1} } 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that these are the only such instances.}}
+\end{proof}<|endoftext|>
+\section{Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes}
+Tags: Prime Numbers
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integers]] cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct|distinct]] [[Definition:Non-Pythagorean Prime|non-pythagorean primes]]:
+:$1, 2, 4, 5, 6, 8, 9, 12, 13, 15, 16, 17, 20, 24, 25, 27, 28, 32, 35, 36, 39, 48, 51, 55$
+{{OEIS|A048262}}
+All [[Definition:Positive Integer|positive integers]] greater than $55$ can be so expressed.
+\end{theorem}<|endoftext|>
+\section{Sets of 4 Integers a, b, c, d for which Every Integer is in form ax^2 + by^2 + cz^2 + du^2}
+Tags: Number Theory, 55
+
+\begin{theorem}
+There are exactly $55$ [[Definition:Set|sets]] of $4$ [[Definition:Integer|integers]] $\left\{ {a, b, c, d}\right\}$ such that all [[Definition:Integer|integers]] can be written in the form:
+:$n = a x^2 + b y^2 + c z^2 + d w^2$
+for [[Definition:Integer|integer]] $x, y, z, w$.
+\end{theorem}<|endoftext|>
+\section{Palindromes Formed by Multiplying by 55}
+Tags: Recreational Mathematics, 55
+
+\begin{theorem}
+$55$ [[Definition:Integer Multiplication|multiplied]] by any of the [[Definition:Odd Integer|odd integers]] between $91$ and $109$ inclusive produces a [[Definition:Palindromic Number|palindromic number]].
+\end{theorem}
+
+\begin{proof}
+We have that:
+:$55 \times 91 = 5005$
+Then we have that:
+:$55 \times 2 = 110$
+Thus:
+:$\forall k: 1 \le k \le 9: 55 \times 2 k = 110k = 110, 220, 330, \ldots, 990$
+Thus for $1 \le k \le 9$:
+:$55 \times \paren {91 + 2 k} = \sqbrk {5kk5}_{10}$
+That is:
+{{begin-eqn}}
+{{eqn | l = 55 \times 91
+ | r = 5005
+}}
+{{eqn | l = 55 \times 93
+ | r = 5115
+}}
+{{eqn | l = 55 \times 95
+ | r = 5225
+}}
+{{eqn | l = 55 \times 97
+ | r = 5335
+}}
+{{eqn | l = 55 \times 99
+ | r = 5445
+}}
+{{eqn | l = 55 \times 101
+ | r = 5555
+}}
+{{eqn | l = 55 \times 103
+ | r = 5665
+}}
+{{eqn | l = 55 \times 105
+ | r = 5775
+}}
+{{eqn | l = 55 \times 107
+ | r = 5885
+}}
+{{eqn | l = 55 \times 109
+ | r = 5995
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Dirichlet Convolution is Commutative}
+Tags: Dirichlet Convolution
+
+\begin{theorem}
+: $f * g = g * f$
+\end{theorem}
+
+\begin{proof}
+From the definition of the [[Definition:Dirichlet Convolution|Dirichlet convolution]]:
+:$\displaystyle \left({f * g}\right) \left({n}\right) = \sum_{a b \mathop = n} f \left({a}\right) g \left({b}\right)$
+By definition, [[Definition:Arithmetic Function|arithmetic functions]] are mappings from the [[Definition:Natural Numbers|natural numbers]] $\N$ to the [[Definition:Complex Number|complex numbers]] $\C$.
+Thus $f \left({a}\right), g \left({b}\right) \in \C$ and commutativity follows from [[Complex Multiplication is Commutative|commutativity of multiplication of complex numbers]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Dirichlet Convolution is Associative}
+Tags: Dirichlet Convolution
+
+\begin{theorem}
+: $\left({f * g}\right) * h = f * \left({g * h}\right)$
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = \left({\left({f * g}\right) * h}\right) \left({n}\right)
+ | r = \sum_{a b \mathop = n} \left({f * g}\right) \left({a}\right) h \left({b}\right)
+ | c =
+}}
+{{eqn | r = \sum_{a b \mathop = n} \ \sum_{c d \mathop = a} f \left({c}\right) g \left({d}\right) h \left({b}\right)
+ | c =
+}}
+{{eqn | r = \sum_{b c d \mathop = n} f \left({c}\right) g \left({d}\right) h \left({b}\right)
+ | c =
+}}
+{{end-eqn}}
+and
+{{begin-eqn}}
+{{eqn | l = \left({f * \left({g * h}\right)}\right) \left({n}\right)
+ | r = \sum_{a b \mathop = n} f \left({a}\right) \left({g * h}\right) \left({b}\right)
+ | c =
+}}
+{{eqn | r = \sum_{a b \mathop = n} f \left({a}\right) \sum_{c d \mathop = b} g \left({c}\right) h \left({d}\right)
+ | c =
+}}
+{{eqn | r = \sum_{a c d \mathop = n} f \left({a}\right) g \left({c}\right) h \left({d}\right)
+ | c =
+}}
+{{end-eqn}}
+and associativity follows.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Identity Element for Dirichlet Convolution}
+Tags: Dirichlet Convolution
+
+\begin{theorem}
+:$\iota * f = f$
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = \map {\paren {\iota * f} } n
+ | r = \sum_{d \mathop \divides n} \delta_{d 1} \map f {\frac n d}
+}}
+{{eqn | r = \map f n
+}}
+{{end-eqn}}
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Upper Adjoint of Galois Connection is Surjection implies Lower Adjoint at Element is Minimum of Preimage of Singleton of Element}
+Tags: Galois Connections
+
+\begin{theorem}
+Let $L = \struct {S, \preceq}, R = \paren {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
+Let $g: S \to T, d:T \to S$ be [[Definition:Mapping|mappings]] such that:
+:$\tuple {g, d}$ is a [[Definition:Galois Connection|Galois connection]]
+and
+:$g$ is a [[Definition:Surjection|surjection]].
+Then
+:$\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {\set t} }$
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Galois Connection|Galois connection]]:
+:$g$ is an [[Definition:Increasing Mapping|increasing mapping]].
+Let $t \in T$.
+By definition of [[Definition:Surjection|surjection]]:
+:$\Img g = T$
+By [[Image of Preimage under Mapping/Corollary]]:
+:$g \sqbrk {g^{-1} \sqbrk {t^\succeq} } = t^\succeq$
+By [[Galois Connection is Expressed by Minimum]]:
+:$\map d t = \min \set {g^{-1} \sqbrk {t^\succeq} }$
+By definition of [[Definition:Min Operation|min operation]]:
+:$\map d t = \inf \set {g^{-1} \sqbrk {t^\succeq} }$ and $\map d t \in g^{-1} \sqbrk {t^\succeq}$
+By definition of [[Definition:Image of Subset under Mapping|image of set]]:
+:$\map g {\map d t} \in g \sqbrk {g^{-1} \sqbrk {t^\succeq} }$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$t \precsim \map g {\map d t}$
+By definition of [[Definition:Minimum Element|minimum element]]:
+:$g^{-1} \sqbrk {t^\succeq}$ admits an [[Definition:Infimum of Set|infimum]].
+By definition of [[Definition:Infimum of Set|infimum]]:
+:$\map d t$ is [[Definition:Lower Bound of Set|lower bound]] for $g^{-1} \sqbrk {t^\succeq}$
+By definition of [[Definition:Surjection|surjection]]:
+:$\exists s \in S: t = \map g s$
+By definition of [[Definition:Singleton|singleton]]:
+:$t \in \set t$
+By [[Set is Subset of Upper Closure]]
+:$\set t \subseteq \set t^\succeq$
+By [[Upper Closure of Singleton]]:
+:$\set t^\succeq = t^\succeq$
+By definition of [[Definition:Image of Subset under Mapping|image of set]]:
+:$s \in g^{-1} \sqbrk {t^\succeq}$
+By definition of [[Definition:Lower Bound of Set|lower bound]]:
+:$\map d t \preceq s$
+By definition of [[Definition:Increasing Mapping|increasing mapping]]:
+:$\map g {\map d t} \precsim t$
+By definition of [[Definition:Antisymmetric Relation|antisymmetry]]:
+:$\map g {\map d t} = t$
+By definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
+:$\map d t \in g^{-1} \sqbrk {\set t}$
+By [[Image of Subset under Relation is Subset of Image/Corollary 3]]:
+:$g^{-1} \sqbrk {\set t} \subseteq g^{-1} \sqbrk {t^\succeq}$
+We will prove that
+:$\map d t$ is an [[Definition:Infimum of Set|infimum]] of $g^{-1} \sqbrk {\set t}$
+Thus by [[Lower Bound is Lower Bound for Subset]]:
+:$\map d t$ is [[Definition:Lower Bound of Set|lower bound]] for $g^{-1} \sqbrk {\set t}$
+Thus by definition:
+:$\forall s \in S: s$ is [[Definition:Lower Bound of Set|lower bound]] for $g^{-1} \sqbrk {\set t} \implies s \preceq \map d t$
+{{qed|lemma}}
+Thus by definition of [[Definition:Min Operation|min operation]]:
+:$\map d t = \min \set {g^{-1} \sqbrk {\set t} }$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Dirichlet Convolution Preserves Multiplicativity}
+Tags: Dirichlet Convolution, Multiplicative Functions
+
+\begin{theorem}
+Let $f, g: \N \to \C$ be [[Definition:Multiplicative Arithmetic Function|multiplicative arithmetic functions]].
+Then their [[Definition:Dirichlet Convolution|Dirichlet convolution]] $f * g$ is again [[Definition:Multiplicative Arithmetic Function|multiplicative]].
+\end{theorem}
+
+\begin{proof}
+Let $m, n$ be [[Definition:Coprime Integers|coprime integers]].
+By definition of [[Definition:Multiplicative Arithmetic Function|multiplicative functions]], we have:
+:$(1): \quad \map f {m n} = \map f m \map f n$
+:$(2): \quad \map g {m n} = \map g m \map g n$
+{{begin-eqn}}
+{{eqn | l = \map {\paren {f * g} } {m n}
+ | r = \sum_{d \mathop \divides m n} \map f d \map g {\frac {m n} d}
+ | c = {{Defof|Dirichlet Convolution}}
+}}
+{{eqn | r = \sum_{\substack {r \mathop \divides m \\ s \mathop \divides n} } \map f {r s} \map g {\frac m r \frac n s}
+ | c = [[Divisors of Product of Coprime Integers]]
+}}
+{{eqn | r = \sum_{\substack {r \mathop \divides m \\ s \mathop \divides n} } \map f r \map f s \map g {\frac m r} \map g {\frac n s}
+ | c = $f, g$ are [[Definition:Multiplicative Arithmetic Function|multiplicative]]
+}}
+{{eqn | r = \paren {\sum_{r \mathop \divides m} \map f r \map g {\frac m r} } \paren {\sum_{s \mathop \divides n} \map f s \map g {\frac n s} }
+ | c =
+}}
+{{eqn | r = \map {\paren {f * g} } m \times \map {\paren {f * g} } n
+ | c = {{Defof|Dirichlet Convolution}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Dirichlet Convolution Preserves Multiplicativity/General Result}
+Tags: Dirichlet Convolution, Multiplicative Functions
+
+\begin{theorem}
+Let $S \subset \N$ be a [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] with the property:
+:$m n \in S, \; \gcd \left({m, n}\right) = 1 \implies m, n \in S$
+Define:
+:$\left({f*_S g}\right) \left({n}\right) = \displaystyle \sum_{\substack {d \mathop \backslash n \\ d \mathop \in S} } f \left({d}\right) g \left({n / d}\right)$
+Then $f*_S g$ is [[Definition:Multiplicative Arithmetic Function|multiplicative]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Dirichlet Convolution]]
+[[Category:Multiplicative Functions]]
+pbtcbkyin7cbj8wlm8zknxwc9n6ydsz
+\end{proof}<|endoftext|>
+\section{Even Integer with Abundancy Index greater than 9}
+Tags: Abundancy
+
+\begin{theorem}
+Let $n \in \Z_{>0}$ have an [[Definition:Abundancy Index|abundancy index]] greater than $9$.
+Then $n$ has at least $35$ [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]].
+\end{theorem}
+
+\begin{proof}
+Since [[Sigma Function is Multiplicative]], it follows easily that [[Definition:Abundancy Index|abundancy index]] is [[Definition:Multiplicative Arithmetic Function|multiplicative]] as well.
+We have for any [[Definition:Prime Number|prime]] $p$ and [[Definition:Positive Integer|positive integer]] $k$:
+{{begin-eqn}}
+{{eqn | l = \frac {\map \sigma {p^k} } {p^k}
+ | r = \frac {p^{k + 1} - 1} {p^k \paren {p - 1} }
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \frac {p - p^{-k} } {p - 1}
+}}
+{{eqn | o = <
+ | r = \frac p {p - 1}
+ | c = as $p^{-k} > 0$
+}}
+{{eqn | r = 1 + \frac 1 {p - 1}
+}}
+{{end-eqn}}
+In fact this is the [[Definition:Limit of Sequence|limit]] of the [[Definition:Abundancy Index|abundancy index]] of a [[Definition:Prime Power|prime power]].
+The greater the [[Definition:Prime Number|prime]], the smaller this value is.
+Therefore finding the least number of [[Definition:Prime Factor|prime factors]] a number needs to have its [[Definition:Abundancy Index|abundancy index]] exceed $9$ is to find the least $m$ such that:
+:$\displaystyle \prod_{i = 1}^m \frac {p_i} {p_i - 1} > 9$
+where $p_i$ is the $i$th [[Definition:Prime Number|prime]] ordered by size.
+The result can be verified by direct computation.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Closed Form for Pentatope Numbers}
+Tags: Closed Forms, Pentatope Numbers
+
+\begin{theorem}
+The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Pentatope Number|pentatope number]] is:
+:$P_n = \dfrac {n \paren {n + 1} \paren {n + 2} \paren {n + 3} } {24}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = P_n
+ | r = \sum_{k \mathop = 1}^n T_k
+ | c = {{Defof|Pentatope Number}}
+}}
+{{eqn | r = \sum_{k \mathop = 1}^n \frac {n \paren {n + 1} \paren {n + 2} } 6
+ | c = [[Closed Form for Tetrahedral Numbers]]
+}}
+{{eqn | r = \sum_{k \mathop = 1}^n \frac {\paren {n^3 + 3 n^2 + 2 n} } 6
+ | c =
+}}
+{{eqn | r = \frac 1 6 \sum_{k \mathop = 1}^n n^3 + \frac 1 2 \sum_{k \mathop = 1}^n n^2 + \frac 1 3 \sum_{k \mathop = 1}^n n
+ | c = [[Summation is Linear]]
+}}
+{{eqn | r = \frac 1 6 \sum_{k \mathop = 1}^n n^3 + \frac 1 2 \sum_{k \mathop = 1}^n n^2 + \frac 1 3 \frac {n \paren {n + 1} } 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{eqn | r = \frac 1 6 \sum_{k \mathop = 1}^n n^3 + \frac 1 2 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 + \frac 1 3 \frac {n \paren {n + 1} } 2
+ | c = [[Sum of Sequence of Squares]]
+}}
+{{eqn | r = \frac 1 6 \dfrac {n^2 \paren {n + 1}^2} 4 + \frac 1 2 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 + \frac 1 3 \frac {n \paren {n + 1} } 2
+ | c = [[Sum of Sequence of Cubes]]
+}}
+{{eqn | r = \dfrac {n^2 \paren {n + 1}^2 + 2 n \paren {n + 1} \paren {2 n + 1} + 4 n \paren {n + 1} } {24}
+ | c = putting over a common [[Definition:Denominator|denominator]]
+}}
+{{eqn | r = \dfrac {n \paren {n + 1} \paren {n \paren {n + 1} + 2 \paren {2 n + 1} + 4} } {24}
+ | c = extracting $n \paren {n + 1}$ as a factor
+}}
+{{eqn | r = \dfrac {n \paren {n + 1} \paren {n^2 + 5 n + 6} } {24}
+ | c = simplifying
+}}
+{{eqn | r = \dfrac {n \paren {n + 1} \paren {n + 2} \paren {n + 3} } {24}
+ | c = factorising
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Relation Between Rank and Exponent of Convergence}
+Tags: Entire Functions
+
+\begin{theorem}
+Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]].
+Let $k$ be its [[Definition:Rank of Entire Function|rank]] and $\tau$ be its [[Definition:Exponent of Convergence|exponent of convergence]].
+Then:
+* $k=\tau=0$ if $f$ has [[Definition:Finitely Many|finitely many]] zeroes.
+* $k<\tau\leq k+1$ otherwise.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Entire Functions]]
+epim9a757ti02kdsc835c98mkdrn4my
+\end{proof}<|endoftext|>
+\section{Necessary and Sufficient Condition for First Order System to be Field for Second Order System}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $\mathbf y$, $\mathbf f$, $\boldsymbol \psi$ be [[Definition:Dimension |N-dimensional]] [[Definition:Vector|vectors]].
+Let $\boldsymbol\psi$ be [[Definition:Continuously Differentiable Vector|continuously differentiable]].
+Then $\forall x \in \closedint a b$ the [[Definition:First Order Ordinary Differential Equation|first-order]] [[Definition:System of Differential Equations|system of differential equations]]:
+:$\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$
+is a [[Definition:Field of Directions|field]] for the [[Definition:Second Order Ordinary Differential Equation|second-order]] [[Definition:System of Differential Equations|system]]
+:$\mathbf y'' = \map {\mathbf f} {x, \mathbf y, \mathbf y'}$
+{{iff}} $\boldsymbol \psi$ satisfies :
+:$\displaystyle \frac {\partial \boldsymbol \psi} {\partial x} + \sum_{i \mathop = 1}^N \frac {\partial \boldsymbol \psi} {\partial y_i} \psi_i = \map {\mathbf f} {x, \mathbf y, \boldsymbol \psi}$
+That is, every [[Definition:Solution to Differential Equation|solution]] to Hamilton-Jacobi [[Definition:System of Differential Equations|system]] is a [[Definition:Field of Directions|field]] for the original [[Definition:System of Differential Equations|system]].
+\end{theorem}
+
+\begin{proof}
+=== Necessary condition ===
+[[Definition:Derivative|Differentiate]] the [[Definition:First Order Ordinary Differential Equation|first-order]] [[Definition:System of Differential Equations|system]] {{WRT|Differentiation}} $x$:
+{{begin-eqn}}
+{{eqn | l = \mathbf y''
+ | r = \frac {\d \boldsymbol \psi} {\d x}
+}}
+{{eqn | r = \frac {\partial \boldsymbol \psi} {\partial x} + \sum_{i \mathop = 1}^N \frac {\partial \boldsymbol \psi} {\partial y_i} \frac {\d y_i} {\d x}
+}}
+{{eqn | r = \frac {\partial \boldsymbol \psi} {\partial x} + \sum_{i \mathop = 1}^N \frac {\partial \boldsymbol \psi} {\partial y_i} \psi_i
+}}
+{{end-eqn}}
+This can be rewritten as the following [[Definition:System of Differential Equations|system of equations]]:
+:$\mathbf y'' = \map {\mathbf f} {x, \mathbf y, \mathbf y'}$
+:$\displaystyle \frac {\partial \boldsymbol \psi} {\partial x} + \sum_{i \mathop = 1}^N \frac {\partial \boldsymbol \psi} {\partial y_i} \psi_i = \map {\mathbf f} {x, \mathbf y, \mathbf y'}$
+By assumption, the [[Definition:First Order Ordinary Differential Equation|first-order]] [[Definition:System of Differential Equations|system]] is valid in $\closedint a b$.
+For the [[Definition:Second Order Ordinary Differential Equation|second-order]] [[Definition:System of Differential Equations|system]] to be valid in the same [[Definition:Closed Real Interval|interval]], the corresponding Hamilton-Jacobi equation has to hold as well.
+{{qed|lemma}}
+=== Sufficient condition ===
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Tetrahedral Number as Sum of Squares}
+Tags: Tetrahedral Numbers, Square Numbers, Tetrahedral Number as Sum of Squares
+
+\begin{theorem}
+:$H_n = \displaystyle \sum_{k \mathop = 0}^{n / 2} \paren {n - 2 k}^2$
+where $H_n$ denotes the $n$th [[Definition:Tetrahedral Number|tetrahedral number]].
+\end{theorem}
+
+\begin{proof}
+Let $n$ be [[Definition:Even Integer|even]] such that $n = 2 m$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{k \mathop = 0}^{n / 2} \paren {n - 2 k}^2
+ | r = \sum_{k \mathop = 0}^m \paren {2 m - 2 k}^2
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 0}^m \paren {2 k}^2
+ | c = [[Permutation of Indices of Summation]]
+}}
+{{eqn | r = \frac {2 m \paren {m + 1} \paren {2 m + 1} } 3
+ | c = [[Sum of Sequence of Even Squares]]
+}}
+{{eqn | r = \frac {2 m \paren {2 m + 1} \paren {2 m + 2} } 6
+ | c =
+}}
+{{eqn | r = H_{2 m}
+ | c = [[Closed Form for Tetrahedral Numbers]]
+}}
+{{end-eqn}}
+Let $n$ be [[Definition:Odd Integer|odd]] such that $n = 2 m + 1$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{k \mathop = 0}^{n / 2} \paren {n - 2 k}^2
+ | r = \sum_{k \mathop = 0}^m \paren {2 m + 1 - 2 k}^2
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 0}^m \paren {2 k + 1}^2
+ | c = [[Permutation of Indices of Summation]]
+}}
+{{eqn | r = \frac {\paren {m + 1} \paren {2 m + 1} \paren {2 m + 3} } 3
+ | c = [[Sum of Sequence of Odd Squares/Formulation 1|Sum of Sequence of Odd Squares: Formulation 1]]
+}}
+{{eqn | r = \frac {\paren {2 m + 1} \paren {2 m + 2} \paren {2 m + 3} } 6
+ | c =
+}}
+{{eqn | r = H_{2 m + 1}
+ | c = [[Closed Form for Tetrahedral Numbers]]
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Tetrahedral Numbers]]
+[[Category:Square Numbers]]
+[[Category:Tetrahedral Number as Sum of Squares]]
+7s51qp2538g2fo2vmilix4kes88mu89
+\end{proof}<|endoftext|>
+\section{Sum of Sequence of Odd Squares/Formulation 1}
+Tags: Sum of Sequence of Odd Squares
+
+\begin{theorem}
+:$\displaystyle \forall n \in \N: \sum_{i \mathop = 0}^n \paren {2 i + 1}^2 = \frac {\paren {n + 1} \paren {2 n + 1} \paren {2 n + 3} } 3$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = 0}^n \paren {2 i + 1}^2
+ | r = \sum_{i \mathop = 0}^n \paren {2 i}^2 + \sum_{i \mathop = 0}^n 4 i + \sum_{i \mathop = 0}^n 1
+ | c =
+}}
+{{eqn | r = \frac {2 n \paren {n + 1} \paren {2 n + 1} } 3 + 4 \sum_{i \mathop = 0}^n i + \sum_{i \mathop = 0}^n 1
+ | c = [[Sum of Sequence of Even Squares]]
+}}
+{{eqn | r = \frac {2 n \paren {n + 1} \paren {2 n + 1} } 3 + 4 \sum_{i \mathop = 1}^n i + \sum_{i \mathop = 0}^n 1
+ | c = adjustment of indices: $4 i = 0$ when $i = 0$
+}}
+{{eqn | r = \frac {2 n \paren {n + 1} \paren {2 n + 1} } 3 + 4 \frac {n \paren {n + 1} } 2 + \sum_{i \mathop = 0}^n 1
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{eqn | r = \frac {2 n \paren {n + 1} \paren {2 n + 1} } 3 + 2 n \paren {n + 1} + \paren {n + 1}
+ | c = further simplification
+}}
+{{eqn | r = \frac {\paren {n + 1} \paren {2 n \paren {2 n + 1} + 6 n + 3} } 3
+ | c = factorising
+}}
+{{eqn | r = \frac {\paren {n + 1} \paren {4 n^2 + 8 n + 3} } 3
+ | c = multiplying out
+}}
+{{eqn | r = \frac {\paren {n + 1} \paren {2 n + 1} \paren {2 n + 3} } 3
+ | c = factorising
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Sum of Sequence of Odd Squares]]
+e1c38kqvljyn1fsv8ucyex9ei9l4jrg
+\end{proof}<|endoftext|>
+\section{Sum of Sequence of Even Squares}
+Tags: Square Numbers, Sums of Sequences
+
+\begin{theorem}
+:$\displaystyle \forall n \in \N: \sum_{i \mathop = 0}^n \left({2 i}\right)^2 = \frac {2 n \left({n + 1}\right) \left({2 n + 1}\right)} 3$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = 0}^n \left({2 i}\right)^2
+ | r = 4 \sum_{i \mathop = 1}^n i^2
+ | c = adjustment of indices: $4 i^2 = 0$ when $i = 0$
+}}
+{{eqn | r = 4 \frac {n \left({n + 1}\right) \left({2 n + 1}\right)} 6
+ | c = [[Sum of Sequence of Squares]]
+}}
+{{eqn | r = \frac {2 n \left({n + 1}\right) \left({2 n + 1}\right)} 3
+ | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Square Numbers]]
+[[Category:Sums of Sequences]]
+dbt2ajh62bj39euk59wtzk7x1nbzprz
+\end{proof}<|endoftext|>
+\section{Killing Form of Orthogonal Lie Algebra}
+Tags: Lie Algebras
+
+\begin{theorem}
+Let $\mathbb K \in \left\{ {\C, \R}\right\}$.
+Let $n$ be a [[Definition:Positive Integer|positive integer]].
+Let $\mathfrak{so}_n \left({\mathbb K}\right)$ be the [[Definition:Lie Algebra|Lie algebra]] of the [[Definition:Special Orthogonal Group|special orthogonal group]] $\operatorname{SO}_n \left({\mathbb K}\right)$.
+Then its [[Definition:Killing Form|Killing form]] is $B: \left({X, Y}\right) \mapsto \left({n - 2}\right) \operatorname {tr} \left({X Y}\right)$.
+\end{theorem}
+
+\begin{proof}
+=== Lemma ===
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $n$ be a [[Definition:Positive Integer|positive integer]].
+Let $E_{ij}$ denote the [[Definition:Matrix|matrix]] with only zeroes except a $1$ at the $(i,j)$th position.
+Then for all $X, Y \in R^{n \times n}$:
+:$\displaystyle \sum_{1 \mathop \le i \mathop < j \le n} \operatorname{tr} \left({\left({X \left({E_{ij} - E_{ji} }\right) Y}\right)^t \left({E_{ij} - E_{ji} }\right)}\right) = \operatorname{tr} \left({Y}\right) \operatorname {tr} \left({X}\right) - \operatorname {tr} \left({Y^t X}\right)$
+=== Proof ===
+Use [[Trace of Alternating Product of Matrices and Almost Zero Matrices]].
+----
+Use [[Definition:Frobenius Inner Product]] and [[Trace in Terms of Orthonormal Basis]] and the fact that the $\left({E_{ij} - E_{ji} }\right) / \sqrt 2$ are an [[Definition:Orthonormal Basis|orthonormal basis]] of $\mathfrak {so}_n$.
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Killing Form of Symplectic Lie Algebra}
+Tags: Lie Algebras
+
+\begin{theorem}
+Let $\mathbb K \in \left\{ {\C, \R}\right\}$.
+Let $n$ be a [[Definition:Positive Integer|positive integer]].
+Let $\mathfrak{sp}_{2 n} \left({\mathbb K}\right)$ be the [[Definition:Lie Algebra|Lie algebra]] of the [[Definition:Symplectic Group|symplectic group]] $\operatorname{Sp} \left({2 n, \mathbb K}\right)$.
+Then its [[Definition:Killing Form|Killing form]] is $B: \left({X, Y}\right) \mapsto \left({2 n + 2}\right) \operatorname{tr} \left({X Y}\right)$.
+\end{theorem}
+
+\begin{proof}
+=== Lemma ===
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $n$ be a [[Definition:Positive Integer|positive integer]].
+Let $E_{ij}$ denote the [[Definition:Matrix|matrix]] with only zeroes except a $1$ at the $\left({i, j}\right)$th position.
+Let $X, Y \in R^{2 n \times 2 n}$.
+Let $X = \begin {pmatrix} X_{11} & X_{12} \\ X_{21} & X_{22} \end{pmatrix}$ and $Y = \begin{pmatrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22}\end{pmatrix}$
+Then:
+:$\displaystyle \sum_{i, j \mathop = 1}^{2 n} \operatorname {tr} \left({\left({X E_{i j} Y}\right)^t E_{i j} }\right) = \operatorname{tr} \left({Y}\right) \operatorname{tr} \left({X}\right)$
+:$\displaystyle \sum_{i, j \mathop = 1}^n \operatorname{tr} \left({\left({X E_{ij} Y}\right)^t E_{j + n, i + n} }\right) = \operatorname{tr} \left({Y_{1 2}^t X_{2 1} }\right)$
+=== Proof ===
+Use [[Trace of Alternating Product of Matrices and Almost Zero Matrices]].
+----
+Use [[Definition:Frobenius Inner Product]] and [[Trace in Terms of Orthonormal Basis]] and the fact that the $\left({E_{i j} }\right)_{i \le n, j \ge n + 1}, \left({E_{i j} }\right)_{i \ge n + 1, j \le n}, \left({E_{i j} - E_{j + n, i + n} }\right) / \sqrt2$ are an [[Definition:Orthonormal Basis|orthonormal basis]] of $\mathfrak{sp}_{2 n}$.
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Trace of Alternating Product of Matrices and Almost Zero Matrices}
+Tags: Matrix Theory
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $n, m$ be [[Definition:Positive Integer|positive integers]].
+Let $E_{ij}$ denote the $n \times n$ [[Definition:Matrix|matrix]] with only zeroes except a $1$ at the $\tuple {i, j}$th [[Definition:Element of Matrix|element]].
+Let $A_1, \ldots, A_m \in R^{n \times n}$.
+Let $i_k, j_k \in \set {1, \ldots, n}$ for $k \in \set {1, \ldots, m}$.
+Let $i_0 = i_m$ and $j_0 = j_m$.
+Then:
+:$\map \tr {A_1 E_{i_1, j_1} A_2 E_{i_2, j_2} \cdots A_m E_{i_m, j_m} } = \displaystyle \prod_{k \mathop = 1}^m \sqbrk {A_k}_{j_{k - 1} i_k}$
+\end{theorem}
+
+\begin{proof}
+Use induction and the facts $E_{i j} A E_{k l} = A_{j k} E_{i l}$ and $\map \tr {A E_{i j} } = A_{j i}$ (induction basis).
+{{ProofWanted}}
+[[Category:Matrix Theory]]
+4qzwd4t34nevpnyuos1ofy2hzki679z
+\end{proof}<|endoftext|>
+\section{Trace in Terms of Orthonormal Basis}
+Tags: Linear Algebra
+
+\begin{theorem}
+Let $\mathbb K \subset \C$ be a [[Definition:Field (Abstract Algebra)|field]].
+Let $\struct {V, \innerprod {\, \cdot \,} {\, \cdot \,} }$ be an [[Definition:Inner Product Space|inner product space]] over $\mathbb K$ of [[Definition:Dimension of Vector Space|dimension]] $n$.
+Let $\tuple {e_1, \ldots, e_n}$ be an [[Definition:Orthonormal Basis|orthonormal basis]] of $V$.
+Let $f: V \to V$ be a [[Definition:Linear Operator|linear operator]].
+Then its [[Definition:Trace of Linear Operator|trace]] equals:
+:$\map \tr f = \displaystyle \sum_{i \mathop = 1}^n \innerprod {\map f {e_i} } {e_i}$
+\end{theorem}
+
+\begin{proof}
+Let $\displaystyle \map f {e_i} = \sum_{j \mathop = 1}^n c_{ij} e_j$
+Let $A$ be the [[Definition:Matrix|matrix]] [[Definition:Relative Matrix|relative]] to the basis $\tuple {e_1, \ldots, e_n}$.
+Then by the above assumption, $A_{ij} = c_{ij}$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \map \tr f
+ | r = \map \tr A
+ | c = {{Defof|Trace of Linear Operator}}
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n A_{ii}
+ | c = {{Defof|Trace of Matrix}}
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n c_{ii}
+ | c = From above
+}}
+{{end-eqn}}
+Now it remains to show that $c_{ii} = \innerprod {\map f {e_i} } {e_i}$:
+{{begin-eqn}}
+{{eqn | l = \innerprod {\map f {e_i} } {e_i}
+ | r = \innerprod {\sum_{j \mathop = 1}^n c_{ij} e_j} {e_i}
+ | c = From above assumption
+}}
+{{eqn | r = \sum_{j \mathop = 1}^n c_{ij} \innerprod {e_j} {e_i}
+ | c = [[Definition:Inner Product|Axiom $(2)$ of Inner Product: Bilinearity]]
+}}
+{{eqn | r = c_{ii} (1)
+ | c = {{Defof|Orthonormal Subset}}: other terms vanish
+}}
+{{eqn | r = c_{ii}
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Galois Connection with Upper Adjoint Surjective implies Scond Ordered Set and Image of Lower Adjoint are Isomorphic}
+Tags: Galois Connections, Order Isomorphisms
+
+\begin{theorem}
+Let $L = \left({S, \preceq}\right), R = \left({T, \precsim}\right)$ be [[Definition:Ordered Set|ordered sets]].
+Ley $g:S \to T, d:T \to S$ be [[Definition:Mapping|mappings]] such that
+:$\left({g, d}\right)$ is [[Definition:Galois Connection|Galois connection]]
+and
+:$g$ is a [[Definition:Surjection|surjection]]/
+Let $N = \left({d\left[{T}\right], \preceq'}\right)$ be an [[Definition:Ordered Subset|ordered subset]] of $L$.
+Then $R$ and $N$ are [[Definition:Order Isomorphism|order isomorphic]].
+\end{theorem}
+
+\begin{proof}
+By [[Galois Connection implies Upper Adjoint is Surjection iff Lower Adjoint is Injection]]:
+:$d$ is an [[Definition:Injection|injection]].
+Define $d' = d:T \to g\left[{T}\right]$
+By definition:
+:$d'$ is an [[Definition:Injection|injection]].
+By definition of [[Definition:Galois Connection|Galois connection]]:
+:$d$ is an [[Definition:Increasing Mapping|increasing mapping]].
+We will prove that
+:$d$ is [[Definition:Order Embedding|order embedding]].
+Let $x, y \in T$.
+By definition of [[Definition:Increasing Mapping|increasing mapping]]:
+:$x \precsim y \implies d\left({x}\right) \preceq d\left({y}\right)$
+Thus by definition of $d'$ and [[Definition:Ordered Subset|ordered subset]]:
+:$x \precsim y \implies d'\left({x}\right) \preceq' d'\left({y}\right)$
+Assume that
+:$d'\left({x}\right) \preceq' d'\left({y}\right)$
+By [[Upper Adjoint of Galois Connection is Surjection implies Lower Adjoint at Element is Minimum of Preimage of Singleton of Element]]
+:$\forall t \in T: d\left({t}\right) = \min\left({g^{-1}\left[{\left\{ {t}\right\} }\right]}\right)$
+By [[Lower Adjoint at Element is Minimum of Preimage of Singleton of Element implies Composition is Identity]]:
+:$g \circ d = I_T$
+By definition of [[Definition:Galois Connection|Galois connection]]:
+:$g$ is an [[Definition:Increasing Mapping|increasing mapping]].
+Thus by definition of $d'$ and [[Definition:Ordered Subset|ordered subset]]:
+:$d\left({x}\right) \preceq d\left({y}\right)$
+By definition of [[Definition:Increasing Mapping|increasing mapping]]:
+:$g\left({d\left({x}\right)}\right) \precsim g\left({d\left({y}\right)}\right)$
+Thus by definitions of [[Definition:Composition of Mappings|composition of mappings]] and [[Definition:Identity Mapping|identity mapping]]:
+:$x \precsim y$
+{{qed|lemma}}
+By definition:
+:$d'$ is a [[Definition:Surjection|surjection]].
+Hence $d'$ is [[Definition:Order Isomorphism|order isomorphism]] between $R$ and $N$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Positive Integer Sum of 2 Fourth Powers in 2 Ways}
+Tags: Fourth Powers
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|positive integer]] which can be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Fourth Power|fourth powers]] in $2$ different ways is $635 \, 318 \, 657$:
+{{begin-eqn}}
+{{eqn | l = 59^4 + 158^4
+ | r = 12 \, 117 \, 361 + 62 \, 201 \, 296
+ | c =
+}}
+{{eqn | r = 635 \, 318 \, 657
+ | c =
+}}
+{{eqn | r = 312 \, 900 \, 721 + 322 \, 417 \, 936
+ | c =
+}}
+{{eqn | r = 133^4 + 134^4
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+The fact that these are the smallest can be demonstrated by calculation.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Trace in Terms of Dual Basis}
+Tags: Linear Algebra
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $M$ be a [[Definition:Free Module|free]] $R$-[[Definition:Module|module]] of [[Definition:Dimension of Module|dimension]] $n$.
+Let $\tuple {e_1, \ldots, e_n}$ be a [[Definition:Basis of Module|basis]] of $M$.
+Let $\tuple {e_1^*,\ldots, e_n^*}$ be its [[Definition:Ordered Dual Basis|dual basis]]
+Let $f: M \to M$ be a [[Definition:Linear Operator|linear operator]].
+Then its [[Definition:Trace of Linear Operator|trace]] equals:
+:$\map \tr f = \displaystyle \sum_{i \mathop = 1}^n e_i^* \paren {\map f {e_i} }$
+\end{theorem}
+
+\begin{proof}
+Let $\displaystyle \map f {e_i} = \sum_{j \mathop = 1}^n c_{ij} e_j$
+Let $A$ be the [[Definition:Matrix|matrix]] [[Definition:Relative Matrix|relative]] to the basis $\tuple {e_1, \ldots, e_n}$.
+Then by the above assumption, $A_{ij} = c_{ij}$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \map \tr f
+ | r = \map \tr A
+ | c = {{Defof|Trace of Linear Operator}}
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n A_{ii}
+ | c = {{Defof|Trace of Matrix}}
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n c_{ii}
+ | c = from above
+}}
+{{end-eqn}}
+Now it remains to show that $c_{ii} = e_i^* \paren {\map f {e_i} }$:
+{{begin-eqn}}
+{{eqn | l = e_i^* \paren {\map f {e_i} }
+ | r = e_i^* \paren {\sum_{j \mathop = 1}^n c_{ij} e_j}
+ | c = From above assumption
+}}
+{{eqn | r = \sum_{j \mathop = 1}^n c_{ij} e_i^* \paren {e_j}
+ | c = $e_i$ is a [[Definition:Linear Form|linear form]]
+}}
+{{eqn | r = \map {c_{ii} } 1
+ | c = {{Defof|Ordered Dual Basis}}: other terms vanish
+}}
+{{eqn | r = c_{ii}
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Internal Angle of Equilateral Triangle}
+Tags: Equilateral Triangles
+
+\begin{theorem}
+The [[Definition:Internal Angle|internal angles]] of an [[Definition:Equilateral Triangle|equilateral triangle]] measure $60^\circ$ or $\dfrac \pi 3$ [[Definition:Radian|radians]].
+\end{theorem}
+
+\begin{proof}
+By definition, an [[Definition:Equilateral Triangle|equilateral triangle]] is a [[Definition:Regular Polygon|regular polygon]] with $3$ [[Definition:Side of Polygon|sides]].
+From [[Internal Angles of Regular Polygon]], the size $A$ of each [[Definition:Internal Angle|internal angle]] of a [[Definition:Regular Polygon|regular $n$-gon]] is given by:
+:$A = \dfrac {\paren {n - 2} 180^\circ} n$
+Thus:
+:$A = \dfrac {180^\circ} n = 60^\circ$
+From [[Value of Degree in Radians]]:
+:$1^\circ = \dfrac {\pi} {180^\circ} \mathrm {rad}$
+and so:
+:$A = 60^\circ \times \dfrac {\pi} {180^\circ} = \dfrac \pi 3 \mathrm {rad}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways}
+Tags: Prime Numbers
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|positive integer]] which is the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Odd Prime|odd primes]] in $6$ different ways is $60$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 60
+ | r = 7 + 53
+ | c = $6$ ways
+}}
+{{eqn | r = 13 + 47
+ | c =
+}}
+{{eqn | r = 17 + 43
+ | c =
+}}
+{{eqn | r = 19 + 41
+ | c =
+}}
+{{eqn | r = 23 + 37
+ | c =
+}}
+{{eqn | r = 29 + 31
+ | c =
+}}
+{{end-eqn}}
+It is determined that there are no smaller numbers with this property by inspection:
+{{begin-eqn}}
+{{eqn | l = 58
+ | r = 53 + 5
+ | c = $3$ ways
+}}
+{{eqn | r = 47 + 11
+ | c =
+}}
+{{eqn | r = 41 + 17
+ | c =
+}}
+{{eqn | r = 29 + 29
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 56
+ | r = 53 + 3
+ | c = $3$ ways
+}}
+{{eqn | r = 43 + 13
+ | c =
+}}
+{{eqn | r = 37 + 19
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 54
+ | r = 47 + 7
+ | c = $5$ ways
+}}
+{{eqn | r = 43 + 11
+ | c =
+}}
+{{eqn | r = 41 + 13
+ | c =
+}}
+{{eqn | r = 37 + 17
+ | c =
+}}
+{{eqn | r = 31 + 23
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 52
+ | r = 47 + 5
+ | c = $3$ ways
+}}
+{{eqn | r = 41 + 11
+ | c =
+}}
+{{eqn | r = 29 + 23
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 50
+ | r = 47 + 3
+ | c = $4$ ways
+}}
+{{eqn | r = 43 + 7
+ | c =
+}}
+{{eqn | r = 37 + 13
+ | c =
+}}
+{{eqn | r = 31 + 19
+ | c =
+}}
+{{end-eqn}}
+From the workings of [[Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways]], $48$ is the smallest [[Definition:Positive Integer|positive integer]] which is the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Odd Prime|odd primes]] in $5$ different ways.
+Thus there is no need to explore smaller numbers.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways}
+Tags: Prime Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] $n$ which are the smallest such that they are the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Odd Prime|odd primes]] in $k$ different ways begins as follows:
+:{| border="1"
+|-
+! align="right" style = "padding: 2px 10px" | $k$
+! align="right" style = "padding: 2px 10px" | $n$
+|-
+| align="right" style = "padding: 2px 10px" | $1$
+| align="right" style = "padding: 2px 10px" | $6$
+|-
+| align="right" style = "padding: 2px 10px" | $2$
+| align="right" style = "padding: 2px 10px" | $10$
+|-
+| align="right" style = "padding: 2px 10px" | $3$
+| align="right" style = "padding: 2px 10px" | $22$
+|-
+| align="right" style = "padding: 2px 10px" | $4$
+| align="right" style = "padding: 2px 10px" | $34$
+|-
+| align="right" style = "padding: 2px 10px" | $5$
+| align="right" style = "padding: 2px 10px" | $48$
+|-
+| align="right" style = "padding: 2px 10px" | $6$
+| align="right" style = "padding: 2px 10px" | $60$
+|}
+{{OEIS|A001172}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 6
+ | r = 3 + 3
+ | c = $1$ way
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 8
+ | r = 5 + 3
+ | c = $1$ way
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 10
+ | r = 7 + 3
+ | c = $2$ ways
+}}
+{{eqn | r = 5 + 5
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 12
+ | r = 7 + 5
+ | c = $1$ way
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 14
+ | r = 11 + 3
+ | c = $2$ ways
+}}
+{{eqn | r = 7 + 7
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 16
+ | r = 13 + 3
+ | c = $2$ ways
+}}
+{{eqn | r = 11 + 5
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 18
+ | r = 13 + 5
+ | c = $2$ ways
+}}
+{{eqn | r = 11 + 7
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 20
+ | r = 17 + 3
+ | c = $2$ ways
+}}
+{{eqn | r = 13 + 7
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 22
+ | r = 19 + 3
+ | c = $3$ ways
+}}
+{{eqn | r = 17 + 5
+ | c =
+}}
+{{eqn | r = 11 + 11
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 24
+ | r = 19 + 5
+ | c = $2$ ways
+}}
+{{eqn | r = 13 + 11
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 26
+ | r = 23 + 3
+ | c = $3$ ways
+}}
+{{eqn | r = 19 + 7
+ | c =
+}}
+{{eqn | r = 13 + 13
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 28
+ | r = 23 + 5
+ | c = $2$ ways
+}}
+{{eqn | r = 17 + 11
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 30
+ | r = 23 + 7
+ | c = $3$ ways
+}}
+{{eqn | r = 19 + 11
+ | c =
+}}
+{{eqn | r = 17 + 13
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 32
+ | r = 29 + 3
+ | c = $2$ ways
+}}
+{{eqn | r = 19 + 13
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 34
+ | r = 31 + 3
+ | c = $4$ ways
+}}
+{{eqn | r = 29 + 5
+ | c =
+}}
+{{eqn | r = 23 + 11
+ | c =
+}}
+{{eqn | r = 17 + 17
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 36
+ | r = 31 + 5
+ | c = $4$ ways
+}}
+{{eqn | r = 29 + 7
+ | c =
+}}
+{{eqn | r = 23 + 13
+ | c =
+}}
+{{eqn | r = 19 + 17
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 38
+ | r = 31 + 7
+ | c = $2$ ways
+}}
+{{eqn | r = 19 + 19
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 40
+ | r = 37 + 3
+ | c = $3$ ways
+}}
+{{eqn | r = 29 + 11
+ | c =
+}}
+{{eqn | r = 23 + 17
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 42
+ | r = 37 + 5
+ | c = $4$ ways
+}}
+{{eqn | r = 31 + 11
+ | c =
+}}
+{{eqn | r = 29 + 13
+ | c =
+}}
+{{eqn | r = 23 + 19
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 44
+ | r = 41 + 3
+ | c = $3$ ways
+}}
+{{eqn | r = 37 + 7
+ | c =
+}}
+{{eqn | r = 31 + 13
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 46
+ | r = 43 + 3
+ | c = $4$ ways
+}}
+{{eqn | r = 41 + 5
+ | c =
+}}
+{{eqn | r = 29 + 17
+ | c =
+}}
+{{eqn | r = 23 + 23
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 48
+ | r = 43 + 5
+ | c = $5$ ways
+}}
+{{eqn | r = 41 + 7
+ | c =
+}}
+{{eqn | r = 37 + 11
+ | c =
+}}
+{{eqn | r = 31 + 17
+ | c =
+}}
+{{eqn | r = 29 + 19
+ | c =
+}}
+{{end-eqn}}
+From [[Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways]], the smallest [[Definition:Positive Integer|positive integer]] which is the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Odd Prime|odd primes]] in $6$ different ways is $60$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Change of Basis Matrix from Basis to Itself is Identity}
+Tags: Change of Basis
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $M$ be a [[Definition:Free Module|free]] $R$-[[Definition:Module|module]] of [[Definition:Dimension (Linear Algebra)|finite dimension]] $n>0$.
+Let $\mathcal B$ be an [[Definition:Ordered Basis|ordered basis]] of $M$.
+Then the [[Definition:Change of Basis Matrix|change of basis matrix]] from $\mathcal B$ to $\mathcal B$ is the $n\times n$ [[Definition:Identity Matrix|identity matrix]]:
+:$\mathbf M_{\mathcal B, \mathcal B} = \mathbf I$
+\end{theorem}
+
+\begin{proof}
+Follows directly from the definition of [[Definition:Change of Basis Matrix|change of basis matrix]].
+{{qed}}
+[[Category:Change of Basis]]
+3ob4isdrjumnvvaa4nbh17c8srbaahz
+\end{proof}<|endoftext|>
+\section{Sum of Sequence of Seventh Powers}
+Tags: Sums of Sequences, Seventh Powers
+
+\begin{theorem}
+:$\displaystyle \sum_{j \mathop = 0}^n j^7 = \dfrac {n^2 \paren {n + 1}^2 \paren {3 n^4 + 6 n^3 - n^2 - 4 n + 2} } {24}$
+\end{theorem}
+
+\begin{proof}
+The proof proceeds by [[Principle of Mathematical Induction|induction]].
+For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
+:$\displaystyle \sum_{j \mathop = 0}^n j^7 = \dfrac {n^2 \paren {n + 1}^2 \paren {3 n^4 + 6 n^3 - n^2 - 4 n + 2} } {24}$
+$\map P 0$ is the case:
+{{begin-eqn}}
+{{eqn | l = \sum_{j \mathop = 0}^0 j^7
+ | r = 0
+ | c =
+}}
+{{eqn | r = \frac {0^2 \paren {0 + 1}^2 \paren {3 \times 0^4 + 6 \times 0^3 - 0^2 - 4 \times 0 + 2} } {24}
+ | c =
+}}
+{{end-eqn}}
+Thus $\map P 0$ is seen to hold.
+=== Basis for the Induction ===
+$\map P 1$ is the case:
+{{begin-eqn}}
+{{eqn | r = \frac {1^2 \paren {1 + 1}^2 \paren {3 \times 1^4 + 6 \times 1^3 - 1^2 - 4 \times 1 + 2} } {24}
+ | o =
+ | c =
+}}
+{{eqn | r = \frac {1 \times 2^2 \paren {3 + 6 - 1 - 4 + 2} } {24}
+ | c =
+}}
+{{eqn | r = \frac {4 \times 6} {24}
+ | c =
+}}
+{{eqn | r = 1
+ | c =
+}}
+{{eqn | r = 1^7
+ | c =
+}}
+{{eqn | r = \sum_{j \mathop = 0}^1 j^7
+ | c =
+}}
+{{end-eqn}}
+Thus $\map P 1$ is seen to hold.
+This is the [[Definition:Basis for the Induction|basis for the induction]].
+=== Induction Hypothesis ===
+Now it needs to be shown that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
+So this is the [[Definition:Induction Hypothesis|induction hypothesis]]:
+:$\displaystyle \sum_{j \mathop = 0}^k j^7 = \dfrac {k^2 \paren {k + 1}^2 \paren {3 k^4 + 6 k^3 - k^2 - 4 k + 2} } {24}$
+from which it is to be shown that:
+:$\displaystyle \sum_{j \mathop = 0}^{k + 1} j^7 = \dfrac {\paren {k + 1}^2 \paren{k + 2}^2 \paren {3 \paren {k + 1}^4 + 6 \paren {k + 1}^3 - \paren {k + 1}^2 - 4 \paren {k + 1} + 2} } {24}$
+=== Induction Step ===
+This is the [[Definition:Induction Step|induction step]]:
+{{begin-eqn}}
+{{eqn | l = \sum_{j \mathop = 0}^{k + 1} j^7
+ | r = \sum_{j \mathop = 0}^k j^7 + \paren {k + 1}^7
+ | c =
+}}
+{{eqn | r = \frac {k^2 \paren {k + 1}^2 \paren {3 k^4 + 6 k^3 - k^2 - 4 k + 2} } {24}
+ | c = [[Sum of Sequence of Seventh Powers#Induction Hypothesis|Induction Hypothesis]]
+}}
+{{eqn | o =
+ | ro= +
+ | r = \paren {k + 1}^2 \paren {k^5 + 5 k^4 + 10 k^3 + 10 k^2 + 5 k + 1}
+ | c = [[Binomial Theorem]]
+}}
+{{eqn | r = \frac {\paren {k + 1}^2} {24} \paren {3 k^6 + 6 k^5 - k^4 - 4 k^3 + 2 k^2 + 24 k^5 + 120 k^4 + 240 k^3 + 240 k^2 + 120 k + 24}
+}}
+{{eqn | r = \frac {\paren {k + 1}^2} {24} \paren {3 k^6 + 30 k^5 + 119 k^4 + 236 k^3 + 242 k^2 + 120 k + 24}
+}}
+{{eqn | r = \frac {\paren {k + 1}^2 \paren {k + 2}^2} {24} \paren {3 k^4 + 18 k^3 + 35 k^2 + 24 k + 6}
+}}
+{{eqn | r = \frac {\paren {k + 1}^2 \paren {k + 2}^2} {24} \paren {3 k^4 + 12 k^3 + 18 k^2 + 12 k + 3 + 6 k^3 + 18 k^2 + 18 k + 6 - k^2 - 2 k - 1 - 4 k - 4 + 2}
+}}
+{{eqn | r = \frac {\paren {k + 1}^2 \paren {k + 2}^2 \paren {3 \paren {k + 1}^4 + 6 \paren {k + 1}^3 - \paren {k + 1}^2 - 4 \paren {k + 1} + 2} } {24}
+ | c = [[Binomial Theorem]]
+}}
+{{end-eqn}}
+So $\map P k \implies \map P {k + 1}$ and the result follows by the [[Principle of Mathematical Induction]].
+Therefore:
+:$\forall n \in \Z_{\ge 0}: \displaystyle \sum_{j \mathop = 0}^n j^7 = \dfrac {n^2 \paren {n + 1}^2 \paren {3 n^4 + 6 n^3 - n^2 - 4 n + 2} } {24}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Change of Coordinate Vector Under Change of Basis}
+Tags: Change of Basis
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $M$ be a [[Definition:Free Module|free $R$-module]] of [[Definition:Dimension (Linear Algebra)|finite dimension]] $n>0$.
+Let $\BB$ and $\CC$ be [[Definition:Basis of Module|bases]] of $M$.
+Let $\mathbf M_{\BB, \CC}$ be the [[Definition:Change of Basis Matrix|change of basis matrix]] from $\BB$ to $\CC$.
+Let $m\in M$.
+Let $\sqbrk m_\BB$ and $\sqbrk m_\CC$ be its [[Definition:Coordinate Vector|coordinate vectors]] relative to $\BB$ and $\CC$ respectively.
+Then $\sqbrk m_\BB = \mathbf M_{\BB, \CC} \cdot \sqbrk m_\CC$.
+\end{theorem}
+
+\begin{proof}
+Intuitively, when we consider $\BB$ and $\CC$ as row vectors, this is because:
+:$\CC = \BB \cdot \mathbf M_{\BB, \CC}$ and:
+:$\BB \cdot \sqbrk m_\BB = \CC \cdot \sqbrk m_\CC$ imply:
+:$\BB \cdot \sqbrk m_\BB = \BB \cdot \mathbf M_{\BB, \CC} \cdot \sqbrk m_\CC$.
+Because $\BB$ is a [[Definition:Basis of Module|basis]], this implies $\sqbrk m_\BB = \mathbf M_{\BB, \CC} \cdot \sqbrk m_\CC$.
+This can be formalized by giving $R \times M$ the structure of a [[Definition:Ring (Abstract Algebra)|ring]]. Alternatively, this can be verified directly, which boils down to re-proving that that [[Definition:Matrix Multiplication|matrix multiplication]] is [[Definition:Associative Operation|associative]].
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Necessary and Sufficient Condition for Boundary Conditions to be Self-adjoint}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $\mathbf p$ be [[Definition:Continuously Differentiable|continuously differentiable]].
+The [[Definition:Boundary Condition|boundary conditions]]
+:$\bigvalueat {\map {\mathbf y} a} {x \mathop = a} = \bigvalueat {\map {\boldsymbol \psi} {\mathbf y} } {x \mathop = a}$
+are [[Definition:Self-Adjoint Boundary Conditions|self-adjoint]] {{iff}}:
+:$\forall i, k \in \N: 1 \le i, k \le N: \valueat {\dfrac {\partial p_i \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } } {\partial y_k} } {x \mathop = a} = \valueat {\dfrac {\partial p_k \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } } {\partial y_i} } {x \mathop = a}$
+\end{theorem}
+
+\begin{proof}
+=== Necessary Condition ===
+By assumption the [[Definition:Boundary Condition|boundary conditions]] are [[Definition:Self-Adjoint Boundary Conditions|self-adjoint]].
+Then there exists $\map g {x, \mathbf y}$ such that:
+:$\map {p_i} {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } = \dfrac {\partial \map g {x \mathbf y} } {\partial y_i}$
+Because $\mathbf p \in C^1$:
+:$g \in C^2$
+{{handwaving|Why?}}
+[[Definition:Derivative|Differentiate]] both sides {{WRT|Differentiation}} $y_k$:
+:$\dfrac {\partial \map {p_i} {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } } {\partial y_k} = \dfrac {\partial^2 \map g {x, \mathbf y} } {\partial y_k \partial y_i}$
+By the [[Schwarz-Clairaut Theorem]], [[Definition:Partial Derivative|partial derivatives]] [[Definition:Commuting Elements|commute]], hence indices can be mutually replaced:
+:$\dfrac {\partial \map {p_i} {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } } {\partial y_k} = \dfrac {\partial \map {p_k} {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } } {\partial y_i}$
+Fixing $x = a$ provides the result.
+{{qed|lemma}}
+=== Sufficient condition ===
+By assumption:
+:$\valueat {\dfrac {\partial p_i} {\partial y_j} } {x \mathop = a} = \valueat {\dfrac {\partial p_j} {\partial y_i} } {x \mathop = a}$
+Then:
+:$\exists \map g {x, \mathbf y} \in C^2: \valueat {\dfrac {\partial p_i} {\partial y_j} } {x \mathop = a} = \valueat {\dfrac {\partial p_j} {\partial y_i} } {x \mathop = a} = \valueat {\dfrac {\partial^2 g} {\partial y_i \partial y_j} } {x \mathop = a}$
+In other words:
+:$\bigvalueat {p_i} {x \mathop = a} = \valueat {\dfrac {\partial g} {\partial y_i} } {x \mathop = a}$
+{{handwaving|why?}}
+Hence, the [[Definition:Boundary Condition|boundary conditions]] are [[Definition:Self-Adjoint Boundary Conditions|self-adjoint]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Reciprocal of 61}
+Tags: 61, Examples of Reciprocals
+
+\begin{theorem}
+The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $61$ has the maximum [[Definition:Period of Recurrence|period]], that is: $60$:
+:$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$
+{{OEIS|A007450}}
+It also contains an equal number ($6$) of each of the [[Definition:Digit|digits]] from $0$ to $9$.
+\end{theorem}
+
+\begin{proof}
+Performing the calculation using [[Definition:Long Division|long division]]:
+
+ 0.01639344262295081967213114754098360655737704918032786885245901...
+ --------------------------------------------------------------------
+61)1.00000000000000000000000000000000000000000000000000000000000000000
+ 61 122 61 61 488 183 183 305
+ -- --- --- -- --- --- --- ---
+ 390 380 590 90 220 470 170 150
+ 366 366 549 61 183 427 122 122
+ --- --- --- -- --- --- --- ---
+ 240 140 410 290 370 430 480 280
+ 183 122 366 244 366 427 427 244
+ --- --- --- --- --- --- --- ---
+ 570 180 440 460 400 300 530 360
+ 549 122 427 427 366 244 488 305
+ --- --- --- --- --- --- --- ---
+ 210 580 130 330 340 560 420 550
+ 183 549 122 305 305 549 366 549
+ --- --- --- --- --- --- --- ---
+ 270 310 80 250 350 110 540 100
+ 244 305 61 244 305 61 488 61
+ --- --- -- --- --- --- --- ---
+ 260 500 190 600 450 490 520 ...
+ 244 488 183 549 427 488 488
+ -- --- --- --- --- --- ---
+ 160 120 70 510 230 200 320
+ 122 61 61 488 183 183 305
+ --- --- -- --- --- --- ---
+
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Composition of Galois Connections is Galois Connection}
+Tags: Galois Connections
+
+\begin{theorem}
+Let $L_1 = \left({S_1, \preceq_1}\right), L_2 = \left({S_2, \preceq_2}\right), L_3 = \left({S_3, \preceq_3}\right)$ be [[Definition:Ordered Set|ordered sets]].
+Let $g_1:S_1 \to S_2, g_2:S_2 \to S_3, d_1:S_2 \to S_1, d_2:S_3 \to S_2$ be [[Definition:Mapping|mappings]] such that
+:$\left({g_1, d_1}\right)$ and $\left({g_2, d_2}\right)$ are [[Definition:Galois Connection|Galois connections]].
+Then $\left({g_2 \circ g_1, d_1 \circ d_2}\right)$ is also [[Definition:Galois Connection|Galois connection]].
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Galois Connection|Galois connection]]:
+:$g_1$, $g_2$, $d_2$, and $d_1$ are [[Definition:Increasing Mapping|increasing mappings]].
+Thus by [[Composition of Increasing Mappings is Increasing]]:
+:$g_2 \circ g_1$ and $d_1 \circ d_2$ are [[Definition:Increasing Mapping|increasing mappings]].
+Let $s \in S_3, t \in S_1$.
+We will prove that
+:$s \preceq_3 \left({g_2 \circ g_1}\right)\left({t}\right) \implies \left({d_1 \circ d_2}\right)\left({s}\right) \preceq_1 t$
+Assume that
+:$s \preceq_3 \left({g_2 \circ g_1}\right)\left({t}\right)$
+By definition of [[Definition:Composition of Mappings|composition of mappings]]:
+:$s \preceq_3 g_2\left({g_1\left({t}\right)}\right)$
+By definition of [[Definition:Galois Connection|Galois connection]]:
+:$d_2\left({s}\right) \preceq_2 g_1\left({t}\right)$
+By definition of [[Definition:Increasing Mapping|increasing mapping]]:
+:$d_1\left({d_2\left({s}\right)}\right) \preceq_1 d_1\left({g_1\left({t}\right)}\right)$
+By [[Galois Connection Implies Order on Mappings]]
+:$d_1 \circ g_1 \preceq_1 I_{S_1}$
+By definitions of [[Definition:Ordering on Mappings|ordering on mappings]] and [[Definition:Composition of Mappings|composition of mappings]]:
+:$g_1\left({d_1\left({t}\right)}\right) \preceq_1 I_{S_1}\left({t}\right)$
+By definition of [[Definition:Identity Mapping|identity mapping]]:
+:$g_1\left({d_1\left({t}\right)}\right) \preceq_1 t$
+By definition of [[Definition:Transitivity|transitivity]]:
+:$d_1\left({d_2\left({s}\right)}\right) \preceq_1 t$
+Thus by definition of [[Definition:Composition of Mappings|composition of mappings]]:
+:$\left({d_1 \circ d_2}\right)\left({s}\right) \preceq_1 t$
+{{qed|lemma}}
+Assume that
+:$\left({d_1 \circ d_2}\right)\left({s}\right) \preceq_1 t$
+By definition of [[Definition:Composition of Mappings|composition of mappings]]:
+:$d_1\left({d_2\left({s}\right)}\right) \preceq_1 t$
+By definition of [[Definition:Galois Connection|Galois connection]]:
+:$d_2\left({s}\right) \preceq_2 g_1\left({t}\right)$
+By definition of [[Definition:Increasing Mapping|increasing mapping]]:
+:$g_2\left({d_2\left({s}\right)}\right) \preceq_1 g_2\left({g_1\left({t}\right)}\right)$
+By [[Galois Connection Implies Order on Mappings]]
+:$I_{S_3} \preceq_3 g_2 \circ d_2$
+By definitions of [[Definition:Ordering on Mappings|ordering on mappings]] and [[Definition:Composition of Mappings|composition of mappings]]:
+:$I_{S_3}\left({s}\right) \preceq_3 g_2\left({d_2\left({s}\right)}\right)$
+By definition of [[Definition:Identity Mapping|identity mapping]]:
+:$s \preceq_3 g_2\left({d_2\left({s}\right)}\right)$
+By definition:Transitivity|transitivity]]:
+:$s \preceq_3 g_2\left({g_1\left({t}\right)}\right)$
+Thus by definition of [[Definition:Composition of Mappings|composition of mappings]]:
+:$s \preceq_3 \left({g_2 \circ g_1}\right)\left({t}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Vector Space has Basis}
+Tags: Bases of Vector Spaces
+
+\begin{theorem}
+Let $K$ be a [[Definition:Division Ring|division ring]].
+Let $V$ be a [[Definition:Vector Space|vector space]] over $K$.
+Then $V$ has a [[Definition:Basis of Vector Space|basis]].
+\end{theorem}
+
+\begin{proof}
+The result follows from [[Vector Space has Basis between Linearly Independent Set and Spanning Set]].
+It suffices to find a [[Definition:Linearly Independent Set|linearly independent]] [[Definition:Subset|subset]] $L \subseteq V$ that is [[Definition:Contain|contained]] in a [[Definition:Spanning Set|spanning set]] $S \subseteq V$.
+By [[Empty Set is Linearly Independent]], $L$ can be taken to be the [[Definition:Empty Set|empty set]].
+Or if $V$ is nonzero, by [[Singleton is Linearly Independent]], $L$ can be taken to be any [[Definition:Singleton|singleton]] of $V$.
+$S$ can be taken to be $V$, since $V$ trivially [[Definition:Spanning Set|spans]] itself.
+Therefore, $L$ and $S$ exist and $L \subseteq S$ so $V$ has a [[Definition:Basis of Vector Space|basis]] $B$ with $L \subseteq B \subseteq S$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Product of Change of Basis Matrices}
+Tags: Change of Basis
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $M$ be a [[Definition:Free Module|free $R$-module]] of [[Definition:Dimension (Linear Algebra)|finite dimension]] $n>0$.
+Let $\mathcal A$, $\mathcal B$ and $\mathcal C$ be [[Definition:Ordered Basis|ordered bases]] of $M$.
+Let $\mathbf M_{\mathcal A,\mathcal B}$, $\mathbf M_{\mathcal B,\mathcal C}$ and $\mathbf M_{\mathcal A,\mathcal C}$ be the [[Definition:Change of Basis Matrix|change of basis matrices]] from $\mathcal A$ to $\mathcal B$, $\mathcal B$ to $\mathcal C$ and $\mathcal A$ to $\mathcal C$ respectively.
+Then $\mathbf M_{\mathcal A,\mathcal C} = \mathbf M_{\mathcal A,\mathcal B} \cdot \mathbf M_{\mathcal B,\mathcal C}$
+\end{theorem}
+
+\begin{proof}
+Let $m\in M$.
+Let $[m]_{\mathcal A}$ be its [[Definition:Coordinate Vector|coordinate vector]] relative to $\mathcal A$, and similary for $\mathcal B$ and $\mathcal C$.
+On the one hand:
+{{begin-eqn}}
+{{eqn | l = [m]_{\mathcal A}
+ | r = \mathbf M_{\mathcal A,\mathcal C} \cdot [m]_{\mathcal C}
+ | c = [[Change of Coordinate Vector Under Change of Basis]]
+}}
+{{end-eqn}}
+On the other hand:
+{{begin-eqn}}
+{{eqn | l = [m]_{\mathcal A}
+ | r = \mathbf M_{\mathcal A,\mathcal B} \cdot [m]_{\mathcal B}
+ | c = [[Change of Coordinate Vector Under Change of Basis]]
+}}
+{{eqn | l =
+ | r = \mathbf M_{\mathcal A,\mathcal B} \cdot \mathbf M_{\mathcal B,\mathcal C}
+ \cdot [m]_{\mathcal C}
+ | c = [[Change of Coordinate Vector Under Change of Basis]]
+}}
+{{end-eqn}}
+Thus $(\mathbf M_{\mathcal A,\mathcal C} - \mathbf M_{\mathcal A,\mathcal B} \cdot \mathbf M_{\mathcal B,\mathcal C}) \cdot [m]_{\mathcal C} = 0$ for all $m\in M$.
+Because $m$ is arbitrary, the result follows.
+{{explain|find a link for this}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Relative Matrix of Composition of Linear Transformations}
+Tags: Linear Transformations, Relative Matrix of Composition of Linear Transformations
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $M, N, P$ be [[Definition:Free Module|free $R$-modules]] of [[Definition:Dimension (Linear Algebra)|finite dimension]] $m, n, p > 0$ respectively.
+Let $\mathcal A,\mathcal B,\mathcal C$ be [[Definition:Ordered Basis (Linear Algebra)|ordered bases]] of $M, N, P$.
+Let $f: M \to N$ and $g : N \to P$ be [[Definition:Linear Transformation|linear transformations]], and $g \circ f$ be their [[Definition:Composition of Mappings|composition]].
+Let $\mathbf M_{f, \mathcal B, \mathcal A}$ and $\mathbf M_{g, \mathcal C, \mathcal B}$ be their [[Definition:Relative Matrix|matrices relative]] to $\mathcal A, \mathcal B$ and $\mathcal B, \mathcal C$ respectively.
+Then the [[Definition:Matrix|matrix]] of $g \circ f$ [[Definition:Relative Matrix|relative to]] $\mathcal A$ and $\mathcal C$ is:
+:$\mathbf M_{g \mathop \circ f, \mathcal C, \mathcal A} = \mathbf M_{g, \mathcal C, \mathcal B}\cdot \mathbf M_{f, \mathcal B, \mathcal A}$
+\end{theorem}
+
+\begin{proof}
+Let:
+: $\mathcal A = \left \langle {a_m} \right \rangle$
+: $\mathcal B = \left \langle {b_n} \right \rangle$
+: $\mathcal C = \left \langle {c_p} \right \rangle$
+Let:
+: $\left[{\alpha}\right]_{m n} = \left[{f; \left \langle {b_n} \right \rangle, \left \langle {a_m} \right \rangle}\right]$
+and:
+: $\left[{\beta}\right]_{n p} = \left[{g; \left \langle {c_p} \right \rangle, \left \langle {b_n} \right \rangle}\right]$
+Then:
+{{begin-eqn}}
+{{eqn | l = \left({g \circ f}\right) \left({a_j}\right)
+ | r = v \left({f \left({a_j}\right)}\right)
+ | c =
+}}
+{{eqn | r = v \left({\sum_{k \mathop = 1}^n \alpha_{k j} b_k}\right)
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^n \alpha_{k j} v \left({b_k}\right)
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^n \alpha_{k j} \left({\sum_{i \mathop = 1}^p \beta_{i k} c_i}\right)
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^n \left({\sum_{i \mathop = 1}^p \alpha_{k j} \beta_{i k} c_i}\right)
+ | c =
+}}
+{{eqn | r = \sum_{i \mathop = 1}^p \left({\sum_{k \mathop = 1}^n \alpha_{k j} \beta_{i k} c_i}\right)
+ | c =
+}}
+{{eqn | r = \sum_{i \mathop = 1}^p \left({\sum_{k \mathop = 1}^n \alpha_{k j} \beta_{i k} }\right) c_i
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}
+
+\begin{proof}
+Let $m\in M$, and $[m]_{\mathcal A}$ be its [[Definition:Coordinate Vector|coordinate vector]] with respect to $\mathcal A$.
+On the one hand:
+{{begin-eqn}}
+{{eqn | l = [g(f(m))]_{\mathcal C}
+ | r = \mathbf M_{(g\mathop\circ f), \mathcal C, \mathcal A} \cdot [m]_{\mathcal A}
+ | c = [[Change of Coordinate Vectors Under Linear Mapping]] applied to $g\circ f$
+}}
+{{end-eqn}}
+On the other hand:
+{{begin-eqn}}
+{{eqn | l = [g(f(m))]_{\mathcal C}
+ | r = \mathbf M_{g, \mathcal C, \mathcal B} \cdot [f(m)]_{\mathcal B}
+ | c = [[Change of Coordinate Vectors Under Linear Mapping]] applied to $g$
+}}
+{{eqn | l =
+ | r = \mathbf M_{g, \mathcal C, \mathcal B} \cdot \mathbf M_{f, \mathcal B, \mathcal A} \cdot [m]_{\mathcal A}
+ | c = [[Change of Coordinate Vectors Under Linear Mapping]] applied to $f$
+}}
+{{end-eqn}}
+Thus $(\mathbf M_{(g\mathop\circ f), \mathcal C, \mathcal A} - \mathbf M_{g, \mathcal C, \mathcal B} \cdot \mathbf M_{f, \mathcal B, \mathcal A}) \cdot [m]_{\mathcal A} = 0$ for all $m\in M$.
+The result follows.
+{{explain|find a link why}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Change of Coordinate Vectors Under Linear Transformation}
+Tags: Linear Transformations
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $M, N$ be [[Definition:Free Module|free $R$-modules]] of [[Definition:Dimension (Linear Algebra)|finite dimension]] $m, n > 0$ respectively.
+Let $\mathcal A, \mathcal B$ be [[Definition:Ordered Basis (Linear Algebra)|ordered bases]] of $M$ and $N$ respectively.
+Let $f: M \to N$ be a [[Definition:Linear Transformation|linear transformation]].
+Let $\mathbf M_{f, \mathcal B, \mathcal A}$ be its [[Definition:Relative Matrix of Linear Transformation|matrix relative to]] $\mathcal A$ and $\mathcal B$.
+Then for all $m \in M$:
+:$\left[{f \left({m}\right)}\right]_{\mathcal B} = \mathbf M_{f, \mathcal B, \mathcal A} \cdot \left[{m}\right]_{\mathcal A}$
+where $\left[{\, \cdot \,}\right]_{-}$ denotes the [[Definition:Coordinate Vector|coordinate vector]] with respect to a [[Definition:Basis (Linear Algebra)|basis]].
+\end{theorem}
+
+\begin{proof}
+Both sides are linear in $m$ and they coincide on the elements of $\mathcal A$ by definition of $\mathbf M_{f, \mathcal B, \mathcal A}$.
+So they are equal for all $m\in M$.
+{{explain|this has to be fleshed out}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Matrix of Bilinear Form Under Change of Basis}
+Tags: Bilinear Forms, Change of Basis
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $M$ be a [[Definition:Free Module|free $R$-module]] of [[Definition:Dimension (Linear Algebra)|finite dimension]] $n>0$.
+Let $\mathcal A$ and $\mathcal B$ be [[Definition:Ordered Basis|ordered bases]] of $M$.
+Let $\mathbf M_{\mathcal A, \mathcal B}$ be the [[Definition:Change of Basis Matrix|change of basis matrix]] from $\mathcal A$ to $\mathcal B$.
+Let $f : M\times M \to R$ be a [[Definition:Bilinear Form|bilinear form]].
+Let $\mathbf M_{f, \mathcal A}$ be its [[Definition:Relative Matrix of Bilinear Form|matrix relative to]] $\mathcal A$.
+Then its [[Definition:Relative Matrix of Bilinear Form|matrix relative to]] $\mathcal B$ equals:
+:$\mathbf M_{f, \mathcal B} = \mathbf M_{\mathcal A, \mathcal B}^\intercal \mathbf M_{f, \mathcal A} \mathbf M_{\mathcal A, \mathcal B}$
+\end{theorem}
+
+\begin{proof}
+Let $m\in M$, and let $[m]_{\mathcal A}$ and $[m]_{\mathcal B}$ denote its [[Definition:Coordinate Vector|coordinate vectors]] relative to $\mathcal A$ and $\mathcal B$.
+We have:
+{{begin-eqn}}
+{{eqn | l = f(m)
+ | r = [m]_{\mathcal A}^\intercal \cdot \mathbf M_{f, \mathcal A} \cdot [m]_{\mathcal A}
+ | c =
+}}
+{{eqn | l =
+ | r = (\mathbf M_{\mathcal A, \mathcal B} \cdot [m]_{\mathcal B})^\intercal \cdot \mathbf M_{f, \mathcal A} \cdot (\mathbf M_{\mathcal A, \mathcal B} \cdot [m]_{\mathcal B})
+ | c = [[Change of Coordinate Vector Under Change of Basis]]
+}}
+{{eqn | l =
+ | r = [m]_{\mathcal B}^\intercal \cdot \mathbf M_{\mathcal A, \mathcal B}^\intercal \cdot \mathbf M_{f, \mathcal A} \cdot \mathbf M_{\mathcal A, \mathcal B} \cdot [m]_{\mathcal B}
+ | c = [[Transpose of Matrix Product]]
+}}
+{{end-eqn}}
+Thus $\mathbf M_{f, \mathcal B} = \mathbf M_{\mathcal A, \mathcal B}^\intercal \mathbf M_{f, \mathcal A} \mathbf M_{\mathcal A, \mathcal B}$.
+{{explain|find a link for this and the first equality above, or make it an alternative definition}}
+{{qed}}
+[[Category:Bilinear Forms]]
+[[Category:Change of Basis]]
+gsen99119fz3pxac3jib09iq7jern0k
+\end{proof}<|endoftext|>
+\section{Symmetric Bilinear Form is Reflexive}
+Tags: Bilinear Forms
+
+\begin{theorem}
+Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
+Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$.
+Let $b$ be a [[Definition:Bilinear Form|bilinear form]] on $V$.
+Let $b$ be [[Definition:Symmetric Bilinear Form|symmetric]].
+Then $b$ is [[Definition:Reflexive Bilinear Form|reflexive]].
+\end{theorem}
+
+\begin{proof}
+Let $\left({v, w}\right) \in V \times V$ with $b \left({v, w}\right) = 0$.
+Because $b$ is [[Definition:Symmetric Bilinear Form|symmetric]], $b \left({w, v}\right) = 0$.
+Because $\left({v, w}\right)$ was arbitrary, $b$ is [[Definition:Reflexive Bilinear Form|reflexive]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Alternating Bilinear Form is Reflexive}
+Tags: Bilinear Forms
+
+\begin{theorem}
+Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
+Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$.
+Let $b$ be a [[Definition:Bilinear Form|bilinear form]] on $V$.
+Let $b$ be [[Definition:Alternating Bilinear Form|alternating]].
+Then $b$ is [[Definition:Reflexive Bilinear Form|reflexive]].
+\end{theorem}
+
+\begin{proof}
+Let $\left({v, w}\right) \in V \times V$ with $b \left({v, w}\right) = 0$.
+We have:
+{{begin-eqn}}
+{{eqn | l = 0
+ | r = b \left({v + w, v + w}\right)
+ | c = $b$ is [[Definition:Alternating Bilinear Form|alternating]]
+}}
+{{eqn | r = b \left({v, v}\right) + b \left({v, w}\right) + b \left({w, v}\right) + b \left({w, w}\right)
+ | c = {{Defof|Bilinear Form}}
+}}
+{{eqn | r = b \left({v, w}\right) + b \left({w, v}\right)
+ | c = $b$ is [[Definition:Alternating Bilinear Form|alternating]]
+}}
+{{eqn | r = b \left({w, v}\right)
+ | c = $b \left({v, w}\right) = 0$
+}}
+{{end-eqn}}
+Because $\left({v, w}\right)$ was arbitrary, $b$ is [[Definition:Reflexive Bilinear Form|reflexive]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Bilinear Form is Reflexive iff Symmetric or Alternating}
+Tags: Bilinear Forms
+
+\begin{theorem}
+Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
+Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$.
+Let $b$ be a [[Definition:Bilinear Form|bilinear form]] on $V$.
+Then the following are [[Definition:Logically Equivalent|equivalent]]:
+:$(1): \quad$ $b$ is [[Definition:Reflexive Bilinear Form|reflexive]]
+:$(2): \quad$ $b$ is [[Definition:Symmetric Bilinear Form|symmetric]] or [[Definition:Alternating Bilinear Form|alternating]]
+\end{theorem}
+
+\begin{proof}
+=== 1 implies 2 ===
+Follows from [[Reflexive Bilinear Form is Symmetric or Alternating]]
+{{qed}}
+=== 2 implies 1 ===
+Follows from:
+:[[Symmetric Bilinear Form is Reflexive]]
+:[[Alternating Bilinear Form is Reflexive]]
+{{qed}}
+[[Category:Bilinear Forms]]
+gcjjc8czzhhl5k20imdbg71du2p651d
+\end{proof}<|endoftext|>
+\section{Reflexive Bilinear Form is Symmetric or Alternating}
+Tags: Bilinear Forms
+
+\begin{theorem}
+Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
+Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$.
+Let $f$ be a [[Definition:Bilinear Form|bilinear form]] on $V$.
+Let $f$ be [[Definition:Reflexive Bilinear Form|reflexive]].
+Then $f$ is [[Definition:Symmetric Bilinear Form|symmetric]] or [[Definition:Alternating Bilinear Form|alternating]].
+\end{theorem}
+
+\begin{proof}
+Let $x, y, z \in V$.
+Because $f$ is [[Definition:Bilinear Form|bilinear]]:
+:$\map f {x, \map f {x, y} z - \map f {x, z} y} = \map f {x, y} \, \map f {x, z} - \map f {x, z} \, \map f {x, y} = 0$
+Because $f$ is [[Definition:Reflexive Bilinear Form|reflexive]]:
+:$\map f {\map f {x, y} z - \map f {x, z} y, x} = 0$
+Because $f$ is [[Definition:Bilinear Form|bilinear]]:
+:$(1): \quad \map f {x, y} \, \map f {z, x} = \map f {x, z} \, \map f {y, x}$
+Letting $z = x$, we obtain:
+:$\map f {x, x} \, \map f {x, y} = \map f {x, x} \, \map f {y, x}$
+Thus:
+:$(2): \quad$ if $\map f {x, x} \ne 0$, then $\map f {x, y} = \map f {y, x}$ for all $y \in V$.
+Suppose $f$ is not [[Definition:Symmetric Bilinear Form|symmetric]].
+We show that $\map f {u, u} = 0$ for all $u \in V$.
+Let $v, w \in V$ with $\map f {v, w} \ne \map f {w, v}$.
+By $(2)$, $\map f {v, v} = \map f {w, w} = 0$.
+Suppose $u \in V$ with $\map f {u, u} \ne 0$.
+By $(2)$, $\map f {u, v} = \map f {v, u}$ and $\map f {u, w} = \map f {w, u}$.
+By $(1)$, we have:
+:$\begin{cases} \map f {v, w} \, \map f {u, v} = \map f {v, u} \, \map f {w, v} \\ \map f {w, v} \, \map f {u, w} = \map f {w, u} \, \map f {v, w} \end{cases}$.
+Because $\map f {v, w} \ne \map f {w, v}$:
+:$\begin{cases} \map f {u, v} = \map f {v, u} = 0 \\ \map f {u, w} = \map f {w, u} = 0 \end{cases}$.
+Because $f$ is [[Definition:Bilinear Form|bilinear]], $\map f {u + v, w} = \map f {v, w} \ne \map f {v, w} = \map f {u + v, w}$.
+By $(2)$, $\map f {u + v, u + v} = 0$.
+Because $f$ is [[Definition:Bilinear Form|bilinear]] and $\map f {u, v} = \map f {v, u} = \map f {v, v} = 0$, $\map f {u, u} = 0$.
+This [[Definition:Contradiction|contradicts]] the assumption $\map f {u, u} \ne 0$.
+Thus $f$ is [[Definition:Alternating Bilinear Form|alternating]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Apéry's Theorem}
+Tags: Apéry's Constant
+
+\begin{theorem}
+[[Definition:Apéry's Constant|Apéry's constant]]:
+:$\map \zeta 3 = \displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$
+is [[Definition:Irrational Number|irrational]].
+\end{theorem}
+
+\begin{proof}
+We have:
+:$\dfrac 6 {\map \zeta 3} = 5 - \cfrac {1^6} {117 - \cfrac {2^6} {535 - \cfrac {\dotsb} {\dotsb - \cfrac {n^6} {34 n^3 + 51 n^2 + 27 n + 5} - \dotsb } } }$
+{{ProofWanted}}
+{{Namedfor|Roger Apéry|cat = Apéry}}
+\end{proof}<|endoftext|>
+\section{Reciprocals whose Decimal Expansion contain Equal Numbers of Digits from 0 to 9}
+Tags: Reciprocals
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integers]] $p$ have [[Definition:Reciprocal|reciprocals]] whose [[Definition:Decimal Expansion|decimal expansions]]:
+:$(1): \quad$ have the maximum [[Definition:Period of Recurrence|period]], that is: $p - 1$
+:$(2): \quad$ have an equal number, $\dfrac {p - 1} {10}$, of each of the [[Definition:Digit|digits]] from $0$ to $9$:
+::$61, 131,\ldots$
+\end{theorem}
+
+\begin{proof}
+From [[Reciprocal of 61]]:
+:$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$
+From [[Reciprocal of 131]]:
+:$\dfrac 1 {131} = 0 \cdotp \dot 00763 \, 35877 \, 86259 \, 54198 \, 47328 \, 24427 \, 48091 \, 60305 \, 34351 \, 14503 \, 81679 \, 38931 \, 29770 \, 99236 \, 64122 \, 13740 \, 45801 \, 52671 \, 75572 \, 51908 \, 39694 \, 65648 \, 85496 \, 18320 \, 61068 \, 7022 \dot 9$
+\end{proof}<|endoftext|>
+\section{Reciprocal of 131}
+Tags: 131, Examples of Reciprocals
+
+\begin{theorem}
+The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $131$ has the maximum [[Definition:Period of Recurrence|period]], that is: $130$:
+:$\dfrac 1 {131} = 0 \cdotp \dot 00763 \, 35877 \, 86259 \, 54198 \, 47328 \, 24427 \, 48091 \, 60305 \, 34351 \, 14503 \, 81679 \, 38931 \, 29770 \, 99236 \, 64122 \, 13740 \, 45801 \, 52671 \, 75572 \, 51908 \, 39694 \, 65648 \, 85496 \, 18320 \, 61068 \, 7022 \dot 9$
+{{OEIS|A007450}}
+It also contains an equal number ($13$) of each of the [[Definition:Digit|digits]] from $0$ to $9$.
+\end{theorem}
+
+\begin{proof}
+Performing the calculation using long division (zoom out for best effect):
+
+ 0.0076335877862595419847328244274809160305343511450381679389312977099236641221374045801526717557251908396946564885496183206106870229
+ --------------------------------------------------------------------------------------------------------------------------------------
+131)1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
+ 917 655 262 131 393 131 786 1179 786
+ --- --- --- --- --- --- --- ---- ---
+ 830 550 980 190 170 290 940 910 240
+ 786 524 917 131 131 262 917 786 131
+ --- --- --- --- --- --- --- --- ---
+ 440 260 630 590 390 280 230 1240 1090
+ 393 131 524 524 262 262 131 1179 1048
+ --- --- --- --- --- --- --- ---- ----
+ 470 1290 1060 660 1280 180 990 610 420
+ 393 1179 1048 655 1179 131 917 524 393
+ --- ---- ---- --- ---- --- --- --- ---
+ 770 1110 1200 500 1010 490 730 860 270
+ 655 1048 1179 393 917 393 655 786 262
+ --- ---- ---- --- --- --- --- --- ---
+ 1150 620 210 1070 930 970 750 740 800
+ 1048 524 131 1048 917 917 655 655 786
+ ---- --- --- ---- --- --- --- --- ---
+ 1020 960 790 220 1300 530 950 850 140
+ 917 917 786 131 1179 524 917 786 131
+ ---- --- --- --- ---- --- --- --- ---
+ 1030 430 400 890 1210 600 330 640 900
+ 917 393 393 786 1179 524 262 524 786
+ ---- --- --- --- ---- --- --- --- ---
+ 1130 370 700 1040 310 760 680 1160 1140
+ 1048 262 655 917 262 655 655 1048 1048
+ ---- --- --- ---- --- --- --- ---- ----
+ 820 1080 450 1230 480 1050 250 1120 920
+ 786 1048 393 1179 393 1048 131 1048 917
+ --- ---- --- ---- --- ---- --- ---- ---
+ 340 320 570 510 870 200 1190 720 300
+ 262 262 524 393 786 131 1179 655 262
+ --- --- --- --- --- --- ---- --- ---
+ 780 580 460 1170 840 690 1100 650 380
+ 655 524 393 1048 786 655 1048 524 262
+ --- --- --- ---- --- --- ---- --- ---
+ 1250 560 670 1220 540 350 520 1260 1180
+ 1179 524 655 1179 524 262 393 1179 1179
+ ---- --- --- ---- --- --- --- ---- ----
+ 710 360 150 410 160 880 1270 810 1000
+ 655 262 131 393 131 786 1179 786 ....
+ --- --- --- --- --- --- ---- ---
+ 550 980 190 170 290 940 910 240
+
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sequences of 4 Consecutive Integers with Rising Sigma}
+Tags: Sigma Function
+
+\begin{theorem}
+The following [[Definition:Ordered Quadruple|ordered quadruples]] of consecutive [[Definition:Integer|integers]] have [[Definition:Sigma Function|sigma]] values which are [[Definition:Strictly Increasing Mapping|strictly increasing]]:
+:$61, 62, 63, 64$
+:$73, 74, 75, 76$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sigma \left({61}\right)
+ | r = 62
+ | c = [[Sigma Function of Prime Number]]: $61$ is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = \sigma \left({62}\right)
+ | r = 96
+ | c = {{SigmaLink|62}}
+}}
+{{eqn | l = \sigma \left({63}\right)
+ | r = 104
+ | c = {{SigmaLink|63}}
+}}
+{{eqn | l = \sigma \left({64}\right)
+ | r = 127
+ | c = {{SigmaLink|64}}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \sigma \left({73}\right)
+ | r = 74
+ | c = [[Sigma Function of Prime Number]]: $73$ is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = \sigma \left({74}\right)
+ | r = 114
+ | c = {{SigmaLink|74}}
+}}
+{{eqn | l = \sigma \left({75}\right)
+ | r = 124
+ | c = {{SigmaLink|75}}
+}}
+{{eqn | l = \sigma \left({76}\right)
+ | r = 140
+ | c = {{SigmaLink|76}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Symmetric Bilinear Form can be Diagonalized}
+Tags: Bilinear Forms
+
+\begin{theorem}
+Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
+Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$ of [[Definition:Dimension (Linear Algebra)|finite dimension]] $n>0$.
+Let $f$ be a [[Definition:Symmetric Bilinear Form|symmetric bilinear form]] on $V$.
+Then there exists an [[Definition:Ordered Basis|ordered basis]] for which the [[Definition:Relative Matrix of Bilinear Form|relative matrix]] of $f$ is [[Definition:Diagonal Matrix|diagonal]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Bilinear Forms]]
+347cej7rct13thjadei7zksp6ejkc16
+\end{proof}<|endoftext|>
+\section{Dimension of Radical of Bilinear Form}
+Tags: Bilinear Forms
+
+\begin{theorem}
+Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
+Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$ of [[Definition:Dimension (Linear Algebra)|finite dimension]] $n > 0$.
+Let $f$ be a [[Definition:Bilinear Form|bilinear form]] on $V$.
+Let $\operatorname{rad} \left({V}\right)$ be the [[Definition:Radical of Bilinear Form|radical]] of $V$.
+Let $\operatorname{rk} \left({f}\right)$ be the [[Definition:Rank of Bilinear Form|rank]] of $f$.
+Then:
+: $\dim \left({\operatorname{rad} \left({V}\right)}\right) = n - \operatorname{rk} \left({f}\right)$
+where $\dim$ denotes [[Definition:Dimension (Linear Algebra)|dimension]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Bilinear Forms]]
+8hy9bc6d969xfy7za814ga2i1fdehey
+\end{proof}<|endoftext|>
+\section{Dimension of Orthogonal Complement With Respect to Bilinear Form}
+Tags: Bilinear Forms
+
+\begin{theorem}
+Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
+Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$ of [[Definition:Dimension (Linear Algebra)|finite dimension]].
+Let $f$ be a [[Definition:Nondegenerate Bilinear Form|nondegenerate bilinear form]] on $V$.
+Let $U\subset V$ be a [[Definition:Vector Subspace|subspace]].
+Let $U^\perp$ be its [[Definition:Orthogonal Complement (Bilinear Form)|orthogonal complement]].
+Then:
+:$\map \dim U + \map \dim U^\perp = \map \dim V$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Bilinear Forms]]
+q2o8hbxerxg39apkrawszwsjkb41w58
+\end{proof}<|endoftext|>
+\section{Pluperfect Digital Invariant has less than 61 Digits}
+Tags: Pluperfect Digital Invariants
+
+\begin{theorem}
+Let $n \in \Z_{>0}$ be a [[Definition:Pluperfect Digital Invariant|pluperfect digital invariant]].
+Then $n$ has less than $61$ [[Definition:Digit|digits]].
+\end{theorem}
+
+\begin{proof}
+We have that:
+:$n \times 9^n < 10^\left({n - 1}\right)$
+when $n > 60$.
+So an [[Definition:Digit|$n$-digit]] [[Definition:Positive Integer|integer]], for $n > 60$, is always greater than the [[Definition:Integer Addition|sum]] of the [[Definition:Integer Power|$n$th powers]] of its [[Definition:Digit|digits]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Anisotropic Vector Gives Composition of Bilinear Space}
+Tags: Bilinear Forms
+
+\begin{theorem}
+Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
+Let $\left({V, f}\right)$ be a [[Definition:Bilinear Space|bilinear space]] over $\mathbb K$.
+Let $v \in V$ be [[Definition:Anisotropic Vector (Bilinear Form)|anisotropic]].
+Let $\left\langle{v}\right\rangle$ be its [[Definition:Linear Span|span]].
+Let $v^\perp$ be its [[Definition:Orthogonal Complement (Bilinear Form)|orthogonal complement]].
+Then $\left({V, f}\right)$ is the [[Definition:Internal Orthogonal Sum (Bilinear Space)|internal orthogonal sum]] of $\left\langle{v}\right\rangle$ and $v^\perp$:
+:$V = \left\langle{v}\right\rangle \oplus v^\perp$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Bilinear Forms]]
+37u4cgm6e1cp6xndr1gf0majk4m78t3
+\end{proof}<|endoftext|>
+\section{Basis of Free Module is No Greater than Generator}
+Tags: Free Modules
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
+Let $M$ be a [[Definition:Free Module|free $R$-module]] with [[Definition:Basis (Linear Algebra)|basis]] $B$.
+Let $S$ be a [[Definition:Generator of Module|generating set]] for $M$.
+Then:
+:$\size B \le \size S$.
+That is, there exists an [[Definition:Injection|injection]] from $B$ to $S$.
+\end{theorem}
+
+\begin{proof}
+Because $S$ is a [[Definition:Generator of Module|generating set]], there is a [[Definition:Surjection|surjective]] [[Definition:Module Homomorphism|homomorphism]]
+:$\phi : R^{\paren S} \to M$
+where $R^{\paren S}$ is the [[Definition:Free Module on Set|free $R$-module]] on $S$.
+Because $B$ is a [[Definition:Basis of Module|basis]], there is an [[Definition:Module Isomorphism|isomorphism]]
+:$\psi : R^{\paren B} \to M$
+Thus $f = \psi^{-1} \circ \phi: R^{\paren S} \to R^{\paren B}$ is a [[Definition:Surjection|surjective]] [[Definition:Module Homomorphism|module homomorphism]].
+By [[Krull's Theorem]], there exists a [[Definition:Maximal Ideal of Ring|maximal ideal]] $M \subset R$.
+By [[Maximal Ideal iff Quotient Ring is Field]], $R / M$ is a [[Definition:Field (Abstract Algebra)|field]].
+Let $k = R / M$.
+Let $\pi: R \to k$ denote the [[Definition:Quotient Mapping|quotient mapping]].
+By some universal properties, there exists a $k$-[[Definition:Module Homomorphism|module homomorphism]] $\bar f: k^{\paren S} \to k^{\paren B}$ such that:
+:$\pi^B \circ f = \bar f \circ \pi^S$
+Where $\pi^B$ denotes [[Definition:Direct Sum of Module Homomorphisms|direct sum of module homomorphisms]].
+Because $\pi^B \circ f$ is [[Definition:Surjection|surjective]], so is $\bar f$.
+Let:
+:$C$ be the [[Definition:Canonical Basis of Free Module on Set|canonical basis]] of $k^{\paren S}$
+:$D$ be the [[Definition:Canonical Basis of Free Module on Set|canonical basis]] of $k^{\paren S}$
+Because $C$ is a [[Definition:Generator of Vector Space|generator]] of $k^{\paren S}$ and $\bar f$ is [[Definition:Surjection|surjective]], $\map {\bar f} C$ is a [[Definition:Generator of Vector Space|generator]] of $k^{\paren B}$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \size B
+ | r = \size D
+ | c = [[Cardinality of Canonical Basis of Free Module on Set]]
+}}
+{{eqn | o = \le
+ | r = \size {\map {\bar f} C}
+ | c = [[Basis of Vector Space Injects into Generator]]
+}}
+{{eqn | o = \le
+ | r = \size C
+ | c = [[Cardinality of Image of Mapping not greater than Cardinality of Domain]]
+}}
+{{eqn | r = \size S
+ | c = [[Cardinality of Canonical Basis of Free Module on Set]]
+}}
+{{end-eqn}}
+{{qed}}
+{{explain|find those universal properties}}
+\end{proof}<|endoftext|>
+\section{Extended Rolle's Theorem}
+Tags: Differential Calculus, Continuous Functions
+
+\begin{theorem}
+Let $f: D \to \R$ be [[Definition:Differentiable on Interval|differentiable]] on a [[Definition:Closed Real Interval|closed interval]] $I \subseteq \R$.
+Let $x_0 < x_1 < \dots < x_n \in I$.
+Let $f \left({x_i}\right) = 0$ for $i = 0, \ldots, n$.
+Then for all $i = 0, \ldots, n-1$:
+: $\exists \xi_i \in \left({x_i \,.\,.\, x_{i+1}}\right): f' \left({\xi_i}\right) = 0$
+\end{theorem}
+
+\begin{proof}
+Since $f$ is [[Definition:Differentiable on Interval|differentiable]] on $I$, f is [[Definition:Differentiable on Interval|differentiable]] on $\left[{x_i \,.\,.\, x_{i+1}}\right]$ for $i = 0, \ldots, n-1$.
+Thus [[Definition:A Fortiori|a fortiori]], $f$ is also [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\left[{x_i \,.\,.\, x_{i+1}}\right]$ and [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\left({x_i \,.\,.\, x_{i+1}}\right)$.
+For $i = 0, \ldots, n$ we have $f \left({x_i}\right) = 0$.
+Hence the conditions of [[Rolle's Theorem]] are fulfilled on each interval which yields the result.
+\end{proof}<|endoftext|>
+\section{Pell's Equation/Examples/61}
+Tags: Pell's Equation, 61
+
+\begin{theorem}
+:$x^2 - 61 y^2 = 1$
+has the smallest [[Definition:Positive Integer|positive integral]] solution:
+:$x = 1 \, 766 \, 319 \, 049$
+:$y = 226 \, 153 \, 980$
+\end{theorem}
+
+\begin{proof}
+From [[Continued Fraction Expansion of Irrational Square Root/Examples/61|Continued Fraction Expansion of $\sqrt {61}$]]:
+:$\sqrt {61} = \left[{7, \left\langle{1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14}\right\rangle}\right]$
+The [[Definition:Cycle of Periodic Continued Fraction|cycle]] is of [[Definition:Cycle Length of Periodic Continued Fraction|length]] is $11$.
+By the solution of [[Pell's Equation]], the only solutions of $x^2 - 61 y^2 = 1$ are:
+:${p_{11 r} }^2 - 61 {q_{11 r} }^2 = \left({-1}\right)^{11 r}$
+for $r = 1, 2, 3, \ldots$
+When $r = 1$ this gives:
+:${p_{11}}^2 - 61 {q_{11}}^2 = -1$
+which is not the solution required.
+When $r = 2$ this gives:
+:${p_{22} }^2 - 61 {q_{22} }^2 = 1$
+From [[Continued Fraction Expansion of Irrational Square Root/Examples/61/Convergents|Convergents of Continued Fraction Expansion of $\sqrt {61}$]]:
+:$p_{22} = 1 \, 766 \, 319 \, 049$
+:$q_{22} = 226 \, 153 \, 980$
+although on that page the numbering goes from $p_0$ to $p_{21}$, and $q_0$ to $q_{21}$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{No Infinitely Descending Membership Chains/Corollary}
+Tags: Axiom of Foundation
+
+\begin{theorem}
+There cannot exist a [[Definition:Sequence|sequence]] $\left\langle{x_n}\right\rangle$ whose [[Definition:Domain (Set Theory)/Mapping|domain]] is $\N_{\gt 0}$ such that:
+:$\forall n \in \N_{\gt 0}: x_{n+1} \in x_n$
+\end{theorem}
+
+\begin{proof}
+Aiming for contradiction, suppose that there is a sequence like that.
+From the definition of a [[Definition:Sequence|sequence]], let $f$ be the the [[Definition:Mapping|mapping]] that is defined by $\left\langle{x_n}\right\rangle$.
+Let $\omega$ denote the [[Definition:Minimal Infinite Successor Set|minimal infinite successor set]].
+Let $g: \omega \to \N_{\gt 0}$ be defined as:
+:$g \left({\alpha}\right) = \alpha + 1$
+Then the [[Definition:Composition of Mappings|composition]] $f \circ g$ is a [[Definition:Mapping|mapping]] whose [[Definition:Domain (Set Theory)/Mapping|domain]] is $\omega$ such that:
+:$\forall n \in \omega: \left({f \circ g}\right) \left({n^+}\right) \in \left({f \circ g}\right) \left({n}\right)$
+But this contradicts [[No Infinitely Descending Membership Chains]].
+Therefore by contradiction there cannot exist such a sequence.
+Hence the result.
+{{qed}}
+[[Category:Axiom of Foundation]]
+al2be08tum7r0r26t5do2vw6aes462f
+\end{proof}<|endoftext|>
+\section{Sequence of Inconsummate Numbers}
+Tags: Inconsummate Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Inconsummate Number|inconsummate numbers]] begins:
+:$62, 63, 65, 75, 84, 95, 161, 173, 195, 216, 261, 266, 272, 276, \ldots$
+{{OEIS|A003635}}
+\end{theorem}<|endoftext|>
+\section{Kaprekar's Process for 2-Digit Numbers}
+Tags: Kaprekar's Process, Reversals
+
+\begin{theorem}
+[[Definition:Kaprekar's Process|Kaprekar's process]], when applied to a non-[[Definition:Repdigit Number|repdigit]] [[Definition:Digit|$2$-digit]] [[Definition:Positive Integer|positive integer]] leads to the cycle:
+:$09 \to 81 \to 63 \to 27 \to 45 \to 09$
+Note that it is important to retain the leading [[Definition:Zero Digit|zero]] on the $9$, or the process trivially terminates in $0$.
+\end{theorem}
+
+\begin{proof}
+Let $n \in \Z_{>0}$ be a [[Definition:Digit|$2$-digit]] [[Definition:Positive Integer|positive integer]].
+{{WLOG}}, let $n$ be expressed in [[Definition:Decimal Notation|decimal notation]] as:
+:$n = 10 x + y$
+where:
+: $x > y$
+:$0 \le x \le 9, 0 \le y \le 9$
+The [[Definition:Reversal|reversal]] of $n$ is $10 y + x$.
+We have:
+{{begin-eqn}}
+{{eqn | l = 10x + y - 10 y - x
+ | r = 9 \left({x - y}\right)
+ | c = where $1 \le x - y \le 9$
+}}
+{{end-eqn}}
+Thus after the first iteration of [[Definition:Kaprekar's Process|Kaprekar's process]], $n$ is one of:
+:$09$
+:$18$
+:$27$
+:$36$
+:$45$
+or their [[Definition:Reversal|reversals]].
+Applying [[Definition:Kaprekar's Process|Kaprekar's process]] to $09$ gives:
+{{begin-eqn}}
+{{eqn | l = 09
+ | r = 90 - 09
+ | c =
+}}
+{{eqn | r = 81
+ | c =
+}}
+{{eqn | o = \to
+ | r = 81 - 18
+ | c =
+}}
+{{eqn | r = 63
+ | c =
+}}
+{{eqn | o = \to
+ | r = 63 - 36
+ | c =
+}}
+{{eqn | r = 27
+ | c =
+}}
+{{eqn | o = \to
+ | r = 72 - 27
+ | c =
+}}
+{{eqn | r = 45
+ | c =
+}}
+{{eqn | o = \to
+ | r = 45 - 54
+ | c =
+}}
+{{eqn | r = 09
+ | c =
+}}
+{{end-eqn}}
+All those [[Definition:Multiple of Integer|multiples]] of $9$ can be seen to be in (or end up in) this loop.
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Necessary and Sufficient Condition for First Order System to be Field for Functional}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $\mathbf y$ be an [[Definition:Dimension|N-dimensional]] [[Definition:Vector|vector]].
+Let $J$ be a [[Definition:Real Functional|(real) functional]] such that:
+:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
+Let the corresponding [[Definition:Canonical Variable|momenta]] and [[Definition:Hamiltonian|Hamiltonian]] be:
+{{begin-eqn}}
+{{eqn | l = \map {\mathbf p} {x, \mathbf y, \mathbf y'}
+ | r = \dfrac {\partial \map F {x, \mathbf y, \mathbf y'} } {\partial \mathbf y'}
+}}
+{{eqn | l = \map H {x, \mathbf y, \mathbf y'}
+ | r = -\map F {x, \mathbf y, \mathbf y'} + \mathbf p \mathbf y'
+}}
+{{end-eqn}}
+Let the following be a family of [[Definition:Boundary Condition|boundary conditions]]:
+:$(1): \quad \map {\mathbf y'} x = \map {\boldsymbol \psi} {x, \mathbf y}$
+Then a family of [[Definition:Boundary Condition|boundary conditions]] is a field for the [[Definition:Real Functional|functional]] $J$ {{iff}} $\forall x \in \closedint a b$ the following [[Definition:Self-Adjoint Boundary Conditions|self-adjointness]] and [[Euler's Equation for Vanishing Variation in Canonical Variables|consistency]] relations hold:
+{{begin-eqn}}
+{{eqn | l = \dfrac {\partial \map {p_i} {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial y_k}
+ | r = \dfrac {\partial \map {p_k} {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial y_i}
+}}
+{{eqn | l = \dfrac {\partial \map {\mathbf p} {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial x}
+ | r = -\dfrac {\partial \map H {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial \mathbf y}
+}}
+{{end-eqn}}
+{{explain|Why "consistency"?}}
+{{explain|Meaning of square brackets, above and throughout}}
+\end{theorem}
+
+\begin{proof}
+=== Necessary Condition ===
+Set $\mathbf y = \map {\boldsymbol \psi} {x, \mathbf y}$ in the definition of [[Definition:Canonical Variable|momenta]] and [[Definition:Hamiltonian|Hamiltonian]].
+Substitute corresponding definitions into the [[Euler's Equation for Vanishing Variation in Canonical Variables|consistency]] relation.
+On the {{LHS}} we have:
+:$\dfrac {\partial \map {p_i} {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial x} = \dfrac {\partial^2 \map F {x, \mathbf y, \map {\boldsymbol \psi} {x,\mathbf y} } } {\partial x \partial y_i'}$
+On the {{RHS}} we have:
+{{begin-eqn}}
+{{eqn | l = -\dfrac {\partial \map H {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial y_i}
+ | r = -\dfrac {\partial \paren {-\map F {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } + \map {\mathbf p} {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } \map {\mathbf y'} x} } {\partial y_i}
+}}
+{{eqn | r = \frac {\partial F} {\partial y_i} - \frac {\partial^2 F} {\partial y_i \partial \mathbf y'} \boldsymbol \psi - \frac {\partial F} {\partial \mathbf y'} \frac {\partial \boldsymbol \psi} {\partial y_i}
+}}
+{{end-eqn}}
+Together they imply:
+:$\dfrac {\partial^2 F} {\partial x \partial y_i'} = \dfrac {\partial F} {\partial y_i} - \dfrac {\partial^2 F} {\partial y_i \partial \mathbf y'} \boldsymbol \psi - \dfrac {\partial F} {\partial \mathbf y'} \dfrac {\partial \boldsymbol \psi} {\partial y_i}$
+By [[Necessary and Sufficient Condition for Boundary Conditions to be Self-adjoint]]:
+:$\dfrac {\partial^2 F} {\partial y_i \partial y_k'} = \dfrac {\partial^2 F} {\partial y_k \partial y_i'}$
+Then:
+:$\dfrac {\partial^2 F} {\partial x \partial y_i'} = \dfrac {\partial F} {\partial y_i} - \dfrac {\partial^2 F} {\partial \mathbf y \partial y_i'} \boldsymbol \psi - \dfrac {\partial F} {\partial \mathbf y'} \dfrac {\partial \boldsymbol \psi} {\partial y_i}$
+$F$ [[Definition:Depend|depends]] on $\mathbf y'$ only through its third [[Definition:Vector|vector]] [[Definition:Variable|variable]], thus $\dfrac {\partial F} {\partial y_k'} = F_{y_k'}$:
+:$(2): \quad \dfrac {\partial F_{y_i'} } {\partial x} = \dfrac {\partial F} {\partial y_i} - \dfrac {\partial F_{y_i'} } {\partial \mathbf y} \boldsymbol \psi - F_{\mathbf y'} \dfrac {\partial \boldsymbol \psi} {\partial y_i}$
+$F$ [[Definition:Depend|depends]] on $x$ directly through its first [[Definition:Variable|variable]] and indirectly through its third [[Definition:Vector|vector]] variable together with [[Definition:Boundary Condition|boundary conditions]] $(1)$:
+:$\dfrac {\partial F_{y_i'} } {\partial x} = F_{y_i' x} + F_{y_i' \mathbf y'} \boldsymbol \psi_x$
+$F$ [[Definition:Depend|depends]] on $\mathbf y$ directly through its second [[Definition:Vector|vector]] [[Definition:Variable|variable]] and indirectly through its third [[Definition:Vector|vector]] [[Definition:Variable|variable]] together with [[Definition:Boundary Condition|boundary conditions]] $(1)$:
+:$\dfrac {\partial F} {\partial y_i} = F_{y_i} + F_{\mathbf y'} \boldsymbol \psi_{y_i}$
+:$\ds \dfrac {\partial F_{y_i'} } {\partial \mathbf y} = F_{y_i' \mathbf y} + \sum_{k \mathop = 1}^N F_{y_i'y_k'} \dfrac {\partial \psi_k} {\partial \mathbf y}$
+Substitution of the last three equations into $\paren 2$ leads to:
+:$\ds F_{y_i' x} + F_{y_i' \mathbf y'} \boldsymbol \psi_x = F_{y_i} + F_{\mathbf y'} \boldsymbol \psi_{y_i} - \paren {F_{y_i' \mathbf y} + \sum_{k \mathop = 1}^N F_{y_i'y_k'} \dfrac {\partial \psi_k} {\partial \mathbf y} } \boldsymbol \psi - F_{\mathbf y'} \dfrac {\partial \boldsymbol \psi} {\partial y_i}$
+which can be simplified to:
+:$F_{y_i} - F_{y_i' x} - F_{y_i' \mathbf y} \boldsymbol \psi - F_{y_i' y_j'} \paren {\dfrac {\partial \psi_j} {\partial x} + \dfrac {\partial \psi_j} {\partial y_j} \psi_j} = 0$
+By assumption:
+:$\dfrac {\d y_k} {\d x} = \psi_k$
+the [[Definition:Second Derivative|second]] [[Definition:Total Derivative|total derivative]] {{WRT}} $x$ of which yields:
+{{begin-eqn}}
+{{eqn | l = \frac {\d^2 y_k} {\d x^2}
+ | r = \frac {\d y_k'} {\d x}
+}}
+{{eqn | r = \frac {\d \psi_k} {\d x}
+}}
+{{eqn | r = \frac {\partial \psi_k} {\partial x} + \frac {\partial \psi_k} {\partial \mathbf y} \frac {\d \mathbf y} {\d x}
+ | c = [[Definition:Total Derivative|Total Derivative]] of $\psi_k$ {{WRT}} $x$
+}}
+{{eqn | r = \frac {\partial \psi_k} {\partial x} + \frac {\partial \psi_k} {\partial \mathbf y} \boldsymbol \psi
+}}
+{{end-eqn}}
+Hence:
+:$F_{y_i} - \paren {F_{y_i' x} + F_{y_i' \mathbf y} \dfrac {\d \mathbf y} {\d x} + F_{y_i' \mathbf y'} \dfrac {\d \mathbf y'} {\d x} } = 0$
+The second term is just a [[Definition:Total Derivative|total derivative]] {{WRT}} $x$, thus:
+:$(3): \quad F_{y_i} - \dfrac {\d F_{y_i'} } {\d x} = 0$
+[[Definition:Boundary Condition|Boundary conditions]] $(1)$ are [[Definition:Mutually Consistent Boundary Conditions|mutually consistent]] {{WRT}} equation $(3)$ because they hold $\forall x \in \closedint a b$.
+By definition, they are [[Definition:Mutually Consistent Boundary Conditions/wrt Functional|consistent {{WRT}} the functional $J$]].
+Since the [[Definition:Boundary Condition|boundary conditions]] are [[Definition:Mutually Consistent Boundary Conditions|consistent {{WRT}} $J$]] and [[Definition:Self-Adjoint Boundary Conditions|self-adjoint]], by definition they constitute a [[Definition:Field of Directions of Functional|field of $J$]].
+{{qed|lemma}}
+=== Sufficient Condition ===
+By assumption, $(1)$ is a [[Definition:Field of Directions of Functional|field of $J$]].
+Hence, $(1)$ is [[Definition:Self-Adjoint Boundary Conditions|self-adjoint]] and [[Definition:Mutually Consistent Boundary Conditions/wrt Functional|mutually consistent {{WRT}} $J$]].
+Thus, by definition, they are [[Definition:Mutually Consistent Boundary Conditions/wrt Functional|consistent]] {{WRT}}:
+:$F_{y_i} - \dfrac {\d F_{y_i'} } {\d x} = 0$
+The {{LHS}} can be rewritten as follows:
+{{begin-eqn}}
+{{eqn | l = F_{y_i} - \frac {\d F_{y_i'} } {\d x}
+ | r = F_{y_i} - \paren {F_{y_i' x} + F_{y_i' \mathbf y} \frac {\d \mathbf y} {\d x} + F_{y_i' \mathbf y'} \frac {\d \mathbf y'} {\d x} }
+}}
+{{eqn | r = F_{y_i} - \paren {F_{y_i' x} + F_{y_i' \mathbf y} \boldsymbol \psi + F_{y_i' \mathbf y'} \paren {\frac {\partial \boldsymbol \psi} {\partial x} + \sum_{k \mathop = 1}^N \frac {\partial \boldsymbol \psi} {\partial y_k} \psi_k} }
+ | c = as $\ds \frac {\d \mathbf y'} {\d x} = \frac {\partial \boldsymbol \psi} {\partial x} + \sum_{k \mathop = 1}^N \frac {\partial \boldsymbol \psi} {\partial y_k} \psi_k$
+}}
+{{eqn | r = \frac {\partial F} {\partial y_i} - F_{\mathbf y'} \boldsymbol \psi_{y_i} - \paren {\frac {\partial F_{y_i'} } {\partial x} + F_{y_i' \mathbf y} \boldsymbol \psi + F_{y_i' \mathbf y'} \sum_{k \mathop = 1}^N \frac {\partial \boldsymbol \psi} {\partial y_k} \psi_k}
+ | c = as $\dfrac {\partial F} {\partial y_i} = F_{y_i} + F_{\mathbf y'} \boldsymbol \psi_{y_i}$
+}}
+{{eqn | r = \frac {\partial F} {\partial y_i} - F_{\mathbf y'} \boldsymbol \psi_{y_i} - \paren {\frac {\partial F_{y_i'} } {\partial x} + \paren { {\frac {\partial F_{y_i'} } {\partial \mathbf y} - \sum_{k \mathop = 1}^N F_{y_i' y_k'} \frac {\partial \psi_k} {\partial \mathbf y} } } \boldsymbol \psi + F_{y_i' \mathbf y'} \sum_{k \mathop = 1}^N \frac {\partial \boldsymbol \psi} {\partial y_k} \psi_k}
+ | c = as $\dfrac {\partial F_{y_i'} } {\partial \mathbf y} = F_{y_i' \mathbf y} + \sum_{k \mathop = 1}^N F_{y_i' y_k'} \frac {\partial \psi_k} {\partial \mathbf y}$
+}}
+{{eqn | r = \frac {\partial F} {\partial y_i} - F_{\mathbf y'} \boldsymbol \psi_{y_i} - \frac {\partial F_{y_i'} } {\partial x} - \frac {\partial F_{y_i'} } {\partial \mathbf y} \boldsymbol \psi
+}}
+{{eqn | r = \frac {\partial F} {\partial y_i} - \frac {\partial F} {\partial \mathbf y'} \frac {\partial \boldsymbol \psi} {\partial y_i} - \frac {\partial^2 F} {\partial x \partial y_i'} - \frac {\partial^2 F} {\partial \mathbf y \partial y_i'} \boldsymbol \psi
+ | c = as $F$ [[Definition:Depend|depends]] on $\mathbf y'$ only through its third [[Definition:Vector|vector]] [[Definition:Variable|variable]]
+}}
+{{eqn | r = \frac {\partial F} {\partial y_i} - \frac {\partial F} {\partial \mathbf y'} \frac {\partial \boldsymbol \psi} {\partial y_i} - \frac {\partial^2 F} {\partial x \partial y_i'} - \frac {\partial^2 F} {\partial y_i \partial \mathbf y'} \boldsymbol \psi
+ | c = By assumption, [[Necessary and Sufficient Condition for Boundary Conditions to be Self-adjoint|boundary conditions are self-adjoint]]
+}}
+{{eqn | r = \frac {\partial F} {\partial y_i} - \mathbf p \frac {\partial \mathbf y'} {\partial y_i} - \frac {\partial p_i} {\partial x} - \frac {\partial \mathbf p} {\partial y_i} \mathbf y'
+}}
+{{eqn | r = \frac {\partial F} {\partial y_i} - \frac {\partial p_i} {\partial x} - \frac {\partial \paren {\mathbf p \mathbf y'} } {\partial y_i}
+}}
+{{eqn | r = -\frac {\partial p_i} {\partial x} - \frac {\partial \paren {-F + \mathbf p \mathbf y'} } {\partial y_i}
+}}
+{{eqn | r = -\frac {\partial p_i} {\partial x} - \frac {\partial H} {\partial y_i}
+}}
+{{end-eqn}}
+The {{RHS}} vanishes.
+Therefore:
+:$\dfrac {\partial \mathbf p} {\partial x} = -\dfrac {\partial H} {\partial \mathbf y}$
+By [[Necessary and Sufficient Condition for Boundary Conditions to be Self-adjoint]]:
+:$\dfrac {\partial^2 F} {\partial y_i \partial y_k'} = \dfrac {\partial^2 F} {\partial y_k \partial y_i'}$
+By definition of [[Definition:Canonical Variable|canonical variable]]:
+:$\mathbf p = \dfrac {\partial F} {\partial \mathbf y'}$
+Hence:
+:$\dfrac {\partial p_k} {\partial y_i} = \dfrac {\partial p_i} {\partial y_k}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers with 6 or more Prime Factors}
+Tags: Prime Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] with $6$ or more [[Definition:Prime Factor|prime factors]] (not necessarily [[Definition:Distinct|distinct]]) begins:
+:$64, 96, 128, 144, 160, 192, 216, 224, 240, 256, \ldots$
+{{OEIS|A046305}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 64
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 2
+ | c =
+}}
+{{eqn | l = 96
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 3
+ | c =
+}}
+{{eqn | l = 128
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times 2}
+ | c =
+}}
+{{eqn | l = 144
+ | r = 2 \times 2 \times 2 \times 2 \times 3 \times 3
+ | c =
+}}
+{{eqn | l = 160
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 5
+ | c =
+}}
+{{eqn | l = 192
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times 3}
+ | c =
+}}
+{{eqn | l = 216
+ | r = 2 \times 2 \times 2 \times 3 \times 3 \times 3
+ | c =
+}}
+{{eqn | l = 224
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 7
+ | c =
+}}
+{{eqn | l = 240
+ | r = 2 \times 2 \times 2 \times 2 \times 3 \times 5
+ | c =
+}}
+{{eqn | l = 256
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times 2 \times 2}
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cube as Sum of Sequence of Centered Hexagonal Numbers}
+Tags: Centered Hexagonal Numbers, Cube Numbers, Cube as Sum of Sequence of Centered Hexagonal Numbers
+
+\begin{theorem}
+:$C_n = \displaystyle \sum_{i \mathop = 1}^n H_i$
+where:
+:$C_n$ denotes the $n$th [[Definition:Cube Number|cube number]]
+:$H_i$ denotes the $i$th [[Definition:Centered Hexagonal Number|centered hexagonal number]].
+\end{theorem}
+
+\begin{proof}
+From [[Closed Form for Centered Hexagonal Numbers]]:
+:$H_n = 3 n \paren {n - 1} + 1$
+Hence:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = 1}^n H_i
+ | r = \sum_{i \mathop = 1}^n \paren {3 i \paren {i - 1} + 1}
+ | c =
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n \paren {3 i^2 - 3 i + 1}
+ | c =
+}}
+{{eqn | r = 3 \sum_{i \mathop = 1}^n i^2 - 3 \sum_{i \mathop = 1}^n i + \sum_{i \mathop = 1}^n 1
+ | c =
+}}
+{{eqn | r = 3 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 - 3 \sum_{i \mathop = 1}^n i + \sum_{i \mathop = 1}^n 1
+ | c = [[Sum of Sequence of Squares]]
+}}
+{{eqn | r = 3 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 - 3 \frac {n \paren {n + 1} } 2 + \sum_{i \mathop = 1}^n 1
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{eqn | r = \frac {n \paren {\paren {n + 1} \paren {2 n + 1} - 3 \paren {n + 1} + 2} } 2
+ | c = simplification
+}}
+{{eqn | r = \frac {n \paren {2 n^2 + 3 n + 1 - 3 n - 3 + 2} } 2
+ | c = multiplying out
+}}
+{{eqn | r = \frac {n \paren {2 n^2} } 2
+ | c = simplification
+}}
+{{eqn | r = n^3
+ | c = simplification
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared}
+Tags: Number Theory, Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared
+
+\begin{theorem}
+Let $p$ be a [[Definition:Prime Number|prime number]].
+Then there exists at least one [[Definition:Positive Integer|positive integer]] $n$ greater than $1$ such that:
+:$n^{p - 1} \equiv 1 \pmod {p^2}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = p^2
+ | o = \equiv
+ | r = 0
+ | rr = \pmod {p^2}
+}}
+{{eqn | l = 1
+ | o = \equiv
+ | r = 1
+ | rr = \pmod {p^2}
+}}
+{{eqn | ll= \leadsto
+ | l = p^2 + 1
+ | o = \equiv
+ | r = 0 + 1
+ | rr= \pmod {p^2}
+ | c = [[Modulo Addition is Well-Defined]]
+}}
+{{eqn | o = \equiv
+ | r = 1
+ | rr = \pmod {p^2}
+}}
+{{eqn | ll= \leadsto
+ | l = \paren {p^2 + 1}^{p - 1}
+ | o = \equiv
+ | r = 1^{p - 1}
+ | rr= \pmod {p^2}
+ | c = [[Congruence of Powers]]
+}}
+{{eqn | o = \equiv
+ | r = 1
+ | rr= \pmod {p^2}
+}}
+{{end-eqn}}
+Hence $p^2 + 1$ fulfils the conditions for the value of $n$ whose existence was required to be demonstrated.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes}
+Tags: Sums of Squares, Sums of Cubes, 65, Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|positive integer]] which is both the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Square Number|square numbers]] in two [[Definition:Distinct|distinct]] ways and also the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Cube Number|cube numbers]] is $65$:
+{{begin-eqn}}
+{{eqn | l = 65
+ | m = 16 + 49
+ | mo= =
+ | r = 4^2 + 7^2
+ | c =
+}}
+{{eqn | m = 1 + 64
+ | mo= =
+ | r = 1^2 + 8^2
+ | c =
+}}
+{{eqn | o =
+ | mo= =
+ | r = 1^3 + 4^3
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+From [[Sum of 2 Squares in 2 Distinct Ways]], the smallest $2$ [[Definition:Positive Integer|positive integer]] which are the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Square Number|square numbers]] in two [[Definition:Distinct|distinct]] ways are $50$ and $65$.
+But $50$ cannot be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Cube Number|cube numbers]]:
+{{begin-eqn}}
+{{eqn | l = 50 - 1^3
+ | r = 49
+ | c = which is not [[Definition:Cube Number|cubic]]
+}}
+{{eqn | l = 50 - 2^3
+ | r = 42
+ | c = which is not [[Definition:Cube Number|cubic]]
+}}
+{{eqn | l = 50 - 3^3
+ | r = 23
+ | c = which is not [[Definition:Cube Number|cubic]]
+}}
+{{eqn | l = 50 - 4^3
+ | r = -14
+ | c = and we have fallen off the end
+}}
+{{end-eqn}}
+Hence $65$ is that smallest number.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Magic Constant of Order 5 Magic Square}
+Tags: Magic Squares, 65, Magic Constant of Order 5 Magic Square
+
+\begin{theorem}
+The [[Definition:Magic Constant|magic constant]] of the [[Magic Square/Examples/Order 5|order $5$ magic square]] is $65$.
+\end{theorem}
+
+\begin{proof}
+Let $M_5$ denote an [[Magic Square/Examples/Order 5|order $5$ magic square]]
+By [[Sum of Terms of Magic Square]], the total of all the entries in $M_5$ is given by:
+:$T_5 = \dfrac {5^2 \left({5^2 + 1}\right)} 2 = \dfrac {25 \times 26} 2 = 325$
+As there are $5$ [[Definition:Row of Matrix|rows]] of $M_5$, the [[Definition:Magic Constant|magic constant]] of $M_5$ is given by:
+:$S_5 = \dfrac {325} 5 = 65$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Triple of Triangular Numbers whose Pairwise Sums are Triangular}
+Tags: Triangular Numbers
+
+\begin{theorem}
+The following [[Definition:Ordered Triple|triplet]] of [[Definition:Triangular Number|triangular numbers]] has the property that the [[Definition:Integer Addition|sum]] of each [[Definition:Ordered Pair|pair]] of them, and their total, are all [[Definition:Triangular Number|triangular numbers]]:
+:$66, 105, 105$
+\end{theorem}
+
+\begin{proof}
+Throughout we use [[Closed Form for Triangular Numbers]], which gives that the $n$th [[Definition:Triangular Number|triangular number]] can be expressed as:
+:$T_n = \dfrac {11 \times 12} 2$
+We have:
+{{begin-eqn}}
+{{eqn | l = 66
+ | r = \frac {11 \times 12} 2
+ | c = and so is [[Definition:Triangular Number|triangular]]
+}}
+{{eqn | l = 105
+ | r = \frac {14 \times 15} 2
+ | c = and so is [[Definition:Triangular Number|triangular]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = 66 + 105
+ | r = 171
+ | c =
+}}
+{{eqn | r = \frac {18 \times 19} 2
+ | c = and so is [[Definition:Triangular Number|triangular]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 105 + 105
+ | r = 210
+ | c =
+}}
+{{eqn | r = \frac {20 \times 21} 2
+ | c = and so is [[Definition:Triangular Number|triangular]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 66 + 105 + 105
+ | r = 210
+ | c =
+}}
+{{eqn | r = \frac {23 \times 24} 2
+ | c = and so is [[Definition:Triangular Number|triangular]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Positive Even Integers as Sum of 2 Composite Odd Integers in 2 Ways}
+Tags: Composite Numbers, Odd Integers
+
+\begin{theorem}
+Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive]] [[Definition:Even Integer|even integer]].
+Let $n$ be such that it cannot be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]] in at least $2$ different ways.
+Then $n$ belongs to the [[Definition:Set|set]]:
+:$\left\{ {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 38, 40, 44, 46, 52, 56, 62, 68}\right\}$
+{{OEIS|A284788}}
+\end{theorem}
+
+\begin{proof}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]] begins:
+:$9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, \ldots$
+which is more than we need for this proof.
+{{OEIS|A071904}}
+Generating the possible [[Definition:Integer Addition|sums]] of these proceeds as follows:
+{{begin-eqn}}
+{{eqn | l = 18
+ | r = 9 + 9
+ | c =
+}}
+{{eqn | l = 24
+ | r = 9 + 15
+ | c =
+}}
+{{eqn | l = 30
+ | r = 9 + 21
+ | c =
+}}
+{{eqn | r = 15 + 15
+ | c =
+}}
+{{eqn | l = 34
+ | r = 9 + 25
+ | c =
+}}
+{{eqn | l = 36
+ | r = 9 + 27
+ | c =
+}}
+{{eqn | r = 15 + 21
+ | c =
+}}
+{{eqn | l = 40
+ | r = 15 + 25
+ | c =
+}}
+{{eqn | l = 42
+ | r = 9 + 33
+ | c =
+}}
+{{eqn | r = 15 + 27
+ | c =
+}}
+{{eqn | r = 21 + 21
+ | c =
+}}
+{{eqn | l = 44
+ | r = 9 + 35
+ | c =
+}}
+{{eqn | l = 46
+ | r = 21 + 25
+ | c =
+}}
+{{eqn | l = 48
+ | r = 9 + 39
+ | c =
+}}
+{{eqn | r = 15 + 33
+ | c =
+}}
+{{eqn | r = 21 + 27
+ | c =
+}}
+{{eqn | l = 50
+ | r = 15 + 35
+ | c =
+}}
+{{eqn | r = 25 + 25
+ | c =
+}}
+{{eqn | l = 52
+ | r = 25 + 27
+ | c =
+}}
+{{eqn | l = 54
+ | r = 9 + 45
+ | c =
+}}
+{{eqn | r = 15 + 39
+ | c =
+}}
+{{eqn | r = 21 + 33
+ | c =
+}}
+{{eqn | r = 27 + 27
+ | c =
+}}
+{{eqn | l = 56
+ | r = 21 + 35
+ | c =
+}}
+{{eqn | l = 58
+ | r = 9 + 49
+ | c =
+}}
+{{eqn | r = 25 + 33
+ | c =
+}}
+{{eqn | l = 60
+ | r = 9 + 51
+ | c =
+}}
+{{eqn | r = 15 + 45
+ | c =
+}}
+{{eqn | r = 21 + 39
+ | c =
+}}
+{{eqn | r = 25 + 35
+ | c =
+}}
+{{eqn | r = 27 + 33
+ | c =
+}}
+{{eqn | l = 62
+ | r = 27 + 35
+ | c =
+}}
+{{eqn | l = 64
+ | r = 9 + 55
+ | c =
+}}
+{{eqn | r = 15 + 49
+ | c =
+}}
+{{eqn | r = 25 + 39
+ | c =
+}}
+{{eqn | l = 66
+ | r = 9 + 57
+ | c =
+}}
+{{eqn | r = 15 + 51
+ | c =
+}}
+{{eqn | r = 21 + 45
+ | c =
+}}
+{{eqn | r = 27 + 39
+ | c =
+}}
+{{eqn | r = 33 + 33
+ | c =
+}}
+{{eqn | l = 68
+ | r = 33 + 35
+ | c =
+}}
+{{end-eqn}}
+This exhausts all possible [[Definition:Integer Addition|sums]] of [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]] of $68$ or less.
+It remains to be demonstrated that all [[Definition:Positive Integer|positive]] [[Definition:Even Integer|even integers]] over $68$ can be so expressed in $2$ different ways.
+Notice that one of $n - 15, n - 25, n - 35$ is [[Definition:Divisor of Integer|divisible]] by $3$.
+Similarly one of $n - 9, n - 15, n - 21, n - 27, n - 33$ is [[Definition:Divisor of Integer|divisible]] by $5$.
+For all above $n - k$ which is [[Definition:Composite Number|composite]], since $k$ is [[Definition:Composite Number|composite]], $\paren {n - k} + k$ is a way to express $n$ as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]].
+Since $n > 68$, these two ways are distinct unless $n - 15$ is [[Definition:Divisor of Integer|divisible]] by $15$.
+In that case, $n - 45$ is also [[Definition:Divisor of Integer|divisible]] by $15$, and $n - 45 \ne 15$.
+Then $n = \paren {n - 15} + 15 = \paren {n - 45} + 45$ are two ways to express $n$ as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Largest Even Integer not expressible as Sum of 2 k Odd Composite Integers}
+Tags: Composite Numbers, Odd Integers
+
+\begin{theorem}
+Let $k \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
+The largest [[Definition:Even Integer|even integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of $2 k$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]] is $18 k + 20$.
+\end{theorem}
+
+\begin{proof}
+Let $n$ be an [[Definition:Even Integer|even integer]] greater than $18 k + 20$.
+Then $n - 9 \paren {2 k - 2}$ is an [[Definition:Even Integer|even integer]] greater than $18 k + 20 - 9 \paren {2 k - 2} = 38$.
+By [[Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers]], every [[Definition:Even Integer|even integer]] greater than $38$ can be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]].
+Thus $n - 9 \paren {2 k - 2}$ can be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]].
+So, let $a$ and $b$ be [[Definition:Odd Integer|odd]] [[Definition:Composite Number|composite integers]] such that $a + b = n - 9 \paren {2 k - 2}$.
+Then:
+:$9 \paren {2 k - 2} + a + b = n$
+This is an [[Definition:Expression|expression]] for $n$ as the [[Definition:Integer Addition|sum]] of $2 k$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]], in which $2 k - 2$ of them are occurrences of $9$.
+Thus such an [[Definition:Expression|expression]] can always be found for $n > 18 k + 20$.
+{{qed|lemma}}
+It remains to be shown that $18 k + 20$ is not expressible as the [[Definition:Integer Addition|sum]] of $2 k$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]].
+The $k = 1$ case is demonstrated in [[Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers]].
+Now suppose $k \ge 2$.
+The two smallest [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]] are $9$ and $15$.
+Suppose $18 k + 20$ is expressible as the [[Definition:Integer Addition|sum]] of $2 k$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]].
+Then at least $2 k - 3$ of them are $9$'s, because:
+:$9 \paren {2 k - 4} + 4 \times 15 = 18 k + 24 > 18 k + 20$
+Then the problem reduces to finding an expression of $18 k + 20 - 9 \paren {2 k - 3} = 47$ as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]].
+The first few [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]] are:
+:$9, 15, 21, 25, 27, 33, 35, 39, 45$
+Their difference with $47$ are:
+:$38, 32, 26, 22, 20, 14, 12, 8, 2$
+The [[Definition:Integer|integers]] above are in the [[Definition:Set|set]] of [[Definition:Integer|integers]] not expressible as a [[Definition:Integer Addition|sum]] of $2$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]].
+The full [[Definition:Set|set]] of these can be found in [[Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers]].
+Thus $18 k + 20$ is not expressible as the [[Definition:Integer Addition|sum]] of $2 k$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]].
+This proves the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Number whose Square and Cube use all Digits Once}
+Tags: Recreational Mathematics, Square Numbers, Cube Numbers, 69
+
+\begin{theorem}
+The only [[Definition:Integer|integer]] whose [[Definition:Square (Algebra)|square]] and [[Definition:Cube (Algebra)|cube]] use each of the [[Definition:Digit|digits]] from $0$ to $9$ exactly once each is $69$.
+\end{theorem}<|endoftext|>
+\section{Cube of 71 is Odd Integers in Sequence}
+Tags: Cube Numbers, Odd Integers, 71
+
+\begin{theorem}
+The [[Definition:Cube (Algebra)|cube]] of $71$, when expressed in [[Definition:Decimal Notation|decimal notation]], is the [[Definition:Odd Integer|odd integers]] from $3$ to $11$ written in sequence:
+::$71^3 = 357 \, 911$
+\end{theorem}
+
+\begin{proof}
+Verified by direct calculation.
+\end{proof}<|endoftext|>
+\section{Prime Numbers which Divide Sum of All Lesser Primes}
+Tags: Prime Numbers
+
+\begin{theorem}
+The following [[Definition:Integer Sequence|sequence]] of [[Definition:Prime Number|prime numbers]] has the property that each is a [[Definition:Divisor of Integer|divisor]] of the [[Definition:Integer Addition|sum]] of all [[Definition:Prime Number|primes]] smaller than them:
+:$2, 5, 71, 369 \, 119, 415 \, 074 \, 643$
+{{OEIS|A007506}}
+As of time of writing (April $2020$), no others are known.
+\end{theorem}
+
+\begin{proof}
+Verified by calculation.
+\end{proof}<|endoftext|>
+\section{4 Positive Integers in Arithmetic Sequence which have Same Euler Phi Value}
+Tags: Euler Phi Function
+
+\begin{theorem}
+The following [[Definition:Set|sets]] of $4$ [[Definition:Positive Integer|positive integers]] which form an [[Definition:Arithmetic Sequence|arithmetic sequence]] are the smallest which all have the same [[Definition:Euler Phi Function|Euler $\phi$ value]]:
+:$72, 78, 84, 90$
+:$216, 222, 228, 234$
+:$76 \, 326, 76 \, 332, 76 \, 338, 76 \, 344$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 78 - 72
+ | r = 6
+ | c =
+}}
+{{eqn | l = 84 - 78
+ | r = 6
+ | c =
+}}
+{{eqn | l = 90 - 84
+ | r = 6
+ | c =
+}}
+{{end-eqn}}
+demonstrating that this is indeed an [[Definition:Arithmetic Sequence|arithmetic sequence]], with a [[Definition:Common Difference|common difference]] of $6$.
+Now we show:
+{{begin-eqn}}
+{{eqn | l = \map \phi {72}
+ | r = 24
+ | c = {{EulerPhiLink|72}}
+}}
+{{eqn | l = \map \phi {78}
+ | r = 24
+ | c = {{EulerPhiLink|78}}
+}}
+{{eqn | l = \map \phi {84}
+ | r = 24
+ | c = {{EulerPhiLink|84}}
+}}
+{{eqn | l = \map \phi {90}
+ | r = 24
+ | c = {{EulerPhiLink|90}}
+}}
+{{end-eqn}}
+{{qed|lemma}}
+{{ProofWanted|It remains to be shown that this is the smallest such arithmetic sequence of $4$ integers.}}
+{{begin-eqn}}
+{{eqn | l = 222 - 216
+ | r = 6
+ | c =
+}}
+{{eqn | l = 228 - 222
+ | r = 6
+ | c =
+}}
+{{eqn | l = 234 - 228
+ | r = 6
+ | c =
+}}
+{{end-eqn}}
+demonstrating that this is indeed an [[Definition:Arithmetic Sequence|arithmetic sequence]], with a [[Definition:Common Difference|common difference]] of $6$.
+Now we show:
+{{begin-eqn}}
+{{eqn | l = \map \phi {216}
+ | r = 72
+ | c = {{EulerPhiLink|216}}
+}}
+{{eqn | l = \map \phi {222}
+ | r = 72
+ | c = {{EulerPhiLink|222}}
+}}
+{{eqn | l = \map \phi {228}
+ | r = 72
+ | c = {{EulerPhiLink|228}}
+}}
+{{eqn | l = \map \phi {234}
+ | r = 72
+ | c = {{EulerPhiLink|234}}
+}}
+{{end-eqn}}
+{{qed|lemma}}
+{{begin-eqn}}
+{{eqn | l = 76 \, 332 - 76 \, 326
+ | r = 6
+ | c =
+}}
+{{eqn | l = 76 \, 338 - 76 \, 332
+ | r = 6
+ | c =
+}}
+{{eqn | l = 76 \, 344 - 76 \, 338
+ | r = 6
+ | c =
+}}
+{{end-eqn}}
+demonstrating that this is indeed an [[Definition:Arithmetic Sequence|arithmetic sequence]], with a [[Definition:Common Difference|common difference]] of $6$.
+Now we show:
+{{begin-eqn}}
+{{eqn | l = \map \phi {76 \, 326}
+ | r = 25 \, 440
+ | c = {{EulerPhiLink|76,326|76 \, 326}}
+}}
+{{eqn | l = \map \phi {76 \, 332}
+ | r = 25 \, 440
+ | c = {{EulerPhiLink|76,332|76 \, 332}}
+}}
+{{eqn | l = \map \phi {76 \, 338}
+ | r = 25 \, 440
+ | c = {{EulerPhiLink|76,338|76 \, 338}}
+}}
+{{eqn | l = \map \phi {76 \, 344}
+ | r = 25 \, 440
+ | c = {{EulerPhiLink|76,344|76 \, 344}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Positive Integers which are Euler Phi Value for 17 Integers}
+Tags: Euler Phi Function
+
+\begin{theorem}
+There are $17$ [[Definition:Positive Integer|positive integers]] which have an [[Definition:Euler Phi Function|Euler $\phi$ value]] of the following:
+:$72, 96, 120, \ldots$
+\end{theorem}
+
+\begin{proof}
+From [[Numbers with Euler Phi Value of 72]], it is seen that $17$ [[Definition:Positive Integer|positive integers]] have an [[Definition:Euler Phi Function|Euler $\phi$ value]] of $72$:
+:$73, 91, 95, 111, 117, 135, 146, 148, 152, 182, 190, 216, 222, 228, 234, 252, 270$
+{{proof wanted|boring}}
+\end{proof}<|endoftext|>
+\section{Euler Phi Function of Non-Square Semiprime}
+Tags: Euler Phi Function, Semiprimes, Euler Phi Function of Non-Square Semiprime
+
+\begin{theorem}
+Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
+Let $\map \phi n$ denote the [[Definition:Euler Phi Function|Euler $\phi$ function]].
+Then:
+:$\map \phi n = \paren {p - 1} \paren {q - 1}$
+\end{theorem}
+
+\begin{proof}
+As $p$ and $q$ are [[Definition:Distinct|distinct]] [[Definition:Prime Number|prime numbers]], it follows that $p$ and $q$ are [[Definition:Coprime Integers|coprime]].
+Thus by [[Euler Phi Function is Multiplicative]]:
+:$\map \phi n = \map \phi p \, \map \phi q$
+From [[Euler Phi Function of Prime]]:
+:$\map \phi p = p - 1$
+:$\map \phi q = q - 1$
+Hence the result.
+{{qed}}
+\end{proof}
+
+\begin{proof}
+A [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] is a [[Definition:Square-Free|square-free]] [[Definition:Integer|integer]].
+:$\map \phi n = \displaystyle \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
+where $p \divides n$ denotes the [[Definition:Prime Number|primes]] which [[Definition:Divisor of Integer|divide]] $n$.
+As there are $2$ [[Definition:Prime Factor|prime factors]]: $p$ and $q$, this devolves to:
+:$\map \phi n = \paren {p - 1} \paren {q - 1}$
+except when $p = 2$, in which case:
+:$\map \phi n = q - 1$
+But when $p = 2$, we have that $p - 1 = 1$ and so:
+:$\paren {p - 1} \paren {q - 1} = q - 1$
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Euler Phi Function of Square-Free Integer}
+Tags: Euler Phi Function, Euler Phi Function of Square-Free Integer
+
+\begin{theorem}
+Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 2$.
+Let $n$ be [[Definition:Square-Free|square-free]].
+Let $\map \phi n$ be the [[Definition:Euler Phi Function|Euler $\phi$ function]] of $n$.
+That is, let $\map \phi n$ be the count of [[Definition:Strictly Positive|strictly positive integers]] less than or equal to $n$ which are [[Definition:Coprime Integers|prime to]] $n$.
+Then:
+:$\map \phi n = \displaystyle \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
+where $p \divides n$ denotes the [[Definition:Prime Number|primes]] which [[Definition:Divisor of Integer|divide]] $n$.
+\end{theorem}
+
+\begin{proof}
+We have that the [[Euler Phi Function is Multiplicative]].
+Let the [[Definition:Prime Decomposition|prime decomposition]] of $n$ be:
+:$\displaystyle n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
+From the definition of [[Definition:Prime Number|prime number]], each of the [[Definition:Prime Factor|prime factors]] of $n$ is [[Definition:Coprime Integers|coprime]] to all other [[Definition:Divisor of Integer|divisors]] of $n$.
+From [[Euler Phi Function of Prime]], we have:
+:$\map \phi {p_i} = \paren {p_i - 1}$
+Thus:
+:$\map \phi n = \displaystyle \prod_{1 \mathop \le i \mathop \le r} \paren {p_i - 1}$
+or:
+:$\map \phi n = \displaystyle \prod_{p \mathop \divides n} \paren {p - 1}$
+where $p$ ranges over all [[Definition:Prime Number|prime numbers]]
+When $p = 2$ we have that:
+:$p - 1 = 1$
+and so:
+:$\displaystyle \prod_{p \mathop \divides n} \paren {p - 1} = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
+Hence the result.
+{{qed}}
+\end{proof}
+
+\begin{proof}
+From [[Euler Phi Function of Integer]]:
+:$\displaystyle \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
+As $n$ is [[Definition:Square-Free Integer|square-free]]:
+:$\displaystyle n = \prod_{p \mathop \divides n} p$
+Hence:
+:$\displaystyle \map \phi n = \prod_{p \mathop \divides n} p \paren {1 - \frac 1 p}$
+and so:
+:$\displaystyle \map \phi n = \prod_{p \mathop \divides n} \paren {p - 1}$
+When $p = 2$ we have that:
+:$p - 1 = 1$
+and so:
+:$\displaystyle \prod_{p \mathop \divides n} \paren {p - 1} = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Euler Phi Function of 2 times Odd Prime}
+Tags: Euler Phi Function, Semiprimes, Euler Phi Function of 2 times Odd Prime
+
+\begin{theorem}
+Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] of the form $2 p$, where $p$ is an [[Definition:Odd Prime|odd prime]].
+Let $\map \phi n$ denote the [[Definition:Euler Phi Function|Euler $\phi$ function]].
+Then:
+:$\map \phi n = p - 1$
+\end{theorem}
+
+\begin{proof}
+By definition $n$ is a [[Definition:Semiprime Number|semiprime]].
+As $p$ is an [[Definition:Odd Prime|odd prime]], $n$ is not [[Definition:Square Number|square]].
+Thus from [[Euler Phi Function of Non-Square Semiprime]]:
+:$\map \phi n = \paren {2 - 1} \paren {p - 1}$
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Set is Not Element of Itself}
+Tags: Axiom of Foundation
+
+\begin{theorem}
+There cannot exist a [[Definition:Set|set]] which is an [[Definition:Element|element]] of itself.
+That is:
+:$\neg \exists a: a \in a$
+\end{theorem}
+
+\begin{proof}
+{{AimForCont}} $a$ is such a set.
+Then $a \in a$ and $a \in \set a$.
+$a \ne \O$ because the [[Definition:Empty Set|empty set]] has no [[Definition:Element|elements]] by definition.
+It is also seen that:
+{{begin-eqn}}
+{{eqn | l = a \cap \set a
+ | r = \set {x: x \in a \land x \in \set a}
+ | c = {{Defof|Set Intersection}}
+}}
+{{eqn | r = \set {x: x \in a \land x = a}
+ | c = {{Defof|Singleton}}
+}}
+{{eqn | r = a
+}}
+{{end-eqn}}
+By the [[Axiom:Axiom of Foundation|Axiom of Foundation]]:
+:$a \cap \set a = \O$
+Thus $a = \O$.
+But it was previously established that $a \ne \O$.
+From this [[Definition:Contradiction|contradiction]] it follows that there cannot exist such a [[Definition:Set|set]].
+Hence the result.
+{{qed}}
+{{AoF}}
+[[Category:Axiom of Foundation]]
+sn2kwjz7ehyjfm6xgawfy59j4ksudqp
+\end{proof}<|endoftext|>
+\section{Product of Number of Edges, Edges per Face and Faces of Tetrahedron}
+Tags: Tetrahedra, 72
+
+\begin{theorem}
+The [[Definition:Integer Multiplication|product]] of the number of [[Definition:Edge of Polyhedron|edges]], [[Definition:Edge of Polyhedron|edges]] per [[Definition:Face of Polyhedron|face]] and [[Definition:Face of Polyhedron|faces]] of a [[Definition:Tetrahedron|tetrahedron]] is $72$.
+\end{theorem}
+
+\begin{proof}
+A [[Definition:Tetrahedron|tetrahedron]] has:
+:$6$ [[Definition:Edge of Polyhedron|edges]]
+and:
+:$4$ [[Definition:Face of Polyhedron|faces]].
+Each [[Definition:Face of Polyhedron|face]] is a [[Definition:Triangle|triangle]], and so has $3$ [[Definition:Edge of Polyhedron|edges]].
+Hence:
+:$6 \times 4 \times 3 = 72$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Product of Number of Edges, Edges per Face and Faces of Cube}
+Tags: Cubes, 288
+
+\begin{theorem}
+The [[Definition:Integer Multiplication|product]] of the number of [[Definition:Edge of Polyhedron|edges]], [[Definition:Edge of Polyhedron|edges]] per [[Definition:Face of Polyhedron|face]] and [[Definition:Face of Polyhedron|faces]] of a [[Definition:Cube (Geometry)|cube]] is $288$.
+\end{theorem}
+
+\begin{proof}
+A [[Definition:Cube (Geometry)|cube]] has:
+:$12$ [[Definition:Edge of Polyhedron|edges]]
+and:
+:$6$ [[Definition:Face of Polyhedron|faces]].
+Each [[Definition:Face of Polyhedron|face]] is a [[Definition:Square (Geometry)|square]], and so has $4$ [[Definition:Edge of Polyhedron|edges]].
+Hence:
+:$12 \times 6 \times 4 = 288$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Product of Number of Edges, Edges per Face and Faces of Regular Octahedron}
+Tags: Regular Octahedra, 288
+
+\begin{theorem}
+The [[Definition:Integer Multiplication|product]] of the number of [[Definition:Edge of Polyhedron|edges]], [[Definition:Edge of Polyhedron|edges]] per [[Definition:Face of Polyhedron|face]] and [[Definition:Face of Polyhedron|faces]] of a [[Definition:Regular Octahedron|regular octahedron]] is $288$.
+\end{theorem}
+
+\begin{proof}
+A [[Definition:Regular Octahedron|regular octahedron]] has:
+:$12$ [[Definition:Edge of Polyhedron|edges]]
+and:
+:$8$ [[Definition:Face of Polyhedron|faces]].
+Each [[Definition:Face of Polyhedron|face]] is a [[Definition:Triangle (Geometry)|triangle]], and so has $3$ [[Definition:Edge of Polyhedron|edges]].
+Hence:
+:$12 \times 8 \times 3 = 288$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Ordinals are Well-Ordered/Corollary}
+Tags: Ordinals, Ordinals are Well-Ordered
+
+\begin{theorem}
+Let $A$ be a [[Definition:Set|set]] of [[Definition:Ordinal|ordinals]].
+Let $\Epsilon {\restriction_A}$ denote the [[Definition:Epsilon Restriction|epsilon restriction]] on $A$.
+Then $A$ is [[Definition:Strict Well-Ordering|strictly well-ordered]] by $\Epsilon {\restriction_A}$.
+\end{theorem}
+
+\begin{proof}
+Let $A$ be a [[Definition:Set|set]] of [[Definition:Ordinal|ordinals]].
+Let $\Epsilon {\restriction_A}$ denote the [[Definition:Epsilon Restriction|epsilon restriction]] on $A$.
+It is to be shown that $\Epsilon {\restriction_A}$ is [[Definition:Antireflexive Relation|antireflexive]].
+Suppose there is a $a \in A$ such that $a \in a$.
+Then by [[Transitive Set is Proper Subset of Ordinal iff Element of Ordinal/Corollary|ordinal is element of ordinal iff proper subset of ordinal]], we have $a \subsetneq a$.
+And so:
+:$\exists x \in a: x \notin a$
+But this is a [[Definition:Contradiction|contradiction]], so $\Epsilon {\restriction_A}$ must be [[Definition:Antireflexive Relation|antireflexive]].
+By [[Strict Subset Relation is Transitive]], it follows that $\Epsilon {\restriction_A}$ is [[Definition:Transitive Relation|transitive]].
+By [[Ordinals are Well-Ordered]], it follows that for any [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] $B \subseteq A$ there is a [[Definition:Smallest Element|smallest element]] $x \in B$ such that:
+:$\forall y \in B: x \in y \lor x = y$
+It has been shown that $\Epsilon {\restriction_A}$ is a [[Definition:Strict Ordering|strict ordering]] such that every [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $A$ has a [[Definition:Smallest Element|smallest element]].
+Hence the result.
+{{qed}}
+[[Category:Ordinals]]
+[[Category:Ordinals are Well-Ordered]]
+806muh777tqq80xmyb86iuar3fnux8f
+\end{proof}<|endoftext|>
+\section{Product of Number of Edges, Edges per Face and Faces of Regular Dodecahedron}
+Tags: Regular Dodecahedra, 1800
+
+\begin{theorem}
+The [[Definition:Integer Multiplication|product]] of the number of [[Definition:Edge of Polyhedron|edges]], [[Definition:Edge of Polyhedron|edges]] per [[Definition:Face of Polyhedron|face]] and [[Definition:Face of Polyhedron|faces]] of a [[Definition:Regular Dodecahedron|regular dodecahedron]] is $1800$.
+\end{theorem}
+
+\begin{proof}
+A [[Definition:Regular Dodecahedron|regular dodecahedron]] has:
+:$30$ [[Definition:Edge of Polyhedron|edges]]
+and:
+:$12$ [[Definition:Face of Polyhedron|faces]].
+Each [[Definition:Face of Polyhedron|face]] is a [[Definition:Pentagon|pentagon]], and so has $5$ [[Definition:Edge of Polyhedron|edges]].
+Hence:
+:$30 \times 12 \times 5 = 1800$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Product of Number of Edges, Edges per Face and Faces of Regular Icosahedron}
+Tags: Regular Icosahedra, 1800
+
+\begin{theorem}
+The [[Definition:Integer Multiplication|product]] of the number of [[Definition:Edge of Polyhedron|edges]], [[Definition:Edge of Polyhedron|edges]] per [[Definition:Face of Polyhedron|face]] and [[Definition:Face of Polyhedron|faces]] of a [[Definition:Regular Icosahedron|regular icosahedron]] is $1800$.
+\end{theorem}
+
+\begin{proof}
+A [[Definition:Regular Icosahedron|regular icosahedron]] has:
+:$30$ [[Definition:Edge of Polyhedron|edges]]
+and:
+:$20$ [[Definition:Face of Polyhedron|faces]].
+Each [[Definition:Face of Polyhedron|face]] is a [[Definition:Triangle (Geometry)|triangle]], and so has $3$ [[Definition:Edge of Polyhedron|edges]].
+Hence:
+:$30 \times 20 \times 3 = 1800$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest 5th Power equal to Sum of 5 other 5th Powers}
+Tags: Fifth Powers, 72
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|positive integer]] whose [[Definition:Fifth Power|fifth power]] can be expressed as the [[Definition:Integer Addition|sum]] of $5$ other [[Definition:Fifth Power|fifth powers]] is $72$:
+:$72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 19^5 + 43^5 + 46^5 + 47^5 + 67^5
+ | r = 2 \, 476 \, 099
+ | c =
+}}
+{{eqn | o =
+ | ro= +
+ | r = 147 \, 008 \, 443
+ | c =
+}}
+{{eqn | o =
+ | ro= +
+ | r = 205 \, 962 \, 976
+ | c =
+}}
+{{eqn | o =
+ | ro= +
+ | r = 229 \, 345 \, 007
+ | c =
+}}
+{{eqn | o =
+ | ro= +
+ | r = 1 \, 350 \, 125 \, 107
+ | c =
+}}
+{{eqn | r = 1 \, 934 \, 917 \, 632
+ | c =
+}}
+{{eqn | r = 72^5
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|Unless there is an analytic proof, the fact that this is indeed the smallest may need to be justified by means of an exhaustive search. We may be justified in posting up a computer program to demonstrate it.}}
+\end{proof}<|endoftext|>
+\section{Smallest Consecutive Even Nontotients}
+Tags: Nontotients
+
+\begin{theorem}
+The smallest [[Definition:Doubleton|pair]] of consecutive [[Definition:Even Integer|even]] [[Definition:Nontotient|nontotients]] is $74$ and $76$.
+\end{theorem}
+
+\begin{proof}
+From the [[Definition:Nontotient/Sequence|sequence of nontotients]]:
+{{:Definition:Nontotient/Sequence}}
+Hence, by inspection, it can be seen that $74$ and $76$ are the smallest such [[Definition:Doubleton|pair]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Positive Integers Expressible by Sum of Integers whose Reciprocals Sum to 1}
+Tags: Reciprocals, Positive Integers Expressible by Sum of Integers whose Reciprocals Sum to 1
+
+\begin{theorem}
+Every [[Definition:Positive Integer|positive integer]] over $77$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Positive Integer|positive integers]] whose [[Definition:Reciprocal|reciprocals]] add up to $1$.
+The full sequence of numbers that cannot be expressed as such is:
+:$2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 33, 34, 35, 36, 39, 40, 41, 42, 44, 46, 47, 48, 49, 51, 56, 58, 63, 68, 70, 72, 77$
+{{OEIS|A051882}}
+\end{theorem}<|endoftext|>
+\section{Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers}
+Tags: Fourth Powers, Hilbert-Waring Theorem, 79
+
+\begin{theorem}
+:$79 = 15 \times 1^4 + 4 \times 2^4$
+\end{theorem}
+
+\begin{proof}
+We have $1^4 = 1, 2^4 = 16, 3^4 = 81 > 79$.
+Hence for each $n < 79$, we can only use $1^4$ and $2^4$ in our [[Definition:Integer Addition|sum]].
+Write $n = 2^4 a + 1^4 b$.
+We can use the [[Definition:Greedy Algorithm|greedy algorithm]] to generate these expressions, since replacing $2^4$ with $16 \times 1^4$ increases the number of [[Definition:Fourth Power|fourth powers]] required.
+Suppose $n < 64$.
+By [[Division Theorem]], there is a unique way to write $n = 16 q + r$, with $q \in \Z$, $0 \le r < 16$.
+{{begin-eqn}}
+{{eqn | l = 16 q + r
+ | r = n
+}}
+{{eqn | ll = \leadsto
+ | l = 16 q + r
+ | o = <
+ | r = 64
+}}
+{{eqn | ll = \leadsto
+ | l = 16 q
+ | o = <
+ | r = 64
+}}
+{{eqn | ll = \leadsto
+ | l = q
+ | o = <
+ | r = 4
+}}
+{{end-eqn}}
+Thus $q + r \le 3 + 15 = 18$.
+It follows that each [[Definition:Positive Integer|positive integer]] less than $64$ can be expressed in not more than $18$ [[Definition:Fourth Power|fourth powers]].
+Suppose $64 \le n \le 78$.
+We cannot use more than $4$ $2^4$, since $5 \times 2^4 = 80 > n$.
+Thus we write $n = 4 \times 2^4 + \paren {n - 64} \times 1^4$.
+Since $n - 64 \le 78 - 64 = 14$, we can use not more than $18$ [[Definition:Fourth Power|fourth powers]] to express $n$.
+For $n = 79$, we still cannot use more than $4$ $2^4$, since $5 \times 2^4 = 80 > n$.
+Thus $n = 4 \times 2^4 + 15 \times 1^4$ uses the least number of [[Definition:Fourth Power|fourth powers]].
+Hence $79$ is the smallest [[Definition:Positive Integer|positive integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of fewer than $19$ [[Definition:Fourth Power|fourth powers]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Reciprocal of 81}
+Tags: 81, Examples of Reciprocals
+
+\begin{theorem}
+The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $81$ has a particularly interesting pattern:
+:$\dfrac 1 {81} = 0 \cdotp \dot 01234 \, 567 \dot 9$
+\end{theorem}
+
+\begin{proof}
+Performing the calculation using [[Definition:Long Division|long division]]:
+
+ 0.0123456790...
+ ----------------
+81)1.0000000000000
+ 81
+ --
+ 190
+ 162
+ ---
+ 280
+ 243
+ ---
+ 370
+ 324
+ ---
+ 460
+ 405
+ ---
+ 550
+ 486
+ ---
+ 640
+ 567
+ --
+ 730
+ 729
+ ---
+ 100
+ 81
+ ---
+ ...
+
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Reciprocal of Square of 1 Less than Number Base}
+Tags: Reciprocals, Reciprocal of Square of 1 Less than Number Base
+
+\begin{theorem}
+Let $b \in \Z$ be an [[Definition:Integer|integer]] such that $b > 2$.
+Let $n = \paren {b - 1}^2$.
+The [[Definition:Reciprocal|reciprocal]] of $n$, expressed in [[Definition:Number Base|base $b$]], recurs with [[Definition:Period of Recurrence|period]] $b - 1$:
+:$\dfrac 1 n = \sqbrk {0.012 \ldots cd012 \ldots}_b$
+where:
+:$c = b - 3$
+:$d = b - 1$
+\end{theorem}<|endoftext|>
+\section{Square of 1 Less than Number Base}
+Tags: Square Numbers, Square of 1 Less than Number Base
+
+\begin{theorem}
+Let $b \in \Z$ be an [[Definition:Integer|integer]] such that $b > 2$.
+Let $n = b - 1$.
+The [[Definition:Square (Algebra)|square]] of $n$ is expressed in [[Definition:Number Base|base $b$]] as:
+:$n^2 = \left[{c1}\right]_b$
+where $c = b - 2$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = n^2
+ | r = \left({b - 1}\right)^2
+ | c =
+}}
+{{eqn | r = b^2 - 2 b + 1
+ | c =
+}}
+{{eqn | r = b \left({b - 2}\right) + 1
+ | c =
+}}
+{{end-eqn}}
+The result follows by definition of [[Definition:Number Base|number base]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Positive Integers whose Square Root equals Sum of Digits}
+Tags: Square Roots
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integers]] have a [[Definition:Square Root|square root]] that equals the [[Definition:Integer Addition|sum]] of their [[Definition:Digit|digits]]:
+:$0, 1, 81$
+and there are no more.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sqrt 0
+ | r = 0
+ | c =
+}}
+{{eqn | l = \sqrt 1
+ | r = 1
+ | c =
+}}
+{{eqn | l = \sqrt {81}
+ | r = 9
+ | c =
+}}
+{{eqn | r = 8 + 1
+ | c =
+}}
+{{end-eqn}}
+By considering the [[Definition:Square Root|square roots]], we are looking for [[Definition:Positive Integer|positive integers]] with a [[Definition:Square Number|square]] for which its [[Definition:Integer Addition|sum]] of [[Definition:Digit|digits]] is equal to the original number.
+It is easy to verify the result up to $36$.
+We prove that for $n > 36$, the [[Definition:Integer Addition|sum]] of [[Definition:Digit|digits]] of $n^2$ cannot equal $n$.
+Let $n$ be a $d$-[[Definition:Digit|digit]] number.
+Then $n < 10^d$.
+Thus $n^2 < 10^{2 d}$.
+Therefore $n^2$ has at most $2 d$ [[Definition:Digit|digits]].
+Hence the [[Definition:Integer Addition|sum]] of [[Definition:Digit|digits]] of $n^2$ is at most $18 d$.
+If $n$ is a $2$-[[Definition:Digit|digit]] number, we have $n > 36 = 18 d$.
+Now suppose $d \ge 3$.
+Then:
+{{begin-eqn}}
+{{eqn | l = n
+ | o = \ge
+ | r = 10^{d - 1}
+}}
+{{eqn | r = 10 \times 10^{d - 2}
+}}
+{{eqn | o = \ge
+ | r = 10 \paren {1 + 9 \paren {d - 2} }
+ | c = [[Bernoulli's Inequality]]
+}}
+{{eqn | r = 90 d - 170
+}}
+{{eqn | o = \ge
+ | r = 18 d + 216 - 170
+ | c = as $d \ge 3$
+}}
+{{eqn | o = >
+ | r = 18 d
+}}
+{{end-eqn}}
+Therefore for $n > 36$, the [[Definition:Integer Addition|sum]] of [[Definition:Digit|digits]] of $n^2$ cannot equal $n$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Closed Form for Heptagonal Numbers}
+Tags: Heptagonal Numbers, Closed Forms
+
+\begin{theorem}
+The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Heptagonal Number|heptagonal number]] is:
+:$H_n = \dfrac {n \paren {5 n - 3} } 2$
+\end{theorem}
+
+\begin{proof}
+[[Definition:Heptagonal Number|Heptagonal numbers]] are [[Definition:Polygonal Number|$k$-gonal numbers]] where $k = 7$.
+From [[Closed Form for Polygonal Numbers]] we have that:
+:$\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$
+Hence:
+{{begin-eqn}}
+{{eqn | l = H_n
+ | r = \frac n 2 \paren {\paren {7 - 2} n - 7 + 4}
+ | c = [[Closed Form for Polygonal Numbers]]
+}}
+{{eqn | r = \dfrac {n \paren {5 n - 3} } 2
+ | c =
+}}
+{{end-eqn}}
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Square Numbers whose Sigma is Square}
+Tags: Sigma Function, Square Numbers, Numbers whose Sigma is Square, Square Numbers whose Sigma is Square
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Square Number|square numbers]] whose [[Definition:Sigma Function|$\sigma$ value]] is [[Definition:Square Number|square]] starts as follows:
+{{begin-eqn}}
+{{eqn | l = \map \sigma {1^2}
+ | r = 1^2
+ | c =
+}}
+{{eqn | l = \map \sigma {9^2}
+ | r = 11^2
+ | c =
+}}
+{{eqn | l = \map \sigma {20^2}
+ | r = 31^2
+ | c =
+}}
+{{eqn | l = \map \sigma {180^2}
+ | r = 341^2
+ | c =
+}}
+{{eqn | l = \map \sigma {1306^2}
+ | r = 1729^2
+ | c =
+}}
+{{end-eqn}}
+{{OEIS|A008847}}
+\end{theorem}<|endoftext|>
+\section{Integers for which Sigma of Phi equals Sigma}
+Tags: Sigma Function, Euler Phi Function
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integers]] have the property that the [[Definition:Sigma Function|$\sigma$ value]] of their [[Definition:Euler Phi Function|Euler $\phi$ value]] equals their [[Definition:Sigma Function|$\sigma$ value]]:
+:$\map \sigma {\map \phi n} = \map \sigma n$
+:$1, 87, 362, 1257, 1798, 5002, 9374, \ldots$
+{{OEIS|A033631}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {\map \phi 1}
+ | r = \map \sigma 1
+ | c = {{EulerPhiLink|1}}
+}}
+{{eqn | r = 1
+ | c = {{SigmaLink|1}}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {\map \phi {87} }
+ | r = \map \sigma {56}
+ | c = {{EulerPhiLink|87}}
+}}
+{{eqn | r = 120
+ | c = {{SigmaLink|56}}
+}}
+{{eqn | r = \map \sigma {87}
+ | c = {{SigmaLink|87}}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {\map \phi {362} }
+ | r = \map \sigma {180}
+ | c = {{EulerPhiLink|362}}
+}}
+{{eqn | r = 546
+ | c = {{SigmaLink|180}}
+}}
+{{eqn | r = \map \sigma {362}
+ | c = {{SigmaLink|362}}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {\map \phi {1257} }
+ | r = \map \sigma {836}
+ | c = {{EulerPhiLink|1257}}
+}}
+{{eqn | r = 1680
+ | c = {{SigmaLink|836}}
+}}
+{{eqn | r = \map \sigma {1257}
+ | c = {{SigmaLink|1257}}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {\map \phi {1798} }
+ | r = \map \sigma {840}
+ | c = {{EulerPhiLink|1798}}
+}}
+{{eqn | r = 2880
+ | c = {{SigmaLink|840}}
+}}
+{{eqn | r = \map \sigma {1798}
+ | c = {{SigmaLink|1798}}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {\map \phi {5002} }
+ | r = \map \sigma {2400}
+ | c = {{EulerPhiLink|5002}}
+}}
+{{eqn | r = 7812
+ | c = {{SigmaLink|2400}}
+}}
+{{eqn | r = \map \sigma {5002}
+ | c = {{SigmaLink|5002}}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {\map \phi {9374} }
+ | r = \map \sigma {4536}
+ | c = {{EulerPhiLink|9374}}
+}}
+{{eqn | r = 14 \, 520
+ | c = {{SigmaLink|4536}}
+}}
+{{eqn | r = \map \sigma {9374}
+ | c = {{SigmaLink|9374}}
+}}
+{{end-eqn}}
+\end{proof}<|endoftext|>
+\section{Consecutive Triplets not Sum of Pentagonal Numbers}
+Tags: Pentagonal Numbers
+
+\begin{theorem}
+The following [[Definition:Ordered Triple|triplets]] of consecutive [[Definition:Positive Integer|positive integers]] are such that none is the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Pentagonal Number|pentagonal numbers]]:
+:$\tuple {19, 20, 21}$
+:$\tuple {88, 89, 90}$
+:$\tuple {99, 100, 101}$
+:$\tuple {111, 112, 113}$
+\end{theorem}<|endoftext|>
+\section{Prime Gaps of 8}
+Tags: Prime Gaps
+
+\begin{theorem}
+The following [[Definition:Ordered Pair|pairs]] of consecutive [[Definition:Prime Number|prime numbers]] are those whose [[Definition:Integer Multiplication|difference]] is $8$:
+:$\left({89, 97}\right), \left({359, 367}\right), \left({389, 397}\right), \left({401, 409}\right), \ldots$
+{{OEIS|A031926}}
+\end{theorem}
+
+\begin{proof}
+Demonstrated by listing the [[Definition:Prime Gap|prime gaps]].
+\end{proof}<|endoftext|>
+\section{Smallest Cunningham Chain of the First Kind of Length 6}
+Tags: Cunningham Chains
+
+\begin{theorem}
+The smallest [[Definition:Cunningham Chain of the First Kind|Cunningham chain of the first kind]] of [[Definition:Length of Sequence|length]] $6$ is:
+:$\tuple {89, 179, 359, 719, 1439, 2879}$
+\end{theorem}
+
+\begin{proof}
+By definition, a [[Definition:Cunningham Chain of the First Kind|Cunningham chain of the first kind]] is a [[Definition:Integer Sequence|sequence]] of [[Definition:Prime Number|prime numbers]] $\tuple {p_1, p_2, \ldots, p_n}$ such that:
+: $p_{k + 1} = 2 p_k + 1$
+: $\dfrac {p_1 - 1} 2$ is not [[Definition:Prime Number|prime]]
+: $2 p_n + 1$ is not [[Definition:Prime Number|prime]].
+Thus each [[Definition:Term of Sequence|term]] except the last is a [[Definition:Sophie Germain Prime|Sophie Germain prime]].
+{{:Definition:Sophie Germain Prime/Sequence}}
+Let $P: \Z \to \Z$ be the [[Definition:Mapping|mapping]] defined as:
+:$\map P n = 2 n + 1$
+Applying $P$ iteratively to each of the smallest [[Definition:Sophie Germain Prime|Sophie Germain primes]] in turn:
+{{begin-eqn}}
+{{eqn | l = \map P 2
+ | r = 5
+ | c =
+}}
+{{eqn | l = \map P 5
+ | r = 11
+ | c =
+}}
+{{eqn | l = \map P {11}
+ | r = 23
+ | c =
+}}
+{{eqn | l = \map P {23}
+ | r = 47
+ | c =
+}}
+{{eqn | l = \map P {47}
+ | r = 95
+ | c = which is not [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+Thus $\tuple {2, 5, 11, 23, 47}$ is a [[Definition:Cunningham Chain of the First Kind|Cunningham chain of the first kind]] of [[Definition:Length of Sequence|length]] $5$.
+{{begin-eqn}}
+{{eqn | l = \map P 3
+ | r = 7
+ | c =
+}}
+{{eqn | l = \map P 7
+ | r = 15
+ | c = which is not [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map P {29}
+ | r = 59
+ | c =
+}}
+{{eqn | l = \map P {59}
+ | r = 119
+ | c = which is not [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map P {41}
+ | r = 83
+ | c =
+}}
+{{eqn | l = \map P {83}
+ | r = 167
+ | c =
+}}
+{{eqn | l = \map P {167}
+ | r = 335
+ | c = which is not [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map P {53}
+ | r = 107
+ | c =
+}}
+{{eqn | l = \map P {107}
+ | r = 215
+ | c = which is not [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map P {83}
+ | r = 167
+ | c =
+}}
+{{eqn | l = \map P {167}
+ | r = 335
+ | c = which is not [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map P {89}
+ | r = 179
+ | c =
+}}
+{{eqn | l = \map P {179}
+ | r = 359
+ | c =
+}}
+{{eqn | l = \map P {359}
+ | r = 719
+ | c =
+}}
+{{eqn | l = \map P {719}
+ | r = 1439
+ | c =
+}}
+{{eqn | l = \map P {1439}
+ | r = 2879
+ | c =
+}}
+{{eqn | l = \map P {2879}
+ | r = 5759
+ | c = which is not [[Definition:Prime Number|prime]].
+}}
+{{end-eqn}}
+It is noted that $\dfrac {89 - 1} 2 = 44$ which is not [[Definition:Prime Number|prime]].
+Hence the [[Definition:Integer Sequence|sequence]] of $6$:
+:$\tuple {89, 179, 359, 719, 1439, 2879}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sequence of Sum of Squares of Digits}
+Tags: Sums of Squares, Recreational Mathematics
+
+\begin{theorem}
+For a [[Definition:Positive Integer|positive integer]] $n$, let $\map f n$ be the [[Definition:Integer|integer]] created by [[Definition:Integer Addition|adding]] the [[Definition:Square (Algebra)|squares]] of [[Definition:Digit|digits]] of $n$.
+Let $m \in \Z_{>0}$ be expressed in [[Definition:Decimal Notation|decimal notation]].
+Let $\sequence {S_m}_{n \mathop \in \Z_{>0} }$ be the [[Definition:Integer Sequence|sequence]] defined as follows:
+:$n_k = \begin{cases} m & : n = 1 \\
+\map f {n_{k - 1} } & : n > 1 \end{cases}$
+Then eventually either $\sequence {S_m}$ sticks at $1$, or goes into the cycle:
+:$\ldots, 4, 16, 37, 58, 89, 145, 42, 20, 4, \ldots$
+\end{theorem}
+
+\begin{proof}
+First note that:
+:$1^2 + 9^2 + 9^2 = 163$
+:$9^2 + 9^2 + 9^2 = 243$
+and it can be seen that for a [[Definition:Positive Integer|positive integer]] $m$ larger than $199$, $\map f m < m$.
+Thus it is necessary to investigate numbers only up as far as that.
+Starting from the bottom, we have that:
+:$\map f 1 = 1^2 = 1$
+and so $\sequence {S_1} = 1, 1, 1, \ldots$
+We note the [[Definition:Integer Sequence|sequence]]:
+{{begin-eqn}}
+{{eqn | ll= \map f 4
+ | lo= =
+ | l = 4^2
+ | r = 16
+}}
+{{eqn | ll= \map f {16}
+ | lo= =
+ | l = 1^2 + 6^2
+ | r = 37
+}}
+{{eqn | ll= \map f {37}
+ | lo= =
+ | l = 3^2 + 7^2
+ | r = 58
+}}
+{{eqn | ll= \map f {58}
+ | lo= =
+ | l = 5^2 + 8^2
+ | r = 89
+}}
+{{eqn | ll= \map f {89}
+ | lo= =
+ | l = 8^2 + 9^2
+ | r = 145
+}}
+{{eqn | ll= \map f {145}
+ | lo= =
+ | l = 1^2 + 4^2 + 5^2
+ | r = 42
+}}
+{{eqn | ll= \map f {42}
+ | lo= =
+ | l = 4^2 + 2^2
+ | r = 20
+}}
+{{eqn | ll= \map f {20}
+ | lo= =
+ | l = 2^2 + 0^2
+ | r = 4
+}}
+{{end-eqn}}
+Hence any $m$ in the [[Definition:Set|set]] $\set {4, 16, 20, 37, 42, 58, 89, 145}$ stays within that [[Definition:Integer Sequence|sequence]], which we will refer to as $\mathcal S$.
+It remains to test the rest.
+Let $\mathcal T$ be the [[Definition:Set|set]] of [[Definition:Positive Integer|integers]] $m$ for which $S_m$ ends up either in $\mathcal S$ or $1$.
+Then the [[Definition:Element|elements]] of $\mathcal S$ are all in $\mathcal T$, as is $1$.
+If $m \in \mathcal T$, then any [[Definition:Positive Integer|integer]] $m'$ whose non-[[Definition:Zero Digit|zero]] [[Definition:Digit|digits]] are the same as those of $m$ is also in $\mathcal T$.
+Thus:
+:$2, 10, 24, 40, 61, 73, 85, 98, 100, 106 \in \mathcal T$
+We use the notation:
+:$a \to b \to c$
+to denote:
+:$\map f a = b, \map f b = c$
+and work progressively through $\Z_{>0}$.
+So:
+{{begin-eqn}}
+{{eqn | l = 3 \to 9 \to 81 \to 65 \to 61
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 18, 30, 56, 90
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 5 \to 25 \to 29 \to 85
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 50, 52, 92
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 6 \to 36 \to 45 \to 41 \to 17 \to 50
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 14, 54, 60, 63, 71
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 7 \to 49 \to 97 \to 130 \to 10
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 13, 31, 70, 79, 94
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 8 \to 64 \to 52
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 46, 80
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 11 \to 2
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 12 \to 5
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 21
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 15 \to 26 \to 40
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 51, 62
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 19 \to 82 \to 68 \to 100
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 28, 86, 91
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 22 \to 8
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 23 \to 13
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 32
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 27 \to 53 \to 34 \to 25
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 35, 43, 72
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 33 \to 18
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 38 \to 73
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 83
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 39 \to 90
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 93
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 44 \to 32
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 47 \to 65
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 74
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 48 \to 80
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 84
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 55 \to 50
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 57 \to 74
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 75
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 59 \to 106
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 95
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 66 \to 72
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 67 \to 85
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 76
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 69 \to 117 \to 51
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 96
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 77 \to 98
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 78 \to 113 \to 11
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | ll= \leadsto
+ | l = 87
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 88 \to 128 \to 69
+ | o = \in
+ | r = \mathcal T
+}}
+{{eqn | l = 99 \to 162 \to 41
+ | o = \in
+ | r = \mathcal T
+}}
+{{end-eqn}}
+{{Finish|It's relatively easy to show that each starting number will decrease up to at least $170$ by considering that the squares of digits will be smaller than the number itself up to around that point (considering $9^2 + 9^2 {{=}} 162$). So manual work is needed "only" below that...
Can it actually be shown that if $m$ is of 3 or more digits, then $\map f m < m$? If so then job done.
Yes, by a series of bounds: $3 \cdot 9^2 {{=}} 243, 2^2 + 2 \cdot 9^2 {{=}} 166, 1 + 6^2 + 9^2 {{=}} 118, 1 + 1 + 9^2 < 100$. Not sure how to formulate nicely, though. For numbers of $n > 3$ digits, $n 9^2 < 10^{n - 1}$}}
+\end{proof}<|endoftext|>
+\section{Necessary and Sufficient Condition for First Order System to be Mutually Consistent}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $\mathbf y$, $\boldsymbol \psi$ be [[Definition:Dimension|N-dimensional]] [[Definition:Vector|vectors]].
+Let $g$ be a twice [[Definition:Differentiable Mapping|differentiable mapping]].
+Let
+:$(1): \quad \map {\boldsymbol \psi} {x, \mathbf y} = \map {\mathbf y'} {x, \mathbf y}$
+:$(2): \quad \mathbf p \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } = \map {g_{\mathbf y} } {x, \mathbf y}$
+where $\mathbf p$ is a [[Definition:Canonical Variable|momentum]].
+Then the [[Definition:Boundary Condition|boundary conditions]] defined by $(1)$ are [[Definition:Mutually Consistent Boundary Conditions|mutually consistent]] {{iff}} the [[Definition:Mapping|mapping]] $\map g {x, \mathbf y}$ satisfies the [[Definition:Hamilton-Jacobi Equation|Hamilton-Jacobi equation]]:
+:$(3): \quad \dfrac {\partial g} {\partial x} + \map H {x, \mathbf y, \dfrac {\partial g} {\partial \mathbf y} } = 0$
+\end{theorem}
+
+\begin{proof}
+=== Necessary condition ===
+Take the [[Definition:Partial Derivative|partial derivative]] of $(3)$ {{WRT}} $x$:
+:$(4): \quad \frac {\partial^2 \map g {x, \mathbf y} } {\partial \mathbf y \partial x} = -\dfrac \partial {\partial \mathbf y} \map H {x, \mathbf y, \dfrac {\partial g} {\partial \mathbf y} }$
+By the [[Schwarz-Clairaut Theorem]], the order of [[Definition:Partial Derivative|partial derivatives]] of $g$ can be exchanged:
+:$(5): \quad \dfrac {\partial^2 \map g {x, \mathbf y} } {\partial \mathbf y \partial x} = \dfrac {\partial \mathbf p \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial x}$
+Since $\map g {x, \mathbf y}$ depends on $\mathbf y$ only in a direct way, it follows that:
+:$\dfrac {\partial g} {\partial \mathbf y} = g_{\mathbf y}$
+Hence, from $(2)$, $(4)$ and $(5)$ it follows that:
+:$\dfrac {\partial \mathbf p \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial x} = -\dfrac \partial {\partial \mathbf y} \map H {x, \mathbf y, \mathbf p \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } }$
+These are the [[Euler's Equation for Vanishing Variation in Canonical Variables|consistency equations]].
+{{qed|lemma}}
+=== Sufficient condition ===
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{2-Digit Numbers forming Longest Reverse-and-Add Sequence}
+Tags: Reverse-and-Add, 89, 98
+
+\begin{theorem}
+Let $m \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]] expressed in [[Definition:Decimal Notation|decimal notation]].
+Let $\map r m$ be the [[Definition:Reverse-and-Add|reverse-and-add process]] on $m$.
+Let $r$ be applied iteratively to $m$.
+The $2$-[[Definition:Digit|digit]] [[Definition:Positive Integer|integers]] $m$ which need the largest number of iterations before reaching a [[Definition:Palindromic Number|palindromic number]] are $89$ and $98$, both needing $24$ iterations.
+\end{theorem}
+
+\begin{proof}
+The [[Definition:Integer Sequence|sequence]] obtained by iterating $r$ on $89$ is:
+:$89, 187, 968, 1837, 9218, 17347, 91718, 173437, 907808, 1716517, 8872688,$
+:$17735476, 85189247, 159487405, 664272356, 1317544822, 3602001953, 7193004016, 13297007933,$
+:$47267087164, 93445163438, 176881317877, 955594506548, 170120002107, 8713200023178$
+{{OEIS|A033670}}
+Note that $\map r {89} = \map r {98} = 187$, so the [[Definition:Integer Sequence|sequence]] obtained by iterating $r$ on $98$ is the same.
+To show that no other $2$-digit [[Definition:Positive Integer|integers]] have longer [[Definition:Integer Sequence|sequences]], we define an [[Definition:Equivalence Relation|equivalence relation]] on the $2$-digit [[Definition:Positive Integer|integers]] as follows:
+:$\sqbrk {a_1 a_2} \sim \sqbrk {b_1 b_2} \iff a_1 + a_2 = b_1 + b_2$
+For each $2$-digit [[Definition:Positive Integer|integer]] $\sqbrk {a_1 a_2}$:
+:$\map r {\sqbrk {a_1 a_2} } = \sqbrk {a_1 a_2} + \sqbrk {a_2 a_1} = 11 \paren {a_1 + a_2}$
+Notice that if $\sqbrk {a_1 a_2} \sim \sqbrk {b_1 b_2}$:
+:$\map r {\sqbrk {a_1 a_2} } = 11 \paren {a_1 + a_2} = 11 \paren {b_1 + b_2} = \map r {\sqbrk {b_1 b_2} }$
+So the value of $r$ is equal in each [[Definition:Equivalence Class|equivalence class]].
+For each $2$-digit [[Definition:Positive Integer|integer]] $\sqbrk {a_1 a_2}$, we have:
+:$18 = 9 + 9 \ge a_1 + a_2 \ge 1 + 0 = 1$
+For $2$-digit [[Definition:Positive Integer|integers]] where $a_1 + a_2 \le 9$:
+:$\map r {\sqbrk {a_1 a_2} } = 11 \paren {a_1 + a_2}$ is a $2$-digit [[Definition:Multiple of Integer|multiple]] of $11$, which is a [[Definition:Palindromic Integer|palindrome]].
+So we consider each [[Definition:Equivalence Class|equivalence class]] of $2$-digit [[Definition:Positive Integer|integers]] with $a_1 + a_2 \ge 10$.
+We have the [[Definition:Integer Sequence|sequences]]:
+:$19, 110, 121$
+:$29, 121$
+:$39, 132, 363$
+:$49, 143, 484$
+:$59, 154, 605, 1111$
+:$69, 165, 726, 1353, 4884$
+:$79, 176, 847, 1595, 7546, 14003, 44044$
+:$89, \cdots, 8713200023178$, with $24$ iterations
+:$99$
+So:
+:every $2$-digit [[Definition:Positive Integer|integer]] not in the [[Definition:Equivalence Class|equivalence class]] of $89$ has a shorter iteration
+and:
+:$\eqclass {89} \sim = \set {89, 98}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Reciprocal of 89}
+Tags: 89, Examples of Reciprocals
+
+\begin{theorem}
+The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $89$ contains the [[Definition:Fibonacci Number|Fibonacci sequence]]:
+:$\dfrac 1 {89} = 0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$
+\end{theorem}
+
+\begin{proof}
+Performing the calculation using [[Definition:Long Division|long division]]:
+
+ 0.0112359550561797752808988764044943820224719101...
+ ------------------------------------------------------
+89)1.0000000000000000000000000000000000000000000000000
+ 89 534 712 801 89
+ -- --- --- --- ---
+ 110 160 880 390 810
+ 89 89 801 356 801
+ --- --- --- --- ---
+ 210 710 790 340 90
+ 178 623 712 267 89
+ --- --- --- --- --
+ 320 870 780 730 100
+ 267 801 712 712 89
+ --- --- --- --- ---
+ 530 690 680 180 ...
+ 445 623 623 178
+ --- --- --- ---
+ 850 670 570 200
+ 801 623 534 178
+ --- --- --- ---
+ 490 470 360 220
+ 445 445 356 178
+ -- --- --- ---
+ 450 250 400 420
+ 445 178 356 356
+ --- --- --- ---
+ 500 720 440 640
+ 445 712 356 623
+ --- --- --- ---
+ 550 800 840 170
+ 534 712 801 89
+
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cauchy Sequence is Bounded/Real Numbers}
+Tags: Real Analysis, Cauchy Sequences, Cauchy Sequence is Bounded
+
+\begin{theorem}
+Every [[Definition:Real Cauchy Sequence|Cauchy sequence in $\R$]] is [[Definition:Bounded Real Sequence|bounded]].
+\end{theorem}<|endoftext|>
+\section{Reciprocal as Sum of Fibonacci Numbers by Negative Powers of 10}
+Tags: 89, Reciprocals, Fibonacci Numbers
+
+\begin{theorem}
+:$\displaystyle \sum_{k \mathop \ge 0} \dfrac {F_k} {10^{k + 1} } = \dfrac 1 {89}$
+where $F_k$ is the $k$th [[Definition:Fibonacci Number|Fibonacci number]]:
+:$F_0 = 0, F_1 = 1, F_k = F_{k - 1} + F_{k - 2}$
+That is:
+
+1 / 89 = 0.0
+ + 0.01
+ + 0.001
+ + 0.0002
+ + 0.00003
+ + 0.000005
+ + 0.0000008
+ + 0.00000013
+ + 0.000000021
+ + 0.0000000034
+ + 0.00000000055
+ + ..............
+
+\end{theorem}
+
+\begin{proof}
+We have that:
+:$89 = 10^2 - 10 - 1$
+So:
+{{begin-eqn}}
+{{eqn | l = \sum_{k \mathop \ge 0} \dfrac {F_k} {10^{k + 1} }
+ | r = \dfrac 1 {10} \sum_{k \mathop \ge 0} F_k \paren {\dfrac 1 {10} }^k
+}}
+{{eqn | r = \dfrac 1 {10} \cdot \dfrac {\dfrac 1 {10} } {1 - \dfrac 1 {10} - \paren {\dfrac 1 {10} }^2 }
+ | c = [[Generating Function for Fibonacci Numbers]]
+}}
+{{eqn | r = \dfrac 1 {10^2 - 10 - 1}
+ | c =
+}}
+{{eqn | r = \dfrac 1 {89}
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Fermat Pseudoprime/Base 3/Examples/91}
+Tags: Fermat Pseudoprimes, 91
+
+\begin{theorem}
+The smallest [[Definition:Fermat Pseudoprime to Base 3|Fermat pseudoprime to base $3$]] is $91$:
+:$3^{91} \equiv 3 \pmod {91}$
+despite the fact that $91$ is not [[Definition:Prime Number|prime]]:
+:$91 = 7 \times 13$
+\end{theorem}
+
+\begin{proof}
+We have that:
+{{begin-eqn}}
+{{eqn | l = 3^{91}
+ | r = 26 \, 183 \, 890 \, 704 \, 263 \, 137 \, 277 \, 674 \, 192 \, 438 \, 430 \, 182 \, 020 \, 124 \, 347
+ | c =
+}}
+{{eqn | r = 26 \, 183 \, 890 \, 704 \, 263 \, 137 \, 277 \, 674 \, 192 \, 438 \, 430 \, 182 \, 020 \, 124 \, 344 + 3
+ | c =
+}}
+{{eqn | r = 91 \times 287 \, 735 \, 062 \, 684 \, 210 \, 299 \, 754 \, 661 \, 455 \, 367 \, 364 \, 637 \, 583 \, 784 + 3
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 3^{91}
+ | o = \equiv
+ | r = 3 \pmod {91}
+ | c =
+}}
+{{end-eqn}}
+Here we define [[Definition:Fermat Pseudoprime to Base 3|Fermat pseudoprime to base $3$]] to be a [[Definition:Composite Number|composite number]] $n$ such that (a slightly stronger result):
+:$3^{n - 1} \equiv 1 \pmod n$
+Otherwise we have:
+:$3^6 = 729 \equiv 3 \pmod 6$
+Following our definition, we see that a [[Definition:Fermat Pseudoprime to Base 3|Fermat pseudoprime to base $3$]] cannot be [[Definition:Divisor of Integer|divisible]] by $3$:
+{{begin-eqn}}
+{{eqn | l = 3^{3 k - 1}
+ | o = \equiv
+ | r = 0
+ | rr = \pmod 3
+}}
+{{eqn | o = \not \equiv
+ | r = 1
+ | rr = \pmod {3 k}
+}}
+{{end-eqn}}
+neither can it be [[Definition:Divisor of Integer|divisible]] by $4$:
+{{begin-eqn}}
+{{eqn | l = 3^{4 k}
+ | r = \paren {3^{2 k} }^2
+}}
+{{eqn | o = \equiv
+ | r = 1
+ | rr = \pmod 4
+ | c = [[Square Modulo 4]]
+}}
+{{eqn | o = \not \equiv
+ | r = 3
+ | rr = \pmod {4 k}
+}}
+{{end-eqn}}
+nor $10$:
+{{begin-eqn}}
+{{eqn | l = 3^{10 k}
+ | r = \paren {3^{5 k} }^2
+}}
+{{eqn | o = \equiv
+ | r = \pm 1
+ | rr = \pmod 5
+ | c = [[Square Modulo 5]]
+}}
+{{eqn | o = \not \equiv
+ | r = 3
+ | rr = \pmod {10 k}
+}}
+{{end-eqn}}
+Also such a number cannot be equal to twice a [[Definition:Prime Number|prime]] greater than $3$:
+{{begin-eqn}}
+{{eqn | l = 3^{2 p}
+ | o = \equiv
+ | r = 3^2
+ | rr = \pmod p
+ | c = [[Fermat's Little Theorem]]
+}}
+{{eqn | o = \not \equiv
+ | r = 3
+ | rr = \pmod p
+ | c = Since $p \nmid 6 = 2 \times 3$
+}}
+{{eqn | ll = \leadsto
+ | l = 3^{2 p}
+ | o = \not \equiv
+ | r = 3
+ | rr = \pmod {2 p}
+}}
+{{end-eqn}}
+nor five times a [[Definition:Prime Number|prime]] greater than $5$:
+{{begin-eqn}}
+{{eqn | l = 3^{5 p}
+ | o = \equiv
+ | r = 3^5
+ | rr = \pmod p
+ | c = [[Fermat's Little Theorem]]
+}}
+{{eqn | o = \not \equiv
+ | r = 3
+ | rr = \pmod p
+ | c = Since $p \nmid 240 = 2^4 \times 3 \times 5$
+}}
+{{eqn | ll = \leadsto
+ | l = 3^{5 p}
+ | o = \not \equiv
+ | r = 3
+ | rr = \pmod {5 p}
+}}
+{{end-eqn}}
+Therefore only the following numbers less than $91$ remain:
+:$25, 49, 77$
+and we check:
+{{begin-eqn}}
+{{eqn | l = 3^{25}
+ | r = 847 \, 288 \, 609 \, 443
+ | c =
+}}
+{{eqn | r = 33 \, 891 \, 544 \, 377 \times 25 + 18
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 3^{25}
+ | o = \equiv
+ | r = 18 \pmod {25}
+ | c =
+}}
+{{eqn | l = 3^{49}
+ | r = 239 \, 299 \, 329 \, 230 \, 617 \, 529 \, 590 \, 083
+ | c =
+}}
+{{eqn | r = 4 \, 883 \, 659 \, 780 \, 216 \, 684 \, 277 \, 348 \times 49 + 31
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 3^{49}
+ | o = \equiv
+ | r = 31 \pmod {49}
+ | c =
+}}
+{{eqn | l = 3^{77}
+ | r = 5 \, 474 \, 401 \, 089 \, 420 \, 219 \, 382 \, 077 \, 155 \, 933 \, 569 \, 751 \, 763
+ | c =
+}}
+{{eqn | r = 71 \, 096 \, 118 \, 044 \, 418 \, 433 \, 533 \, 469 \, 557 \, 578 \, 827 \, 944 \times 77 + 75
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 3^{77}
+ | o = \equiv
+ | r = 75 \pmod {77}
+ | c =
+}}
+{{end-eqn}}
+hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{91 is Pseudoprime to 35 Bases less than 91}
+Tags: Fermat Pseudoprimes, 91
+
+\begin{theorem}
+$91$ is a [[Definition:Fermat Pseudoprime|Fermat pseudoprime]] in $35$ bases less than itself:
+:$3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, 90$
+\end{theorem}
+
+\begin{proof}
+By definition of a [[Definition:Fermat Pseudoprime|Fermat pseudoprime]], we need to check for $a < 91$:
+:$a^{90} \equiv 1 \pmod {91}$
+is satisfied or not.
+Note that:
+{{begin-eqn}}
+{{eqn | l = \paren {91 - a}^{90}
+ | r = {-a}^{90}
+ | c = [[Congruence of Powers]]
+}}
+{{eqn | r = a^{90}
+}}
+{{end-eqn}}
+So we essentially need to check $2 \le a \le 45$ only.
+Since $91 = 7 \times 13$, we can also exclude all multiples of $7$ and $13$.
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Even Integers not Sum of 2 Twin Primes}
+Tags: Twin Primes, 94
+
+\begin{theorem}
+The following [[Definition:Even Integer|even integers]] cannot be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Prime Number|prime numbers]] which are each one of a [[Definition:Twin Primes|pair of twin primes]]:
+:$2, 4, 94, 96, 98, 400, 402, 404, \ldots$
+{{OEIS|A007534}}
+{{expand|add a page to the effect that it is conjectured that this list is complete.}}
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Can be demonstrated by brute force. A pseudocode program and a printout may be adequate, but first we need to develop a standard language-agnostic {{ProofWiki}} pseudocode.}}
+\end{proof}<|endoftext|>
+\section{Reciprocal of 97}
+Tags: 97, Examples of Reciprocals
+
+\begin{theorem}
+The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $97$ has the maximum [[Definition:Period of Recurrence|period]], that is: $96$:
+:$\dfrac 1 {97} = 0 \cdotp \dot 01030 \, 92783 \, 50515 \, 46391 \, 75257 \, 73195 \, 87628 \, 86597 \, 93814 \, 43298 \, 96907 \, 21649 \, 48453 \, 60824 \, 74226 \, 80412 \, 37113 \, 40206 \, 18556 \, \dot 7$
+{{OEIS|A007450}}
+\end{theorem}
+
+\begin{proof}
+Performing the calculation using [[Definition:Long Division|long division]]:
+
+ 0.01030927835051546391752577319587628865979381443298969072164948453608247422680412371134020618556701...
+ ------------------------------------------------------------------------------------------------------
+97)1.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
+ 97 485 873 679 582 679 291 679 776 388 388 291 485
+ -- --- --- --- --- --- --- --- --- --- --- --- ---
+ 300 500 170 310 280 910 290 210 440 720 120 390 650
+ 291 485 97 291 194 873 194 194 388 679 97 388 582
+ --- --- --- --- --- --- --- --- --- --- --- --- ---
+ 900 150 730 190 860 370 960 160 520 410 230 200 680
+ 873 97 679 97 776 291 983 97 485 388 194 194 679
+ --- --- --- --- --- --- --- --- --- --- --- --- ---
+ 270 530 510 930 840 790 870 630 350 220 360 600 100
+ 194 485 485 873 776 776 776 582 291 194 291 582 97
+ --- --- --- --- --- --- --- --- --- --- --- --- ---
+ 760 450 250 570 640 140 940 480 590 260 690 180 ...
+ 679 388 194 485 582 97 873 388 582 194 679 97
+ --- --- --- --- --- --- --- --- --- --- --- ---
+ 810 620 560 850 580 430 670 920 800 660 110 830
+ 776 582 485 776 485 388 582 873 776 582 97 776
+ --- --- --- --- --- --- --- --- --- --- --- ---
+ 340 380 750 740 950 420 880 470 240 780 130 540
+ 291 291 679 679 873 388 873 388 194 776 97 485
+ --- --- --- --- --- --- --- --- --- --- --- ---
+ 490 890 710 610 770 320 700 820 460 400 330 550
+ 485 873 679 582 679 291 679 776 388 388 291 485
+
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Central Field is Field of Functional}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $\mathbf y$ be an [[Definition:Dimension|$N$-dimensional]] [[Definition:Vector|vector]].
+Let $J$ be a [[Definition:Real Functional|functional]], such that:
+:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
+Let the following be a [[Definition:Central Field|central field]]:
+:$\map {\mathbf y'} x = \map {\boldsymbol \psi} {x, \mathbf y}$
+Then this [[Definition:Central Field|central field]] is a [[Definition:Field of Directions of Functional|field of functional $J$]].
+\end{theorem}
+
+\begin{proof}
+Suppose:
+:$\ds \map g {x, \mathbf y} = \int_c^{\paren {x, \mathbf y} } \map F {x, \hat {\mathbf y}, \hat {\mathbf y}'} \rd x$
+where $\hat {\mathbf y}$ is an [[Definition:Extremum of Functional|extremal]] of $J$ connecting [[Definition:Point|points]] $c$ and $\tuple {x, \mathbf y}$.
+{{explain|Extremum or extremal? Can this be made consistent?}}
+Define a [[Definition:Field of Directions of Functional|field of directions]] in $R$ by the following:
+:$\map {\mathbf p} {x, \mathbf y, \mathbf y'} = \map {g_{\mathbf y} } {x, \mathbf y}$
+where $\mathbf p$ stands for [[Definition:Canonical Variable|momentum]].
+Note, that it does not depend on the path defined by $\hat {\mathbf y}$, only on its endpoint at $\tuple {x, \mathbf y}$.
+By definition, $ \map g {x, \mathbf y}$ is a [[Definition:Geodetic Distance|geodetic distance]] $S$.
+Hence, $g_\mathbf y$ does not explicitly depend on $\mathbf y'$.
+Then:
+:$\map {g_{\mathbf y} } {x, \mathbf y} = \map {\mathbf p} {x, \mathbf y, \mathbf z}$
+where $\mathbf z = \map {\mathbf z} {x, \mathbf y}$ denotes [[Definition:Slope|slope]] of the [[Definition:Curve|curve]] joining $c$ and $\tuple {x, \mathbf y}$ at the [[Definition:Point|point]] $\tuple {x, \mathbf y}$.
+Furthermore, since $g$ is a [[Definition:Geodetic Distance|geodetic distance]], it satisfies [[Definition:Hamilton-Jacobi Equation|Hamilton-Jacobi equation]]:
+:$\ds \frac {\partial \map g {x, \mathbf y} } {\partial x} + \map H {x, \mathbf y, \map {g_{\mathbf y} } {x, \mathbf y} } = 0$
+which together with the previous relation results in a system of equations:
+:$\begin {cases}
+\dfrac {\partial \map g {x, \mathbf y} } {\partial x} + \map H {x, \mathbf y, \map {\mathbf p} {x, \mathbf y, \mathbf z} } = 0 \\
+\map {\mathbf z} {x, \mathbf y} = \map{\mathbf y'} x
+\end {cases}$
+[[Definition:Derivative|Differentiation]] {{WRT|Differentiation}} $\mathbf y$ yields:
+:$\begin {cases}
+\dfrac {\partial \mathbf p} {\partial x} = -\dfrac {\partial H} {\partial \mathbf y} \\
+\map {\mathbf z} {x,\mathbf y} = \map{\mathbf y'} x
+\end {cases}$
+These are [[Euler's Equation for Vanishing Variation in Canonical Variables|Euler's equations in canonical coordinates]] with a constraint on [[Definition:Derivative|derivative]].
+{{explain|the last part is not exactly clear; where is the second Euler equation?}}
+Hence, the previously defined [[Definition:Field of Directions of Functional|field of directions]] coincides with the field of [[Definition:Real Functional|functional]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Continued Fraction Expansion of Fourth Power of Pi}
+Tags: Pi, Fourth Powers, Continued Fractions
+
+\begin{theorem}
+The [[Definition:Continued Fraction Expansion of Real Number|continued fraction expansion]] of the [[Definition:Fourth Power|$4$th power]] of [[Definition:Pi|$\pi$]] begins:
+:$\pi^4 \approx 97 + \cfrac 1 {2 + \cfrac 1 {2 + \cfrac 1 {3 + \cfrac 1 {1 + \cfrac 1 {16 \, 539} } } } }$
+{{OEIS|A058286}}
+\end{theorem}<|endoftext|>
+\section{Decimal Expansion of Fourth Power of Pi}
+Tags: Pi, Fourth Powers
+
+\begin{theorem}
+The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Fourth Power|$4$th power]] of [[Definition:Pi|$\pi$]] begins:
+:$97 \cdotp 40909 \, 10340 \, 0 \ldots$
+{{OEIS|A092425}}
+\end{theorem}<|endoftext|>
+\section{Reciprocal of 98}
+Tags: 98, Examples of Reciprocals
+
+\begin{theorem}
+The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $98$ starts with the [[Definition:Integer Power|powers]] of $2$:
+:$\dfrac 1 {98} = 0 \cdotp 0 \dot 1020 \, 40816 \, 32653 \, 06122 \, 44897 \, 95918 \, 36734 \, 69387 \, 75 \dot 5$
+{{OEIS|A021102}}
+\end{theorem}
+
+\begin{proof}
+Performing the calculation using long division:
+
+ 0.01020408163265306122448979591836734693877551...
+ --------------------------------------------------
+98)1.00000000000000000000000000000000000000000000000
+ 98 196 196 490 294 686
+ -- --- --- --- --- ---
+ 200 640 440 900 460 540
+ 196 588 392 882 392 490
+ --- --- --- --- --- ---
+ 400 520 480 180 680 500
+ 392 490 392 98 588 490
+ --- --- --- --- --- ---
+ 800 300 880 820 920 100
+ 784 294 784 784 882 98
+ --- --- --- --- --- ---
+ 160 600 960 360 380 ...
+ 98 588 882 294 294
+ --- --- --- --- ---
+ 620 120 780 660 860
+ 588 98 686 588 784
+ --- --- --- --- ---
+ 320 220 940 720 760
+ 294 196 882 686 686
+ --- --- --- --- ---
+ 260 240 580 340 740
+ 196 196 490 294 686
+
+We have that:
+:$0 \cdotp 01 + 0 \cdotp 0002 + 0 \cdotp 000004 + 0 \cdotp 00000008 + \cdots$
+is nothing else but:
+:$\dfrac 1 2 \paren {\dfrac 1 {50} + \paren {\dfrac 1 {50} }^2 + \paren {\dfrac 1 {50} }^3 + \paren {\dfrac 1 {50} }^4 + \cdots} = \displaystyle \dfrac 1 2 \sum_{k \mathop \ge 1} \paren {\dfrac 1 {50} }^k$
+Hence:
+{{begin-eqn}}
+{{eqn | l = \dfrac 1 2 \sum_{k \mathop \ge 1} \paren {\dfrac 1 {50} }^k
+ | r = \dfrac 1 2 \dfrac {1 / 50} {1 - 1 / 50}
+ | c = [[Sum of Infinite Geometric Sequence/Corollary 1|Sum of Infinite Geometric Sequence: Corollary 1]]
+}}
+{{eqn | r = \dfrac 1 2 \dfrac 1 {50 - 1}
+ | c =
+}}
+{{eqn | r = \dfrac 1 2 \dfrac 1 {49}
+ | c =
+}}
+{{eqn | r = \dfrac 1 {98}
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Even Integers not Expressible as Sum of 3, 5 or 7 with Prime}
+Tags: 98, Prime Numbers
+
+\begin{theorem}
+The [[Definition:Even Integer|even integers]] that cannot be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Prime Number|prime numbers]] where one of those [[Definition:Prime Number|primes]] is $3$, $5$ or $7$ begins:
+:$98, 122, 124, 126, 128, 148, 150, \ldots$
+{{OEIS|A283555}}
+\end{theorem}
+
+\begin{proof}
+These are the [[Definition:Prime Number|primes]] which coincide with the upper end of a [[Definition:Prime Gap|prime gap]] greater than $6$.
+These can be found at:
+:$89$ to $97$: [[Definition:Prime Gap|prime gap]] of $8$
+:$113$ to $127$: [[Definition:Prime Gap|prime gap]] of $14$
+:$139$ to $149$: [[Definition:Prime Gap|prime gap]] of $10$
+and so on.
+We have that:
+{{begin-eqn}}
+{{eqn | l = 98
+ | r = 19 + 79
+}}
+{{eqn | l = 122
+ | r = 13 + 109
+}}
+{{eqn | l = 124
+ | r = 11 + 113
+}}
+{{eqn | l = 126
+ | r = 13 + 113
+}}
+{{eqn | l = 128
+ | r = 19 + 109
+}}
+{{end-eqn}}
+and so on.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Reciprocal of 99}
+Tags: 99, Examples of Reciprocals
+
+\begin{theorem}
+The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $99$ is:
+:$\dfrac 1 {99} = 0 \cdotp \dot 0 \dot 1$
+{{OEIS|A000035}}
+\end{theorem}
+
+\begin{proof}
+Performing the calculation using [[Definition:Long Division|long division]]:
+
+ 0.0101...
+ ----------
+99)1.0000000
+ 99
+ --
+ 100
+ 99
+ ---
+ ...
+
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Repdigit Number consisting of Instances of 9 is Kaprekar}
+Tags: Kaprekar Numbers, Repdigit Numbers
+
+\begin{theorem}
+A [[Definition:Repdigit Number|repdigit number]] that consists entirely of the [[Definition:Digit|digit]] $9$ is a [[Definition:Kaprekar Number|Kaprekar number]].
+\end{theorem}
+
+\begin{proof}
+We note as examples:
+{{begin-eqn}}
+{{eqn | l = 9^2
+ | r = 81
+}}
+{{eqn | l = 8 + 1
+ | r = 9
+}}
+{{eqn | l = 99^2
+ | r = 9801
+}}
+{{eqn | l = 98 + 01
+ | r = 99
+}}
+{{eqn | l = 999^2
+ | r = 998001
+}}
+{{eqn | l = 998 + 001
+ | r = 999
+}}
+{{end-eqn}}
+Now we consider:
+{{begin-eqn}}
+{{eqn | l = \paren {\sum_{k \mathop = 0}^n 9 \times 10^k}^2
+ | r = \paren {10^{n + 1} - 1}^2
+ | c = [[Sum of Geometric Sequence]]
+}}
+{{eqn | r = 10^{n + 1} \paren {10^{n + 1} - 2} + 1
+}}
+{{end-eqn}}
+So $\paren {10^{n + 1} - 1}^2$ can be split as:
+:$\paren {10^{n + 1} - 2} + 1 = 10^{n + 1} - 1 = \displaystyle \sum_{k \mathop = 0}^n 9 \times 10^k$
+Thus $\displaystyle \sum_{k \mathop = 0}^n 9 \times 10^k$ is a [[Definition:Kaprekar Number|Kaprekar number]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers equal to Sum of Primes not Greater than its Prime Counting Function Value}
+Tags: Prime Numbers, Prime-Counting Function
+
+\begin{theorem}
+Let $\pi \left({n}\right): \Z_{\ge 0} \to \Z_{\ge 0}$ denote the [[Definition:Prime-Counting Function|prime-counting function]]:
+:$\pi \left({n}\right) =$ the count of the [[Definition:Prime Number|primes]] less than $n$
+Consider the equation:
+:$n = \displaystyle \sum_{p \mathop \le \pi \left({n}\right)} p$
+where $p \le \pi \left({n}\right)$ denotes the [[Definition:Prime Number|primes]] not greater than $\pi \left({n}\right)$.
+Then $n$ is one of:
+:$5, 17, 41, 77, 100$
+{{OEIS|A091864}}
+\end{theorem}
+
+\begin{proof}
+We have that:
+{{begin-eqn}}
+{{eqn | l = \pi \left({5}\right)
+ | r = 3
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 2 + 3
+ | r = 5
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \pi \left({17}\right)
+ | r = 7
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 2 + 3 + 5 + 7
+ | r = 17
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \pi \left({41}\right)
+ | r = 13
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 2 + 3 + 5 + 7 + 11 + 13
+ | r = 41
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \pi \left({77}\right)
+ | r = 21
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19
+ | r = 77
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \pi \left({100}\right)
+ | r = 25
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23
+ | r = 100
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that there are no others.}}
+\end{proof}<|endoftext|>
+\section{Smallest Seventh Power which is Sum of 8 other Seventh Powers}
+Tags: Seventh Powers, 102
+
+\begin{theorem}
+The smallest [[Definition:Seventh Power|seventh power]] that can be expressed as the [[Definition:Integer Addition|sum]] of $8$ other [[Definition:Seventh Power|seventh powers]] is $102^7$:
+:$102^7 = 12^7 + 35^7 + 53^7 + 58^7 + 64^7 + 83^7 + 85^7 + 90^7$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Brute force}}
+\end{proof}<|endoftext|>
+\section{Reciprocal of 103}
+Tags: 103, Examples of Reciprocals
+
+\begin{theorem}
+:$\dfrac 1 {103} = 0 \cdotp \dot 00970 \, 87378 \, 64077 \, 66990 \, 29126 \, 21359 \, 223 \dot 3$
+\end{theorem}
+
+\begin{proof}
+Performing the calculation using [[Definition:Long Division|long division]]:
+
+ 0.0097087378640776699029126213592233009...
+ --------------------------------------------
+103)1.0000000000000000000000000000000000000...
+ 927 618 927 103 309
+ --- --- --- --- ---
+ 730 420 300 370 1000
+ 721 412 206 309 927
+ --- --- --- --- ----
+ 900 800 940 610 ....
+ 824 721 927 515
+ --- --- --- ---
+ 760 790 130 950
+ 721 721 103 927
+ --- --- --- ---
+ 390 690 270 230
+ 309 618 206 206
+ --- --- --- ---
+ 810 720 640 240
+ 721 618 618 206
+ --- --- --- ---
+ 890 1020 220 340
+ 824 927 206 309
+ --- ---- --- ---
+ 660 930 140 310
+ 618 927 103 309
+
+{{ProofWanted|Still to be demonstrated that it is the smallest.}}
+\end{proof}<|endoftext|>
+\section{Consecutive Integers with Same Euler Phi Value}
+Tags: Euler Phi Function, Consecutive Integers with Same Euler Phi Value
+
+\begin{theorem}
+Let $\phi: \Z_{>0} \to \Z_{>0}$ be the [[Definition:Euler Phi Function|Euler $\phi$ function]], defined on the [[Definition:Strictly Positive Integer|strictly positive integers]].
+The equation:
+:$\map \phi n = \map \phi {n + 1}$
+is satisfied by [[Definition:Strictly Positive Integer|integers]] in the [[Definition:Integer Sequence|sequence]]:
+:$1, 3, 15, 104, 164, 194, 255, 495, 584, 975, 2204, \ldots$
+{{OEIS|A001274}}
+\end{theorem}<|endoftext|>
+\section{Odd Integers whose Smaller Odd Coprimes are Prime}
+Tags: Coprime Integers, 105
+
+\begin{theorem}
+Let $n \in \Z_{>0}$ be an [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive integer]] such that all smaller [[Definition:Odd Integer|odd integers]] greater than $1$ which are [[Definition:Coprime Integers|coprime]] to it are [[Definition:Prime Number|prime]].
+The complete list of such $n$ is as follows:
+:$1, 3, 5, 7, 9, 15, 21, 45, 105$
+{{OEIS|A327823}}
+\end{theorem}
+
+\begin{proof}
+First it is demonstrated that $105$ itself satisfies this property.
+Let $d \in \Z_{> 1}$ be [[Definition:Odd Integer|odd]] and [[Definition:Coprime Integers|coprime]] to $105$.
+Then $d$ does not have $3$, $5$ or $7$ as a [[Definition:Prime Factor|prime factor]].
+Thus $d$ must have at least one [[Definition:Odd Prime|odd prime]] as a [[Definition:Divisor of Integer|divisor]] which is $11$ or greater.
+The smallest such [[Definition:Composite Number|composite number]] is $11^2$.
+But $11^2 = 121 > 105$.
+Thus $d$ must be [[Definition:Odd Prime|prime]].
+Thus it has been demonstrated that all [[Definition:Odd Integer|odd integers]] greater than $1$ and smaller than $105$ which are [[Definition:Coprime Integers|coprime]] to $105$ are [[Definition:Prime Number|prime]].
+{{qed|lemma}}
+Using an argument similar to the above, we see that for an [[Definition:Integer|integer]] to have this property,
+if it is greater than $p^2$ for some [[Definition:Odd Prime|odd prime]] $p$, then it must be [[Definition:Divisor of Integer|divisible]] by $p$.
+If not, it will be [[Definition:Coprime Integers|coprime]] to $p^2$, a [[Definition:Composite Number|composite number]].
+Let $p_n$ denote the $n$th [[Definition:Prime Number|prime]].
+Suppose $N$ has this property.
+By the argument above, if $p_{n + 1}^2 \ge N > p_n^2$, we must have $p_2 p_3 \cdots p_n \divides N$.
+By [[Absolute Value of Integer is not less than Divisors]], we have $p_2 p_3 \cdots p_n \le N$.
+[[Bertrand-Chebyshev Theorem]] asserts that there is a [[Definition:Prime Number|prime]] between $p_n$ and $2 p_n$.
+Thus we have $2 p_n > p_{n + 1}$.
+Hence for $n \ge 5$:
+{{begin-eqn}}
+{{eqn | l = N
+ | o = \ge
+ | r = p_2 p_3 \cdots p_n
+}}
+{{eqn | r = 3 \times 5 p_4 \cdots p_n
+}}
+{{eqn | o = >
+ | r = 8 p_{n - 1} p_n
+}}
+{{eqn | o = >
+ | r = 4 p_n^2
+ | c = [[Bertrand-Chebyshev Theorem]]
+}}
+{{eqn | o = >
+ | r = p_{n + 1}^2
+ | c = [[Bertrand-Chebyshev Theorem]]
+}}
+{{eqn | o = \ge
+ | r = N
+ | c = From assumption
+}}
+{{end-eqn}}
+This is a [[Definition:Contradiction|contradiction]].
+Hence we must have $N \le p_5^2 = 121$.
+From the argument above we also have:
+:$3 \divides N$ for $9 < N \le 25$
+:$3, 5 \divides N$ for $25 < N \le 49$
+:$3, 5, 7 \divides N$ for $49 < N \le 121$
+So we end up with the list $N = 1, 3, 5, 7, 9, 15, 21, 45, 105$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sequences of 3 Consecutive Integers with Rising Phi}
+Tags: Euler Phi Function
+
+\begin{theorem}
+The following [[Definition:Ordered Triple|ordered triples]] of consecutive [[Definition:Integer|integers]] have [[Definition:Euler Phi Function|$\phi$]] values which are [[Definition:Strictly Increasing Mapping|strictly increasing]]:
+:$105, 106, 107$
+:$165, 166, 167$
+:$315, 316, 317$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \map \phi {105}
+ | r = 48
+ | c = {{EulerPhiLink|105}}
+}}
+{{eqn | l = \map \phi {106}
+ | r = 52
+ | c = {{EulerPhiLink|106}}
+}}
+{{eqn | l = \map \phi {107}
+ | r = 106
+ | c = [[Euler Phi Function of Prime]]: $107$ is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \phi {165}
+ | r = 80
+ | c = {{EulerPhiLink|165}}
+}}
+{{eqn | l = \map \phi {166}
+ | r = 82
+ | c = {{EulerPhiLink|166}}
+}}
+{{eqn | l = \map \phi {167}
+ | r = 166
+ | c = [[Euler Phi Function of Prime]]: $167$ is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \phi {315}
+ | r = 144
+ | c = {{EulerPhiLink|315}}
+}}
+{{eqn | l = \map \phi {316}
+ | r = 156
+ | c = {{EulerPhiLink|316}}
+}}
+{{eqn | l = \map \phi {317}
+ | r = 316
+ | c = [[Euler Phi Function of Prime]]: $317$ is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integers whose Sigma equals Half Phi times Tau}
+Tags: Sigma Function, Euler Phi Function, Tau Function
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integers]] $n$ have the property where:
+:$\sigma \left({n}\right) = \dfrac {\phi \left({n}\right) \times \tau \left({n}\right)} 2$
+where:
+:$\sigma \left({n}\right)$ denotes the [[Definition:Sigma Function|$\sigma$ function]]: the [[Definition:Integer Addition|sum]] of the [[Definition:Divisor of Integer|divisors]] of $n$
+:$\phi \left({n}\right)$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ function]]: the count of [[Definition:Positive Integer|positive integers]] smaller than of $n$ which are [[Definition:Coprime Integers|coprime]] to $n$
+:$\tau \left({n}\right)$ denotes the [[Definition:Tau Function|$\tau$ function]]: the count of the [[Definition:Divisor of Integer|divisors]] of $n$:
+These [[Definition:Positive Integer|positive integers]] are:
+:$35, 105, \ldots$
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = \phi \left({35}\right)
+ | r = 24
+ | c = [[Euler Phi Function of 35]]
+}}
+{{eqn | l = \tau \left({35}\right)
+ | r = 4
+ | c = [[Tau Function of 35]]
+}}
+{{eqn | ll= \leadsto
+ | l = \phi \left({35}\right) \times \tau \left({35}\right)
+ | r = \dfrac {24 \times 4} 2
+ | c =
+}}
+{{eqn | r = 48
+ | c =
+}}
+{{eqn | r = \sigma \left({35}\right)
+ | c = [[Sigma Function of 35]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \phi \left({105}\right)
+ | r = 48
+ | c = [[Euler Phi Function of 105]]
+}}
+{{eqn | l = \tau \left({105}\right)
+ | r = 8
+ | c = [[Tau Function of 105]]
+}}
+{{eqn | ll= \leadsto
+ | l = \phi \left({105}\right) \times \tau \left({105}\right)
+ | r = \dfrac {48 \times 8} 2
+ | c =
+}}
+{{eqn | r = 192
+ | c =
+}}
+{{eqn | r = \sigma \left({105}\right)
+ | c = [[Sigma Function of 105]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Reciprocals of Odd Numbers adding to 1}
+Tags: Reciprocals, Unit Fractions
+
+\begin{theorem}
+$105$ is the smallest [[Definition:Positive Integer|positive integer]] $n$ such that $1$ can be expressed as the [[Definition:Rational Addition|sum]] of [[Definition:Reciprocal|reciprocals]] of [[Definition:Distinct|distinct]] [[Definition:Odd Integer|odd integers]] such that none are less than $\dfrac 1 n$:
+:$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {33} + \dfrac 1 {35} + \dfrac 1 {45} + \dfrac 1 {55} + \dfrac 1 {77} + \dfrac 1 {105}$
+{{OEIS|A238795}}
+There are $5$ ways of expressing $1$ as the [[Definition:Rational Addition|sum]] of [[Definition:Reciprocal|reciprocals]] of only $9$ [[Definition:Distinct|distinct]] [[Definition:Odd Integer|odd integers]], but then the least term is less than $\dfrac 1 {105}$.
+The $9$-term solutions are as follows:
+:$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {35} + \dfrac 1 {45} + \dfrac 1 {231}$
+{{OEIS|A201644}}
+:$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {21} + \dfrac 1 {231} + \dfrac 1 {315}$
+{{OEIS|A201648}}
+:$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {33} + \dfrac 1 {45} + \dfrac 1 {385}$
+{{OEIS|A201649}}
+:$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {21} + \dfrac 1 {165} + \dfrac 1 {693}$
+{{OEIS|A201647}}
+:$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {21} + \dfrac 1 {135} + \dfrac 1 {10 \, 395}$
+{{OEIS|A201646}}
+\end{theorem}<|endoftext|>
+\section{Reciprocals of Odd Numbers adding to 1/Historical Note}
+Tags: Historical Notes
+
+\begin{theorem}
+{{AuthorRef|David Wells}} reports in his {{BookLink|Curious and Interesting Numbers|David Wells}} of $1986$ that this result was reported by {{AuthorRef|Friend Hans Kierstead, Jr.}} and {{AuthorRef|Harry Lewis Nelson}} in Volume $10$ of the {{JournalLink|name = Journal of Recreational Mathematics}}, but no information has so far been found to back this up.
+It is reported in the various relevant pages in the [http://oeis.org/ On-Line Encyclopedia of Integer Sequences] that this set was apparently discovered by '''S. Yamashita''' in $1976$, but once more corroborative evidence is hard to come by.
+\end{theorem}<|endoftext|>
+\section{Smallest Polyomino with Hole}
+Tags: Polyominoes, Heptominoes
+
+\begin{theorem}
+The smallest [[Definition:Polyomino|polyomino]] with a hole is the [[Definition:Heptomino|heptomino]] in the form of a $3 \times 3$
+[[Definition:Square (Geometry)|square]] with the center $1 \times 1$ [[Definition:Square (Geometry)|square]] and a corner $1 \times 1$ [[Definition:Square (Geometry)|square]] missing:
+:[[File:HeptominoWithHole.png|200px]]
+\end{theorem}
+
+\begin{proof}
+From [[35 Hexominoes]] and by inspection, none of the $35$ [[Definition:Hexomino|hexominoes]] has a hole.
+From [[Number of Heptominoes]] and by inspection, exactly one of the $108$ [[Definition:Heptomino|heptominoes]] has a hole.
+This is the smallest [[Definition:Polyomino|polyomino]] with a hole.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integers whose Sigma Value is Cube}
+Tags: Sigma Function, Cube Numbers, Numbers whose Sigma is Cubic
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integers]] are those whose [[Definition:Sigma Function|$\sigma$ value]] is a [[Definition:Cube Number|cube]]:
+:$1, 7, 102, 110, 142, 159, 187, 381, 690, 714, 770, 994, 1034, \ldots$
+{{OEIS|A020477}}
+\end{theorem}<|endoftext|>
+\section{Magic Constant of Smallest Prime Magic Square}
+Tags: Prime Magic Squares, 111
+
+\begin{theorem}
+The [[Definition:Magic Constant|magic constant]] of the smallest [[Definition:Prime Magic Square|prime magic square]] is $111$.
+\end{theorem}
+
+\begin{proof}
+The [[Prime Magic Square/Examples/Order 3/Smallest|smallest prime magic square (including $1$)]] is:
+{{:Prime Magic Square/Examples/Order 3/Smallest}}
+As can be seen by inspection, the [[Definition:Integer Addition|sums]] of the [[Definition:Element of Array|elements]] in the [[Definition:Row of Array|rows]], [[Definition:Column of Array|columns]] and [[Definition:Diagonal of Array|diagonals]] are $111$:
+{{begin-eqn}}
+{{eqn | l = 67 + 1 + 43
+ | r = 111
+}}
+{{eqn | l = 13 + 37 + 61
+ | r = 111
+}}
+{{eqn | l = 31 + 73 + 7
+ | r = 111
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 67 + 13 + 31
+ | r = 111
+}}
+{{eqn | l = 1 + 37 + 73
+ | r = 111
+}}
+{{eqn | l = 43 + 61 + 7
+ | r = 111
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 67 + 37 + 7
+ | r = 111
+}}
+{{eqn | l = 43 + 37 + 31
+ | r = 111
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sequence of Palindromic Lucky Numbers}
+Tags: Palindromic Numbers, Lucky Numbers
+
+\begin{theorem}
+The The [[Definition:Integer Sequence|sequence]] of [[Definition:Lucky Number|lucky numbers]] which are also [[Definition:Palindromic Number|palindromic]] begins:
+:$1, 3, 7, 9, 33, 99, 111, 141, 151, 171, \ldots$
+{{OEIS|A031161}}
+\end{theorem}
+
+\begin{proof}
+{{:Definition:Lucky Number/Sequence}}
+Of these, the [[Definition:Palindromic Number|palindromic]] ones can be picked out by inspection.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sequence of Square Lucky Numbers}
+Tags: Lucky Numbers, Square Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Lucky Number|lucky numbers]] which are also [[Definition:Square Number|square]] begins:
+:$1, 9, 25, 49, 169, 289, 361, 529, \ldots$
+{{OEIS|A031162}}
+\end{theorem}
+
+\begin{proof}
+{{:Definition:Lucky Number/Sequence}}
+Of these, the [[Definition:Square Number|square]] ones can be picked out by inspection.
+{{qed}}
+[[Category:Lucky Numbers]]
+[[Category:Square Numbers]]
+926u5u5kr0ql3e1xd83oaoyatuxjcbw
+\end{proof}<|endoftext|>
+\section{Conditions for Extremal Embedding in Field of Functional}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $J$ be a [[Definition:Real Functional|functional]] such that:
+:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
+Let $\gamma$ be an [[Definition:Extremum of Functional|extremal]] of $J$, defined by $\mathbf y = \map {\mathbf y} x$ for $x \in \closedint a b$.
+Suppose:
+:$\forall x \in \closedint a b: \det \paren {F_{\mathbf y' \mathbf y'} } \ne 0$
+Suppose no [[Definition:Conjugate Point|points conjugate]] to $\paren {a, \map {\mathbf y} a}$ lie on $\gamma$.
+Then $\gamma$ can be [[Definition:Extremal Embedding in Field of Functional|embedded in a field]].
+\end{theorem}
+
+\begin{proof}
+Let $c \in \R$ be [[Definition:Conjugate Point|conjugate]] to $a$, such that $c < a$.
+By assumption:
+:$c \notin \closedint a b$
+Hence, there exists a [[Definition:Set|set]] $\closedint c b$ such that:
+:$\closedint c b = \closedint c a \cup \closedint a b$
+where $\size {c - a} > 0$.
+By [[there exists a real point between two real points]]:
+:$\exists \epsilon: \size {c - a} > \epsilon > 0$
+Therefore, there exists $\epsilon > 0$ such that:
+:$\gamma$ can be extended onto the whole [[Definition:Closed Interval|interval]] $\closedint {a - \epsilon} b$, where extension means [[Definition:Definition|definition]] of some [[Definition:Mapping|mapping]] in $\hointr {a - \epsilon} a$
+{{explain|the author provides this as a condition, but does a definability affect existence ? }}
+:the [[Definition:Closed Interval|interval]] $\closedint {a - \epsilon} b$ contains no [[Definition:Conjugate Point|points conjugate]] to $a$.
+Consider a [[Definition:Family|family]] of [[Definition:Extremum of Functional|extremals]] leaving the [[Definition:Point|point]] $\tuple {a - \epsilon, \map {\mathbf y} {a - \epsilon} }$.
+There are no [[Definition:Conjugate Point|points conjugate]] to $a - \epsilon$ in $\closedint {a - \epsilon} b$.
+Hence, for $x \in \closedint a b$ no two [[Definition:Extremum of Functional|extremals]] in this [[Definition:Family|family]] which are sufficiently close to the original [[Definition:Extremum of Functional|extremal]] $\gamma$ can [[Definition:Set Intersection|intersect]].
+{{explain|Follows from the definition, but a picture or a better explanation would help}}
+Since all the functions are [[Definition:Extremum of Functional|extremals]], they satisfy same [[Definition:Differential Equation|differential equations]].
+Lack of [[Definition:Set Intersection|intersection]] implies different [[Definition:Boundary Condition|boundary conditions]].
+Denote these conditions collectively as $\map {\boldsymbol \psi} {x, \mathbf y} = \map {\mathbf y'} x$.
+Thus, in some [[Definition:Region|region]] $R$ [[Definition:Contain|containing]] $\gamma$ [[Definition:Extremum of Functional|extremals]] sufficiently close to $\gamma$ define a [[Definition:Central Field|central field]] in which $\gamma$ is [[Definition:Extremal Embedding in Field of Functional|embedded]].
+By [[Central Field is Field of Functional]], $\gamma$ can be [[Definition:Extremal Embedding in Field of Functional|embedded]] in the [[Definition:Field of Directions of Functional|field of functional]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sequence of Smallest Consecutive Composite Numbers longer than 100}
+Tags: Prime Gaps
+
+\begin{theorem}
+The $1$st [[Definition:Prime Gap|prime gap]] greater than $100$ is between $370 \, 261$ and $370 \, 373$, of length $112$.
+That is, the [[Definition:Integer Sequence|sequence]] of the smallest consecutive [[Definition:Composite Number|composite]] [[Definition:Positive Integer|positive integers]] longer than $100$ is that of $111$ such, from $370 \, 262$ to $370 \, 372$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Brute force?}}
+\end{proof}<|endoftext|>
+\section{Difference between Two Squares equal to Repunit}
+Tags: Square Numbers, Repunits, Difference between Two Squares equal to Repunit
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Difference of Two Squares|differences of two squares]] that each make a [[Definition:Repunit|repunit]] begins:
+{{begin-eqn}}
+{{eqn | l = 1^2 - 0^2
+ | r = 1
+ | c =
+}}
+{{eqn | l = 6^2 - 5^2
+ | r = 11
+ | c =
+}}
+{{eqn | l = 20^2 - 17^2
+ | r = 111
+ | c =
+}}
+{{eqn | l = 56^2 - 45^2
+ | r = 1111
+ | c =
+}}
+{{eqn | l = 56^2 - 55^2
+ | r = 111
+ | c =
+}}
+{{eqn | l = 156^2 - 115^2
+ | r = 11 \, 111
+ | c =
+}}
+{{eqn | l = 340^2 - 67^2
+ | r = 111 \, 111
+ | c =
+}}
+{{eqn | l = 344^2 - 65^2
+ | r = 111 \, 111
+ | c =
+}}
+{{eqn | l = 356^2 - 125^2
+ | r = 111 \, 111
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+Let $x^2 - y^2 = R_n$ for some $n$, where $R_n$ denotes the [[Definition:Digit|$n$-digit]] [[Definition:Repunit|repunit]].
+From [[Integer as Difference between Two Squares]]:
+:$R_n$ has at least two [[Definition:Distinct|distinct]] [[Definition:Divisor of Integer|divisors]] of the same [[Definition:Parity|parity]].
+Then from [[Difference of Two Squares]]:
+:$x = \dfrac {a + b} 2$
+:$y = \dfrac {a - b} 2$
+where:
+:$R_n = a b$
+for all $a, b$ where:
+:$a b = R_n$
+:$a$ and $b$ are of the same [[Definition:Parity|parity]].
+Here we have that $R_n$ is [[Definition:Odd Integer|odd]].
+So both $a$ and $b$ are always [[Definition:Odd Integer|odd]] and therefore always of the same [[Definition:Parity|parity]].
+It remains to perform the calculations and evaluate the examples.
+\end{proof}<|endoftext|>
+\section{Smallest Equilateral Triangle with Internal Point at Integer Distances from Vertices}
+Tags: Equilateral Triangles, 112, 57, 65, 73
+
+\begin{theorem}
+The smallest [[Definition:Equilateral Triangle|equilateral triangle]] with [[Definition:Side of Polygon|sides]] of [[Definition:Integer|integer]] [[Definition:Length of Line|length]] which contains a [[Definition:Point|point]] which is an [[Definition:Integer|integer]] [[Definition:Distance (Linear Measure)|distance]] from each [[Definition:Vertex of Polygon|vertex]] has a [[Definition:Side of Polygon|side]] [[Definition:Length of Line|length]] $112$:
+:[[File:112EquilateralTriangle.png|500px]]
+There exists a [[Definition:Point|point]] inside it which is $57$, $65$ and $73$ away from the three [[Definition:Vertex of Polygon|vertices]].
+\end{theorem}<|endoftext|>
+\section{Smallest 3-Digit Permutable Prime}
+Tags: Permutable Primes, 113
+
+\begin{theorem}
+The smallest [[Definition:Digit|$3$-digit]] [[Definition:Permutable Prime|permutable prime]] is $113$.
+\end{theorem}
+
+\begin{proof}
+:$113$ is [[Definition:Prime Number|prime]].
+:$131$ is [[Definition:Prime Number|prime]].
+:$311$ is [[Definition:Prime Number|prime]].
+Consider the $3$-digit [[Definition:Prime Number|primes]] smaller than $113$:
+:$101, 103, 107, 109$
+They all contain a [[Definition:Zero Digit|zero]].
+Thus, for each of these, at least one [[Definition:Permutation|permutation]] ends in a [[Definition:Zero Digit|zero]].
+Hence it is [[Definition:Divisor of Integer|divisible]] by $10$ and so is not [[Definition:Prime Number|prime]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Infinite Group has Infinite Number of Subgroups}
+Tags: Infinite Groups
+
+\begin{theorem}
+Let $\struct {G, \circ}$ be an [[Definition:Infinite Group|infinite group]].
+Then $\struct {G, \circ}$ has an [[Definition:Infinite Set|infinite]] number of [[Definition:Distinct|distinct]] [[Definition:Subgroup|subgroups]].
+\end{theorem}
+
+\begin{proof}
+There are two cases to consider: either $\struct {G, \circ}$ has an [[Definition:Infinite Cyclic Group|infinite cyclic]] [[Definition:Subgroup|subgroup]], or it does not.
+=== Case 1 ===
+Suppose that $\struct {G, \circ}$ has an [[Definition:Infinite Cyclic Group|infinite cyclic]] [[Definition:Subgroup|subgroup]] denoted as $H$.
+Let $a \in G$ be the [[Definition:Element|element]] of $G$ such that $\gen a = H$.
+Then by [[Distinct Subgroups of Infinite Cyclic Group]], $\sequence {\gen {a^k} }_{k \mathop \in \N^*}$ is a [[Definition:Sequence|sequence]] of [[Definition:Distinct|distinct]] subgroups of $H$.
+So, by [[Subgroup of Subgroup is Subgroup]], it follows that $\struct {G, \circ}$ has an [[Definition:Infinite Set|infinite number]] of [[Definition:Subgroup|subgroups]].
+=== Case 2 ===
+Suppose that $\struct {G, \circ}$ does ''not'' have an [[Definition:Infinite Cyclic Group|infinite cyclic]] [[Definition:Subgroup|subgroup]].
+Then for every $x \in G$, we have that $\gen x$ is a [[Definition:Finite Group|finite group]].
+Let $a, b \in G$.
+Let $\sim$ be the [[Definition:Equivalence Relation|equivalence relation]] on $G$ defined as:
+:$a \sim b \iff \gen a = \gen b$
+Let $\eqclass a \sim$ denote the [[Definition:Equivalence Class|equivalence class]] of $a$ under $\sim$.
+Let $x \sim a$.
+Then:
+:$\gen x = \gen a$
+Since $x \in \gen x$, it follows that:
+:$x \in \gen a$
+It has been shown that:
+:$\eqclass a \sim \subseteq \gen a$
+By [[Subset of Finite Set is Finite]], every [[Definition:Equivalence Class|equivalence class]] under $\sim$ is [[Definition:Finite Set|finite]].
+By [[Equivalence Class holds Equivalent Elements]]:
+:$\eqclass a \sim \ne \eqclass b \sim \implies a \not \sim b \implies \gen a \ne \gen b$
+So every [[Definition:Equivalence Class|equivalence class]] generates a unique [[Definition:Subgroup|subgroup]] of $\struct {G, \circ}$.
+By [[Union of Equivalence Classes is Whole Set]], the [[Definition:Union of Family|union]] of all [[Definition:Equivalence Class|equivalence classes]] under $\sim$ must equal $G$.
+From [[Finite Union of Finite Sets is Finite]], it follows that this must be a [[Definition:Countable Union|(countably) infinite union]].
+Thus there must exist an [[Definition:Infinite Set|infinite number]] of [[Definition:Equivalence Class|equivalence classes]], and hence an infinite number of [[Definition:Subgroup|subgroups]].
+The result then follows from [[Proof by Cases]].
+{{qed}}
+[[Category:Infinite Groups]]
+897cv70vtox46af0sbs3np9gwkuncrc
+\end{proof}<|endoftext|>
+\section{Group is Finite iff Finite Number of Subgroups}
+Tags: Finite Groups
+
+\begin{theorem}
+Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
+Then $G$ is [[Definition:Finite Set|finite]] {{iff}} $\struct {G, \circ}$ has a [[Definition:Finite Set|finite number]] of [[Definition:Subgroup|subgroups]].
+\end{theorem}
+
+\begin{proof}
+=== Necessary Condition ===
+Suppose that $\struct {G, \circ}$ is a [[Definition:Finite Group|finite group]].
+Let $\struct {H, \circ}$ be a [[Definition:Subgroup|subgroup]] of $\struct {G, \circ}$.
+$H \subseteq G$ by definition.
+Therefore:
+: $H \in \powerset G$
+where $\powerset G$ denotes the [[Definition:Power Set|power set]] of $G$.
+By [[Power Set of Finite Set is Finite]], $\powerset G$ is [[Definition:Finite Set|finite]].
+It is seen that the set of all subgroups form a [[Definition:Subset|subset]] of $\powerset G$.
+The result then follows from [[Subset of Finite Set is Finite]].
+{{qed|lemma}}
+=== Sufficient Condition ===
+Suppose that $\struct {G, \circ}$ is a [[Definition:Group|group]] with only a [[Definition:Finite Set|finite]] number of [[Definition:Subgroup|subgroups]].
+It is noted that an [[Infinite Group has Infinite Number of Subgroups]].
+The result then follows from the [[Rule of Transposition]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{3-Digit Permutable Primes}
+Tags: Permutable Primes
+
+\begin{theorem}
+The [[Definition:Digit|$3$-digit]] [[Definition:Permutable Prime|permutable primes]] are:
+:$311, 199, 337$
+and their [[Definition:Anagram|anagrams]], and no other.
+\end{theorem}
+
+\begin{proof}
+It is confirmed that:
+:$113, 131, 311$ are all [[Definition:Prime Number|prime]]
+:$199, 919, 991$ are all [[Definition:Prime Number|prime]]
+:$337, 373, 733$ are all [[Definition:Prime Number|prime]].
+From [[Digits of Permutable Prime]], all [[Definition:Permutable Prime|permutable primes]] contain [[Definition:Digit|digits]] in the [[Definition:Set|set]]:
+:$\left\{ {1, 3, 7, 9}\right\}$
+The [[Definition:Integer Addition|sum]] of a [[Definition:Digit|$3$-digit]] [[Definition:Repdigit Number|repdigit number]] is [[Definition:Divisor of Integer|divisible]] by $3$.
+By [[Divisibility by 3]] it follows that all [[Definition:Digit|$3$-digit]] [[Definition:Repdigit Number|repdigit numbers]] are [[Definition:Divisor of Integer|divisible]] by $3$ and therefore [[Definition:Composite Number|composite]].
+Hence the possibly [[Definition:Permutable Prime|permutable primes]] are:
+:$113, 117, 119, 133, 137, 139, 177, 179, 199, 337, 339, 377, 379, 399, 779, 799$
+and their [[Definition:Anagram|anagrams]].
+We eliminate $113, 199, 337$ from this list, as it has been established that they are [[Definition:Permutable Prime|permutable primes]].
+Of those remaining, the following are [[Definition:Composite Number|composite]]:
+{{begin-eqn}}
+{{eqn | l = 117
+ | r = 3^2 \times 13
+}}
+{{eqn | l = 119
+ | r = 7 \times 17
+}}
+{{eqn | l = 133
+ | r = 7 \times 19
+}}
+{{eqn | l = 177
+ | r = 3 \times 59
+}}
+{{eqn | l = 339
+ | r = 3 \times 113
+}}
+{{eqn | l = 377
+ | r = 13 \times 29
+}}
+{{eqn | l = 779
+ | r = 19 \times 41
+}}
+{{eqn | l = 799
+ | r = 17 \times 47
+}}
+{{end-eqn}}
+It remains to demonstrate that at least one [[Definition:Anagram|anagram]] of the remaining numbers:
+:$137, 139, 179, 379$
+is [[Definition:Composite Number|composite]].
+We find that:
+{{begin-eqn}}
+{{eqn | l = 371
+ | r = 7 \times 53
+}}
+{{eqn | l = 319
+ | r = 11 \times 29
+}}
+{{eqn | l = 791
+ | r = 7 \times 113
+}}
+{{eqn | l = 793
+ | r = 13 \times 61
+}}
+{{end-eqn}}
+All contenders are eliminated except for the established [[Definition:Permutable Prime|permutable primes]] $113, 119, 337$ and their [[Definition:Anagram|anagrams]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Digits of Permutable Prime}
+Tags: Permutable Primes
+
+\begin{theorem}
+Let $p$ be a [[Definition:Permutable Prime|permutable prime]] with more than $1$ [[Definition:Digit|digit]].
+Then $p$ contains only [[Definition:Digit|digits]] from the [[Definition:Set|set]]:
+:$\left\{ {1, 3, 7, 9}\right\}$
+\end{theorem}
+
+\begin{proof}
+First note that from [[3-Digit Permutable Primes]], both $337$ and $199$ are [[Definition:Permutable Prime|permutable primes]].
+Hence it follows that all the [[Definition:Element|elements]] of $\left\{ {1, 3, 7, 9}\right\}$ appear in at least one [[Definition:Permutable Prime|permutable prime]].
+Let $p$ contain an [[Definition:Even Integer|even]] [[Definition:Digit|digit]] $d$.
+Then at least one [[Definition:Anagram|anagram]] $p'$ of $p$ has $d$ at the end.
+From [[Divisibility by 2]], it follows that $p'$ is [[Definition:Even Integer|even]].
+As $p$ has more than $1$ [[Definition:Digit|digit]] it follows that $p'$ is [[Definition:Composite|composite]].
+Let $p$ contain the [[Definition:Digit|digit]] $5$.
+Then at least one [[Definition:Anagram|anagram]] $p'$ of $p$ has $5$ at the end.
+From [[Divisibility by 5]], it follows that $p'$ is [[Definition:Divisor of Integer|divisible]] by $5$.
+Except for $5$ itself, all [[Definition:Integer|integers]] which are [[Definition:Divisor of Integer|divisible]] by $5$ are [[Definition:Composite|composite]].
+As $p$ has more than $1$ [[Definition:Digit|digit]] it follows that $p'$ is [[Definition:Composite|composite]].
+The result follows by definition of [[Definition:Permutable Prime|permutable prime]].
+{{qed}}
+[[Category:Permutable Primes]]
+fg1nktn31ejo7yn7ruv3oli7m6hih4k
+\end{proof}<|endoftext|>
+\section{Divisibility by 5}
+Tags: Divisibility Tests, 5
+
+\begin{theorem}
+An [[Definition:Integer|integer]] $N$ expressed in [[Definition:Decimal Notation|decimal notation]] is [[Definition:Divisor of Integer|divisible]] by $5$ {{iff}} the [[Definition:Units Digit|units digit]] of $N$ is [[Definition:Divisor of Integer|divisible]] by $5$.
+That is:
+:$N = \sqbrk {a_n \ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is [[Definition:Divisor of Integer|divisible]] by $5$
+{{iff}}:
+:$a_0$ is [[Definition:Divisor of Integer|divisible]] by $5$.
+\end{theorem}
+
+\begin{proof}
+Let $N$ be [[Definition:Divisor of Integer|divisible]] by $5$.
+Then:
+{{begin-eqn}}
+{{eqn | l = N
+ | o = \equiv
+ | r = 0 \pmod 5
+}}
+{{eqn | ll= \leadstoandfrom
+ | l = \sum_{k \mathop = 0}^n a_k 10^k
+ | o = \equiv
+ | r = 0 \pmod 5
+}}
+{{eqn | ll= \leadstoandfrom
+ | l = a_0 + 10 \sum_{k \mathop = 1}^n a_k 10^{k - 1}
+ | o = \equiv
+ | r = 0 \pmod 5
+}}
+{{eqn | ll= \leadstoandfrom
+ | l = a_0
+ | o = \equiv
+ | r = 0 \pmod 5
+ | c = as $10 \equiv 0 \pmod 5$
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Divisibility Tests]]
+[[Category:5]]
+3tvnw0m20osbfmwu5igoai5vzdssl35
+\end{proof}<|endoftext|>
+\section{Prime Repdigit Number is Repunit}
+Tags: Repdigit Numbers, Repunits, Repunit Primes
+
+\begin{theorem}
+Let $b \in \Z_{>0}$ be an [[Definition:Integer|integer]] greater than $1$.
+Let $n \in \Z$ expressed in [[Definition:Number Base|base $b$]] be a [[Definition:Repdigit Number|repdigit number]] with more than $1$ [[Definition:Digit|digit]].
+Let $n$ be [[Definition:Prime Number|prime]].
+Then $n$ is a [[Definition:Repunit|repunit]] (in [[Definition:Number Base|base $b$]]).
+\end{theorem}
+
+\begin{proof}
+Let $n$ be a [[Definition:Repdigit Number|repdigit number]] with $k$ [[Definition:Digit|digits]].
+Then by the [[Basis Representation Theorem]]:
+:$\displaystyle n = \sum_{j \mathop = 0}^k m b^j$
+for some $m$ such that $1 \le m < b$.
+Let $m \ge 2$.
+Then:
+:$\displaystyle n = m \sum_{j \mathop = 0}^k b^j$
+and so has $m$ as a [[Definition:Divisor of Integer|divisor]].
+Hence $n$ is not [[Definition:Prime Number|prime]].
+The result follows by the [[Rule of Transposition]].
+{{qed}}
+[[Category:Repdigit Numbers]]
+[[Category:Repunits]]
+[[Category:Repunit Primes]]
+35tgqldbngld9g08gyfadp7qrakygnz
+\end{proof}<|endoftext|>
+\section{2-Digit Permutable Primes}
+Tags: Permutable Primes
+
+\begin{theorem}
+The [[Definition:Digit|$2$-digit]] [[Definition:Permutable Prime|permutable primes]] are:
+:$11, 13, 17, 37, 79$
+and their [[Definition:Anagram|anagrams]], and no other.
+\end{theorem}
+
+\begin{proof}
+It is confirmed that:
+:$13$ and $31$ are both [[Definition:Prime Number|prime]]
+:$17$ and $71$ are both [[Definition:Prime Number|prime]]
+:$37$ and $73$ are both [[Definition:Prime Number|prime]].
+:$79$ and $97$ are both [[Definition:Prime Number|prime]].
+From [[Digits of Permutable Prime]], all [[Definition:Permutable Prime|permutable primes]] contain [[Definition:Digit|digits]] in the [[Definition:Set|set]]:
+:$\left\{ {1, 3, 7, 9}\right\}$
+By [[Prime Repdigit Number is Repunit]], all [[Definition:Digit|$2$-digit]] [[Definition:Repdigit Number|repdigit numbers]] are [[Definition:Composite Number|composite]] except for $11$ which is [[Definition:Prime Number|prime]].
+From [[Repunit Prime is Permutable Prime]], it follows that $11$ is a [[Definition:Permutable Prime|permutable prime]].
+Apart from $11$ whose status is known, the possibly [[Definition:Permutable Prime|permutable primes]] are:
+:$13, 17, 19, 37, 39, 79$
+and their [[Definition:Anagram|anagrams]].
+We eliminate $13, 17, 37, 79$ from this list, as it has been established that they are [[Definition:Permutable Prime|permutable primes]].
+Of those remaining, the following is [[Definition:Composite Number|composite]]:
+{{begin-eqn}}
+{{eqn | l = 39
+ | r = 3 \times 13
+}}
+{{end-eqn}}
+It remains to demonstrate that the [[Definition:Anagram|anagram]] of $19$, that is, $91$, [[Definition:Composite Number|composite]].
+We find that:
+{{begin-eqn}}
+{{eqn | l = 91
+ | r = 7 \times 13
+}}
+{{end-eqn}}
+All contenders are eliminated except for the established [[Definition:Permutable Prime|permutable primes]] $11, 13, 17, 37, 79$ and their [[Definition:Anagram|anagrams]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Prime Gaps of 14}
+Tags: Prime Gaps
+
+\begin{theorem}
+The following [[Definition:Ordered Pair|pairs]] of consecutive [[Definition:Prime Number|prime numbers]] are those whose [[Definition:Integer Multiplication|difference]] is $14$:
+:$\tuple {113, 127}, \tuple {293, 307}, \tuple {317, 331}, \ldots$
+{{OEIS|A031932}}
+\end{theorem}
+
+\begin{proof}
+Demonstrated by listing the [[Definition:Prime Gap|prime gaps]].
+\end{proof}<|endoftext|>
+\section{Prime Gaps of 18}
+Tags: Prime Gaps
+
+\begin{theorem}
+After the [[Definition:Prime Gap|prime gap]] of $14$ between the [[Definition:Ordered Pair|pairs]] of consecutive [[Definition:Prime Number|prime numbers]]:
+:$\left({113, 127}\right), \left({293, 307}\right), \left({317, 331}\right), \ldots$
+the next [[Definition:Prime Gap|prime gap]] which is greater than $14$ is between the [[Definition:Ordered Pair|pair]] of consecutive [[Definition:Prime Number|prime numbers]]:
+:$\left({523, 541}\right)$
+for a [[Definition:Prime Gap|prime gap]] of $18$.
+\end{theorem}
+
+\begin{proof}
+Demonstrated by listing the [[Definition:Prime Gap|prime gaps]].
+\end{proof}<|endoftext|>
+\section{Number of Different Ways to Colour the Faces of Cube with 3 Colours}
+Tags: Cubes, Combinatorics, 114
+
+\begin{theorem}
+The number of different ways to colour the [[Definition:Face of Polyhedron|faces]] of a [[Definition:Cube (Geometry)|cube]] with $3$ given colours, one colour per [[Definition:Face of Polyhedron|face]], is $114$.
+\end{theorem}<|endoftext|>
+\section{Number of Different Ways to Colour the Faces of Cube with 4 Colours}
+Tags: Cubes, Combinatorics, 2652
+
+\begin{theorem}
+The number of different ways to colour the [[Definition:Face of Polyhedron|faces]] of a [[Definition:Cube (Geometry)|cube]] with $4$ given colours, one colour per [[Definition:Face of Polyhedron|face]], is $2652$.
+\end{theorem}<|endoftext|>
+\section{Number of Different Ways to Colour the Faces of Cube with 5 Colours}
+Tags: Cubes, Combinatorics, 29,660
+
+\begin{theorem}
+The number of different ways to colour the [[Definition:Face of Polyhedron|faces]] of a [[Definition:Cube (Geometry)|cube]] with $5$ given colours, one colour per [[Definition:Face of Polyhedron|face]], is $29 \, 660$.
+\end{theorem}<|endoftext|>
+\section{Smallest Number which is Sum of 4 Triples with Equal Products}
+Tags: 118
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|positive integer]] which is the [[Definition:Integer Addition|sum]] of $4$
+[[Definition:Distinct|distinct]] [[Definition:Ordered Triple|ordered triples]], each of which has the same [[Definition:Integer Multiplication|product]], is $118$:
+{{begin-eqn}}
+{{eqn | l = 118
+ | r = 14 + 50 + 54
+}}
+{{eqn | r = 15 + 40 + 63
+ | c =
+}}
+{{eqn | r = 18 + 30 + 70
+ | c =
+}}
+{{eqn | r = 21 + 25 + 72
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 14 \times 50 \times 54
+ | r = \paren {2 \times 7} \times \paren {2 \times 5^2} \times \paren {2 \times 3^3}
+}}
+{{eqn | r = 2^3 \times 3^3 \times 5^2 \times 7
+ | c =
+}}
+{{eqn | r = 37 \, 800
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 15 \times 40 \times 63
+ | r = \paren {3 \times 5} \times \paren {2^3 \times 5} \times \paren {3^2 \times 7}
+}}
+{{eqn | r = 2^3 \times 3^3 \times 5^2 \times 7
+ | c =
+}}
+{{eqn | r = 37 \, 800
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 18 \times 30 \times 70
+ | r = \paren {2 \times 3^2} \times \paren {2 \times 3 \times 5} \times \paren {2 \times 5 \times 7}
+}}
+{{eqn | r = 2^3 \times 3^3 \times 5^2 \times 7
+ | c =
+}}
+{{eqn | r = 37 \, 800
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 21 \times 25 \times 72
+ | r = \paren {3 \times 7} \times \paren {5^2} \times \paren {2^3 \times 3^2}
+}}
+{{eqn | r = 2^3 \times 3^3 \times 5^2 \times 7
+ | c =
+}}
+{{eqn | r = 37 \, 800
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that $118$ is the smallest.}}
+\end{proof}<|endoftext|>
+\section{Interval containing Prime Number of forms 4n - 1, 4n + 1, 6n - 1, 6n + 1}
+Tags: Prime Numbers
+
+\begin{theorem}
+Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n \ge 118$.
+Then between $n$ and $\dfrac {4 n} 3$ there exists at least one [[Definition:Prime Number|prime number]] of each of the forms:
+:$4 m - 1, 4 m + 1, 6 m - 1, 6 m + 1$
+\end{theorem}
+
+\begin{proof}
+{{questionable|See below.}}
+It is demonstrated that the result is true for $n = 118$:
+:$\dfrac {4 \times 118} 3 = 157 \cdotp \dot 3$
+The [[Definition:Prime Number|primes]] between $118$ and $157$ are:
+{{begin-eqn}}
+{{eqn | l = 127
+ | r = 4 \times 32 - 1
+}}
+{{eqn | l =
+ | r = 6 \times 21 + 1
+}}
+{{eqn | l = 131
+ | r = 4 \times 33 - 1
+}}
+{{eqn | l =
+ | r = 6 \times 22 - 1
+}}
+{{eqn | l = 137
+ | r = 4 \times 34 + 1
+}}
+{{eqn | l =
+ | r = 6 \times 23 - 1
+}}
+{{eqn | l = 139
+ | r = 4 \times 35 - 1
+}}
+{{eqn | l =
+ | r = 6 \times 23 + 1
+}}
+{{eqn | l = 149
+ | r = 4 \times 37 + 1
+}}
+{{eqn | l =
+ | r = 6 \times 25 - 1
+}}
+{{eqn | l = 151
+ | r = 4 \times 38 - 1
+}}
+{{eqn | l =
+ | r = 6 \times 25 + 1
+}}
+{{eqn | l = 157
+ | r = 4 \times 39 + 1
+}}
+{{eqn | l =
+ | r = 6 \times 26 + 1
+}}
+{{end-eqn}}
+So it can be seen that:
+:the [[Definition:Prime Number|primes]] of the form $4 m - 1$ are:
+::$127, 131, 139, 151$
+:the [[Definition:Prime Number|primes]] of the form $4 m + 1$ are:
+::$137, 139, 157$
+:the [[Definition:Prime Number|primes]] of the form $6 m - 1$ are:
+::$131, 137, 149$
+:the [[Definition:Prime Number|primes]] of the form $6 m + 1$ are:
+::$127, 139, 151, 157$
+and so the conditions of the result are fulfilled for $118$.
+But then note we have:
+:$\dfrac {4 \times 117} 3 = 156$
+and so the [[Definition:Prime Number|primes]] between $117$ and $157$ are the same ones as between $118$ and $157$, excluding $157$ itself.
+Thus the conditions also hold for $117$.
+It is puzzling as to why the condition $n \ge 118$ has been applied to the initial statement.
+\end{proof}<|endoftext|>
+\section{Sum of Cubes of 5 Consecutive Integers which is Square}
+Tags: Sums of Cubes
+
+\begin{theorem}
+The following [[Definition:Integer Sequence|sequences]] of $5$ consecutive [[Definition:Positive Integer|positive integers]] have [[Definition:Cube Number|cubes]] that [[Definition:Integer Addition|sum]] to [[Definition:Square Number|squares]]:
+:$1, 2, 3, 4, 5$
+:$96, 96, 98, 99, 100$
+:$118, 119, 120, 121, 122$
+No other such [[Definition:Integer Sequence|sequence]] of $5$ consecutive [[Definition:Positive Integer|positive integers]] has the same property.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1^3 + 2^3 + 3^3 + 4^3 + 5^3
+ | r = 1 + 8 + 27 + 64 + 125
+ | c =
+}}
+{{eqn | r = 225
+ | c =
+}}
+{{eqn | r = 15^2
+ | c = also see [[Sum of Sequence of Cubes]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 96^3 + 97^3 + 98^3 + 99^3 + 100^3
+ | r = 884 \, 736 + 912 \, 673 + 941 \, 192 + 970 \, 299 + 1 \, 000 \, 000
+ | c =
+}}
+{{eqn | r = 4 \, 708 \, 900
+ | c =
+}}
+{{eqn | r = 2170^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 118^3 + 119^3 + 120^3 + 121^3 + 122^3
+ | r = 1 \, 643 \, 032 + 1 \, 685 \, 159 + 1 \, 728 \, 000 + 1 \, 771 \, 561 + 1 \, 815 \, 848
+ | c =
+}}
+{{eqn | r = 8 \, 643 \, 600
+ | c =
+}}
+{{eqn | r = 2940^2
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that these are the only ones.}}
+\end{proof}<|endoftext|>
+\section{Smallest Number to appear 6 Times in Pascal's Triangle}
+Tags: Pascal's Triangle, 120
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|positive integer]] greater than $1$ to appear $6$ times in [[Definition:Pascal's Triangle|Pascal's Triangle]] is $120$.
+\end{theorem}
+
+\begin{proof}
+We have:
+:$\dbinom {120} 1 = \dbinom {16} 2 = \dbinom {10} 3 = \dbinom {10} 7 = \dbinom {16} {14} = \dbinom {120} {119} = 120$
+To verify that this is the smallest, we look at [[Definition:binomial Coefficient|binomial coefficients]] that are no more than $120$.
+Observe that for $n > 120$, $1 \le k \le n - 1$:
+:$\dbinom n k \ge \dbinom n 1 = n > 120$
+For $n > 16$, $2 \le k \le n - 2$:
+:$\dbinom n k \ge \dbinom n 2 = \dfrac {n \paren {n - 1}} {2!} > \dfrac {16 \paren {15}} {2!} = 120$
+For $n > 10$, $3 \le k \le n - 3$:
+:$\dbinom n k \ge \dbinom n 3 = \dfrac {n \paren {n - 1} \paren {n - 2}} {3!} > \dfrac {10 \paren 9 \paren 8} {3!} = 120$
+Any number, except $2$, appears twice in the form $\dbinom n 1$ and $\dbinom n {n - 1}$.
+Therefore we scout for numbers appearing four times, not as those forms, by listing $\dbinom n k$ for:
+:$n \le 10$ and $2 \le k \le n - 2$;
+:$n \le 16$ and $k = 2$ or $n - 2$
+They are:
+:$6$
+:$10, 10$
+:$15, 20, 15$
+:$21, 35, 35, 21$
+:$28, 56, 70, 56, 28$
+:$36, 84, 126, 126, 84, 36$
+:$45, 120, *, *, *, 120, 45$
+:$55, 55$
+:$66, 66$
+:$78, 78$
+:$91, 91$
+:$105, 105$
+:$120, 120$
+and we see that the only number appearing four times is $120$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest n such that 6 n + 1 and 6 n - 1 are both Composite}
+Tags: Prime Numbers, 20
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|positive integer]] $n$ such that $6 n + 1$ and $6 n - 1$ are both [[Definition:Composite Number|composite]] is $20$.
+\end{theorem}
+
+\begin{proof}
+Running through the [[Definition:Positive Integer|positive integers]] in turn:
+{{begin-eqn}}
+{{eqn | l = 6 \times 1 - 1
+ | r = 5
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 1 + 1
+ | r = 7
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 2 - 1
+ | r = 11
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 2 + 1
+ | r = 13
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 3 - 1
+ | r = 17
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 3 + 1
+ | r = 19
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 4 - 1
+ | r = 23
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 4 + 1
+ | r = 25 = 5^2
+ | c = and so is [[Definition:Composite Number|composite]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 5 - 1
+ | r = 29
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 5 + 1
+ | r = 31
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 6 - 1
+ | r = 35 = 5 \times 7
+ | c = and so is [[Definition:Composite Number|composite]]
+}}
+{{eqn | l = 6 \times 5 + 1
+ | r = 37
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 7 - 1
+ | r = 41
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 7 + 1
+ | r = 43
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 8 - 1
+ | r = 47
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 8 + 1
+ | r = 49 = 7^2
+ | c = and so is [[Definition:Composite Number|composite]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 9 - 1
+ | r = 53
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 9 + 1
+ | r = 55 = 5 \times 11
+ | c = and so is [[Definition:Composite Number|composite]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 10 - 1
+ | r = 59
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 10 + 1
+ | r = 61
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 11 - 1
+ | r = 65 = 5 \times 13
+ | c = and so is [[Definition:Composite Number|composite]]
+}}
+{{eqn | l = 6 \times 11 + 1
+ | r = 67
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 12 - 1
+ | r = 71
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 12 + 1
+ | r = 73
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 13 - 1
+ | r = 77 = 7 \times 11
+ | c = and so is [[Definition:Composite Number|composite]]
+}}
+{{eqn | l = 6 \times 13 + 1
+ | r = 79
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 14 - 1
+ | r = 83
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 14 + 1
+ | r = 85 = 5 \times 17
+ | c = and so is [[Definition:Composite Number|composite]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 15 - 1
+ | r = 89
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 15 + 1
+ | r = 91 = 7 \times 13
+ | c = and so is [[Definition:Composite Number|composite]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 16 - 1
+ | r = 95 = 5 \times 19
+ | c = and so is [[Definition:Composite Number|composite]]
+}}
+{{eqn | l = 6 \times 16 + 1
+ | r = 97
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 17 - 1
+ | r = 101
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 17 + 1
+ | r = 103
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 18 - 1
+ | r = 107
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 18 + 1
+ | r = 109
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 19 - 1
+ | r = 113
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6 \times 19 + 1
+ | r = 115 = 5 \times 23
+ | c = and so is [[Definition:Composite Number|composite]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 20 - 1
+ | r = 119 = 7 \times 17
+ | c = and so is [[Definition:Composite Number|composite]]
+}}
+{{eqn | l = 6 \times 20 + 1
+ | r = 121 = 11^2
+ | c = and so is [[Definition:Composite Number|composite]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Number with 16 Divisors}
+Tags: Tau Function, 120
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|positive integer]] with $16$ [[Definition:Divisor of Integer|divisors]] is $120$.
+\end{theorem}
+
+\begin{proof}
+From [[Tau Function of 120]]:
+:$\map \tau {120} = 16$
+The result is a specific instance of [[Smallest Number with 2^n Divisors|Smallest Number with $2^n$ Divisors]]:
+:$120 = 2 \times 3 \times 4 \times 5$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Fermat Set is Diophantine Quadruple}
+Tags: Diophantine m-Tuples
+
+\begin{theorem}
+The [[Definition:Fermat Set|Fermat set]] $F = \left\{{1, 3, 8, 120}\right\}$ is a [[Definition:Diophantine Quadruple|Diophantine quadruple]]:
+:$\forall a, b \in F: a \ne b: a b + 1 = n^2$
+for some $n \in \Z$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1 \times 3 + 1
+ | r = 4
+ | c =
+}}
+{{eqn | r = 2^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 1 \times 8 + 1
+ | r = 9
+ | c =
+}}
+{{eqn | r = 3^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 1 \times 120 + 1
+ | r = 121
+ | c =
+}}
+{{eqn | r = 11^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 3 \times 8 + 1
+ | r = 25
+ | c =
+}}
+{{eqn | r = 5^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 3 \times 120 + 1
+ | r = 361
+ | c =
+}}
+{{eqn | r = 19^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 8 \times 120 + 1
+ | r = 961
+ | c =
+}}
+{{eqn | r = 31^2
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Fermat Set cannot be Extended to Diophantine Quintuple}
+Tags: Diophantine m-Tuples
+
+\begin{theorem}
+The [[Definition:Fermat Set|Fermat set]] $F = \left\{{1, 3, 8, 120}\right\}$ cannot be extended to a [[Definition:Diophantine Quintuple|Diophantine quintuple]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Can be found in the link below.}}
+\end{proof}<|endoftext|>
+\section{Ratio of 360 to Aliquot Sum}
+Tags: 360, Aliquot Sums
+
+\begin{theorem}
+$360$ has the property that its [[Definition:Ratio|ratio]] to its [[Definition:Aliquot Sum|aliquot sum]] is $4 : 9$.
+\end{theorem}
+
+\begin{proof}
+The [[Definition:Aliquot Sum|aliquot sum]] of an [[Definition:Integer|integer]] $n$ is the [[Definition:Integer Addition|integer sum]] of the [[Definition:Aliquot Part|aliquot parts]] of $n$.
+That is, the [[Definition:Aliquot Sum|aliquot sum]] of $360$ is the [[Definition:Sigma Function|$\sigma$ function]] of $360$ minus $360$.
+Thus:
+{{begin-eqn}}
+{{eqn | l = \map \sigma {360} - 360
+ | r = 1170 - 360
+ | c = {{SigmaLink|360}}
+}}
+{{eqn | r = 810
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 4 \times \paren {\map \sigma {360} - 360}
+ | r = 3240
+ | c =
+}}
+{{eqn | r = 9 \times 360
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Are All Triperfect Numbers Even?/Progress/Minimum Size}
+Tags: Are All Triperfect Numbers Even?
+
+\begin{theorem}
+It has been established that an [[Definition:Odd Integer|odd]] [[Definition:Triperfect Number|triperfect number]], if one were to exist, would be greater than $10^{70}$.
+If it does not have $3$ as a [[Definition:Prime Factor|prime factor]], then it is greater than $10^{108}$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Details}}
+\end{proof}<|endoftext|>
+\section{Are All Triperfect Numbers Even?/Progress/Prime Factors}
+Tags: Are All Triperfect Numbers Even?
+
+\begin{theorem}
+An [[Definition:Odd Integer|odd]] [[Definition:Triperfect Number|triperfect number]] has:
+:at least $11$ [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]]
+:at least $32$ [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] if $3$ is not one of them.
+\end{theorem}<|endoftext|>
+\section{Are All Triperfect Numbers Even?/Progress/Form}
+Tags: Are All Triperfect Numbers Even?
+
+\begin{theorem}
+An [[Definition:Odd Integer|odd]] [[Definition:Triperfect Number|triperfect number]] is [[Definition:Square Number|square]].
+\end{theorem}<|endoftext|>
+\section{Multiply Perfect Number of Order 8}
+Tags: Multiply Perfect Numbers
+
+\begin{theorem}
+The number defined as:
+:$n = 2^{65} \times 3^{23} \times 5^9 \times 7^{12} \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29^2 \times 31^2$
+::$\times 37 \times 41 \times 53 \times 61 \times 67^2 \times 71^2 \times 73 \times 83 \times 89 \times 103 \times 127 \times 131$
+::$\times 149 \times 211 \times 307 \times 331 \times 463 \times 521 \times 683 \times 709 \times 1279 \times 2141 \times 2557 \times 5113$
+::$\times 6481 \times 10 \, 429 \times 20 \, 857 \times 110 \, 563 \times 599 \, 479 \times 16 \, 148 \, 168 \, 401$
+is [[Definition:Multiply Perfect Number|multiply perfect]] of [[Definition:Order of Multiply Perfect Number|order]] $8$.
+\end{theorem}
+
+\begin{proof}
+From [[Sigma Function is Multiplicative]], we may take each [[Definition:Prime Factor|prime factor]] separately and form $\map \sigma n$ as the [[Definition:Integer Multiplication|product]] of the [[Definition:Sigma Function|$\sigma$ function]] of each.
+Each of the [[Definition:Prime Factor|prime factors]] which occur with [[Definition:Multiplicity of Prime Factor|multiplicity $1$]] will be treated first.
+A [[Definition:Prime Factor|prime factor]] $p$ contributes towards the combined $\sigma$ a [[Definition:Divisor of Integer|factor]] $p + 1$.
+Hence we have:
+{{begin-eqn}}
+{{eqn | l = \map \sigma {23}
+ | r = 24
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^3 \times 3
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {37}
+ | r = 38
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 19
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {41}
+ | r = 42
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 3 \times 7
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {53}
+ | r = 54
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 3^3
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {61}
+ | r = 62
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 31
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {73}
+ | r = 74
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 37
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {83}
+ | r = 84
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^2 \times 3 \times 7
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {89}
+ | r = 90
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 3^2 \times 5
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {103}
+ | r = 104
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^3 \times 13
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {127}
+ | r = 128
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^7
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {131}
+ | r = 132
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^2 \times 3 \times 11
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {149}
+ | r = 150
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 3 \times 5^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {211}
+ | r = 212
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^2 \times 53
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {307}
+ | r = 308
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^2 \times 7 \times 11
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {331}
+ | r = 332
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^2 \times 83
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {463}
+ | r = 464
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^4 \times 29
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {521}
+ | r = 522
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 3^2 \times 29
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {683}
+ | r = 684
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^2 \times 3^2 \times 19
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {709}
+ | r = 710
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 5 \times 71
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {1279}
+ | r = 1280
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^8 \times 5
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {2141}
+ | r = 2142
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 3^2 \times 7 \times 17
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {2557}
+ | r = 2558
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 1279
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {5113}
+ | r = 5114
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 2557
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {6481}
+ | r = 6482
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 7 \times 463
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {10 \, 429}
+ | r = 10 \, 430
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 5 \times 7 \times 149
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {20 \, 857}
+ | r = 20 \, 858
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 10 \, 429
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {110 \, 563}
+ | r = 110 \, 564
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^2 \times 131 \times 211
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {599 \, 479}
+ | r = 599 \, 480
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^3 \times 5 \times 7 \times 2141
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {16 \, 148 \, 168 \, 401}
+ | r = 16 \, 148 \, 168 \, 402
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 103 \times 709 \times 110 \, 563
+ | c =
+}}
+{{end-eqn}}
+The remaining factors are treated using [[Sigma Function of Power of Prime]]:
+:$\map \sigma {p^k} = \dfrac {p^{k + 1} - 1} {p - 1}$
+Thus:
+{{begin-eqn}}
+{{eqn | l = \map \sigma {2^{65} }
+ | r = 2 \times 2^{65} - 1
+ | c = [[Sigma Function of Power of 2]]
+}}
+{{eqn | r = 73 \, 786 \, 976 \, 294 \, 838 \, 206 \, 463
+ | c =
+}}
+{{eqn | r = 3^2 \times 7 \times 23 \times 67 \times 89 \times 683 \times 20 \, 857 \times 599 \, 479
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {3^{23} }
+ | r = \dfrac {3^{24} - 1} {3 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {282 \, 429 \, 536 \, 481 - 1} 2
+ | c =
+}}
+{{eqn | r = 141 \, 214 \, 768 \, 240
+ | c =
+}}
+{{eqn | r = 2^4 \times 5 \times 7 \times 13 \times 41 \times 73 \times 6481
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {5^9}
+ | r = \dfrac {5^{10} - 1} {5 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {9 \, 765 \, 625 - 1} 4
+ | c =
+}}
+{{eqn | r = 2 \, 441 \, 406
+ | c =
+}}
+{{eqn | r = 2 \times 3 \times 11 \times 71 \times 521
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {7^{12} }
+ | r = \dfrac {7^{13} - 1} {7 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {96 \, 889 \, 010 \, 407 - 1} 6
+ | c =
+}}
+{{eqn | r = 16 \, 148 \, 168 \, 401
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {11^3}
+ | r = \dfrac {11^4 - 1} {11 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {14 \, 641 - 1} {10}
+ | c =
+}}
+{{eqn | r = 1464
+ | c =
+}}
+{{eqn | r = 2^3 \times 3 \times 61
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {13^3}
+ | r = \dfrac {13^4 - 1} {13 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {28 \, 561 - 1} {12}
+ | c =
+}}
+{{eqn | r = 2380
+ | c =
+}}
+{{eqn | r = 2^2 \times 5 \times 7 \times 17
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {17^2}
+ | r = \dfrac {17^3 - 1} {17 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {4913 - 1} {16}
+ | c =
+}}
+{{eqn | r = 307
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {19^2}
+ | r = \dfrac {19^3 - 1} {19 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {6859 - 1} {18}
+ | c =
+}}
+{{eqn | r = 381
+ | c =
+}}
+{{eqn | r = 3 \times 127
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {29^2}
+ | r = \dfrac {29^3 - 1} {29 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {24 \, 389 - 1} {28}
+ | c =
+}}
+{{eqn | r = 871
+ | c =
+}}
+{{eqn | r = 13 \times 67
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {31^2}
+ | r = \dfrac {31^3 - 1} {31 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {29 \, 791 - 1} {30}
+ | c =
+}}
+{{eqn | r = 993
+ | c =
+}}
+{{eqn | r = 3 \times 331
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {67^2}
+ | r = \dfrac {67^3 - 1} {67 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {300 \, 763 - 1} {66}
+ | c =
+}}
+{{eqn | r = 4557
+ | c =
+}}
+{{eqn | r = 3 \times 7^2 \times 31
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {71^2}
+ | r = \dfrac {71^3 - 1} {71 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {357 \, 911 - 1} {66}
+ | c =
+}}
+{{eqn | r = 5113
+ | c =
+}}
+{{end-eqn}}
+Gathering up the [[Definition:Prime Factor|prime factors]], we have:
+:$\sigma \left({n}\right) = 2^{68} \times 3^{23} \times 5^9 \times 7^{12} \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29^2 \times 31^2$
+::$\times 37 \times 41 \times 53 \times 61 \times 67^2 \times 71^2 \times 73 \times 83 \times 89 \times 103 \times 127 \times 131$
+::$\times 149 \times 211 \times 307 \times 331 \times 463 \times 521 \times 683 \times 709 \times 1279 \times 2141 \times 2557 \times 5113$
+::$\times 6481 \times 10 \, 429 \times 20 \, 857 \times 110 \, 563 \times 599 \, 479 \times 16 \, 148 \, 168 \, 401$
+By inspection of the [[Definition:Multiplicity of Prime Factor|multiplicities]] of the [[Definition:Prime Factor|prime factors]] of $n$ and $\map \sigma n$, it can be seen that they match for all except for $2$.
+It follows that $\map \sigma n = 2^3 \times n = 8 n$.
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{121 is Square Number in All Bases greater than 2}
+Tags: Square Numbers, Number Bases
+
+\begin{theorem}
+Let $b \in \Z$ be an [[Definition:Integer|integer]] such that $b \ge 3$.
+Let $n$ be a [[Definition:Positive Integer|positive integer]] which can be expressed in [[Definition:Number Base|base $b$]] as $121_b$.
+Then $n$ is a [[Definition:Square Number|square number]].
+\end{theorem}
+
+\begin{proof}
+Consider $11_b$.
+By the [[Basis Representation Theorem]]:
+:$11_b = b + 1$
+Thus:
+{{begin-eqn}}
+{{eqn | l = {11_b}^2
+ | r = \paren {b + 1}^2
+ | c =
+}}
+{{eqn | r = b^2 + 2 b + 1
+ | c = [[Square of Sum]]
+}}
+{{eqn | r = 121_b
+ | c =
+}}
+{{end-eqn}}
+Thus:
+:$121_b = {11_b}^2$
+and so is a [[Definition:Square Number|square number]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cube of 11 is Palindromic}
+Tags: Cube Numbers, 11, 1331, Palindromic Numbers
+
+\begin{theorem}
+::$11^3 = 1331$
+\end{theorem}
+
+\begin{proof}
+From [[Square of Small-Digit Palindromic Number is Palindromic/Examples/11|Square of 11 is Palindromic]]:
+:$11^2 = 121$
+Thus:
+
+ 121
+x 11
+-----
+ 121
+1210
+-----
+1331
+
{{qed}}
+\end{proof}<|endoftext|>
+\section{Fourth Power of 11 is Palindromic}
+Tags: Fourth Powers, 11, 14,641, Palindromic Numbers
+
+\begin{theorem}
+::$11^4 = 14641$
+\end{theorem}
+
+\begin{proof}
+From [[Square of Small-Digit Palindromic Number is Palindromic/Examples/11|Square of 11 is Palindromic]]:
+:$11^2 = 121$
+Then from [[Square of Small-Digit Palindromic Number is Palindromic/Examples/121|Square of 121 is Palindromic]]:
+:$121^2 = 14641 = \paren {11^2}^2 = 11^4$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Squares which are 4 Less than Cubes}
+Tags: 4, 121, Square Numbers, Cube Numbers
+
+\begin{theorem}
+The only two [[Definition:Square Number|square numbers]] which are $4$ less than a [[Definition:Cube Number|cube]] are:
+:$2^2 + 4 = 2^3$
+:$11^2 + 4 = 5^3$
+\end{theorem}<|endoftext|>
+\section{Integer Greater than 121 is Sum of Distinct Primes of form 4 n + 1}
+Tags: 121, Prime Numbers
+
+\begin{theorem}
+Let $n$ be an [[Definition:Integer|integer]] greater than $121$.
+Then $n$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct|distinct]] [[Definition:Prime Number|prime numbers]] of the form $4 n + 1$.
+\end{theorem}<|endoftext|>
+\section{Palindromes in Base 10 and Base 3}
+Tags: Palindromic Numbers, 3, 10
+
+\begin{theorem}
+The following $n \in \Z$ are [[Definition:Palindromic Number|palindromic]] in both [[Definition:Decimal Notation|decimal]] and [[Definition:Ternary Notation|ternary]]:
+:$0, 1, 2, 4, 8, 121, 151, 212, 242, 484, 656, 757, \ldots$
+{{OEIS|A007633}}
+\end{theorem}
+
+\begin{proof}
+:{| border="1"
+|-
+! align="center" style = "padding: 2px 10px" | $n_{10}$
+! align="center" style = "padding: 2px 10px" | $n_3$
+|-
+| align="right" style = "padding: 2px 10px" | $0$
+| align="right" style = "padding: 2px 10px" | $0$
+|-
+| align="right" style = "padding: 2px 10px" | $1$
+| align="right" style = "padding: 2px 10px" | $1$
+|-
+| align="right" style = "padding: 2px 10px" | $2$
+| align="right" style = "padding: 2px 10px" | $2$
+|-
+| align="right" style = "padding: 2px 10px" | $4$
+| align="right" style = "padding: 2px 10px" | $11$
+|-
+| align="right" style = "padding: 2px 10px" | $8$
+| align="right" style = "padding: 2px 10px" | $22$
+|-
+| align="right" style = "padding: 2px 10px" | $121$
+| align="right" style = "padding: 2px 10px" | $11 \, 111$
+|-
+| align="right" style = "padding: 2px 10px" | $151$
+| align="right" style = "padding: 2px 10px" | $12 \, 121$
+|-
+| align="right" style = "padding: 2px 10px" | $212$
+| align="right" style = "padding: 2px 10px" | $21 \, 212$
+|-
+| align="right" style = "padding: 2px 10px" | $242$
+| align="right" style = "padding: 2px 10px" | $22 \, 222$
+|-
+| align="right" style = "padding: 2px 10px" | $484$
+| align="right" style = "padding: 2px 10px" | $122 \, 221$
+|-
+| align="right" style = "padding: 2px 10px" | $656$
+| align="right" style = "padding: 2px 10px" | $220 \, 022$
+|-
+| align="right" style = "padding: 2px 10px" | $757$
+| align="right" style = "padding: 2px 10px" | $1 \, 001 \, 001$
+|}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Square Number which is Sum of Consecutive Powers}
+Tags: Square Numbers, 121
+
+\begin{theorem}
+The only [[Definition:Square Number|square number]] which is the [[Definition:Integer Addition|sum]] of consecutive [[Definition:Integer Power|powers]] of a [[Definition:Positive Integer|positive integer]] is $121$:
+:$121 = 3^0 + 3^1 + 3^2 + 3^3 + 3^4$
+\end{theorem}
+
+\begin{proof}
+:$121 = 1 + 3 + 9 + 27 + 81$
+{{ProofWanted|It remains to be shown that this is the only such [[Definition:Square Number|square number]].}}
+\end{proof}<|endoftext|>
+\section{Numbers whose Difference equals Difference between Cube and Seventh Power}
+Tags: Powers
+
+\begin{theorem}
+The following $2$ pairs of [[Definition:Integer|integers]] are the only ones known which exhibit this pattern:
+:$\left\vert {5^3 - 2^7}\right\vert = 5 - 2$
+:$\left\vert {13^3 - 3^7}\right\vert = 13 - 3$
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = \left\vert {5^3 - 2^7}\right\vert
+ | r = \left\vert {125 - 128}\right\vert
+ | c =
+}}
+{{eqn | r = 3
+ | c =
+}}
+{{eqn | r = 5 - 2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \left\vert {13^3 - 3^7}\right\vert
+ | r = \left\vert {2197 - 2187}\right\vert
+ | c =
+}}
+{{eqn | r = 10
+ | c =
+}}
+{{eqn | r = 13 - 3
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Reciprocal of 8}
+Tags: 8
+
+\begin{theorem}
+The [[Definition:Reciprocal|reciprocal]] of $8$ can be expressed as the [[Definition:Summation|summation]] of the [[Definition:Integer Power|powers]] of $2$ [[Definition:Integer Multiplication|multiplied]] by the [[Definition:Reciprocal|reciprocal]] [[Definition:Integer Power|powers]] of $10$:
+:$\dfrac 1 8 = \displaystyle \sum_{k \mathop \ge 0} \dfrac {2^k} {10^{k + 1} }$
+
+.1
+ 2
+ 4
+ 8
+ 16
+ 32
+ 64
+ 128
+ 256
+ 512
+ 1024
+ ....
+-------------
+.12499999744
+-------------
+
+\end{theorem}
+
+\begin{proof}
+We have that:
+:$\dfrac {2^k} {10^{k + 1} } = \dfrac 1 {10} \paren {\dfrac 1 5}^k$
+and so by [[Sum of Infinite Geometric Sequence]]:
+{{begin-eqn}}
+{{eqn | l = \sum_{k \mathop \ge 0} \dfrac {2^k} {10^{k + 1} }
+ | r = \dfrac {1/10} {1 - 1/5}
+ | c =
+}}
+{{eqn | r = \dfrac 1 {10 \paren {4 / 5} }
+ | c =
+}}
+{{eqn | r = \dfrac 5 {40}
+ | c =
+}}
+{{eqn | r = \dfrac 1 8
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Triangles with Integer Area and Integer Sides in Arithmetical Sequence}
+Tags: Areas of Triangles
+
+\begin{theorem}
+The [[Definition:Triangle (Geometry)|triangles]] with the following [[Definition:Side of Polygon|sides]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] have [[Definition:Integer|integer]] [[Definition:Area|areas]]:
+:$3, 4, 5$
+:$13, 14, 15$
+:$15, 28, 41$
+:$15, 26, 37$
+Their [[Definition:Area|areas]] are:
+:$6, 84, 126, 156$
+\end{theorem}
+
+\begin{proof}
+From [[Heron's Formula]], the [[Definition:Area|area]] $A$ of $\triangle ABC$ is given by:
+:$A = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
+where $s = \dfrac{a + b + c} 2$ is the [[Definition:Semiperimeter|semiperimeter]] of $\triangle ABC$.
+For $3, 4, 5$:
+{{begin-eqn}}
+{{eqn | l = s
+ | r = \frac {3 + 4 + 5} 2
+ | c =
+}}
+{{eqn | r = 6
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = A
+ | r = \sqrt {6 \paren {6 - 3} \paren {6 - 4} \paren {6 - 5} }
+ | c =
+}}
+{{eqn | r = \sqrt {6 \times 3 \times 2 \times 1}
+ | c =
+}}
+{{eqn | r = \sqrt {6 \times 6}
+ | c =
+}}
+{{eqn | r = 6
+ | c =
+}}
+{{end-eqn}}
+For $13, 14, 15$:
+{{begin-eqn}}
+{{eqn | l = s
+ | r = \frac {13 + 14 + 15} 2
+ | c =
+}}
+{{eqn | r = 21
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = A
+ | r = \sqrt {21 \paren {21 - 13} \paren {21 - 14} \paren {21 - 15} }
+ | c =
+}}
+{{eqn | r = \sqrt {21 \times 8 \times 7 \times 6}
+ | c =
+}}
+{{eqn | r = \sqrt {\paren {3 \times 7} \times 2^3 \times 7 \times \paren {2 \times 3} }
+ | c =
+}}
+{{eqn | r = 3 \times 7 \times 2^2
+ | c =
+}}
+{{eqn | r = 84
+ | c =
+}}
+{{end-eqn}}
+For $15, 28, 41$:
+{{begin-eqn}}
+{{eqn | l = s
+ | r = \frac {15 + 28 + 41} 2
+ | c =
+}}
+{{eqn | r = 42
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = A
+ | r = \sqrt {42 \paren {42 - 15} \paren {42 - 28} \paren {42 - 41} }
+ | c =
+}}
+{{eqn | r = \sqrt {42 \times 27 \times 14 \times 1}
+ | c =
+}}
+{{eqn | r = \sqrt {\paren {2 \times 3 \times 7} \times 3^3 \times \paren {2 \times 7} }
+ | c =
+}}
+{{eqn | r = 2 \times 3^2 \times 7
+ | c =
+}}
+{{eqn | r = 126
+ | c =
+}}
+{{end-eqn}}
+For $15, 26, 37$:
+{{begin-eqn}}
+{{eqn | l = s
+ | r = \frac {15 + 26 + 37} 2
+ | c =
+}}
+{{eqn | r = 39
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = A
+ | r = \sqrt {39 \paren {39 - 15} \paren {39 - 26} \paren {39 - 37} }
+ | c =
+}}
+{{eqn | r = \sqrt {39 \times 24 \times 13 \times 2}
+ | c =
+}}
+{{eqn | r = \sqrt {\paren {3 \times 13} \times \paren {2^3 \times 3} \times 13 \times 2}
+ | c =
+}}
+{{eqn | r = 2^2 \times 3 \times 13
+ | c =
+}}
+{{eqn | r = 156
+ | c =
+}}
+{{end-eqn}}
+{{expand|The citation below generalises this result, and this page could be turned into it.}}
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Weierstrass E-Function}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $\mathbf y, \mathbf z, \mathbf w$ be [[Definition:Dimension|$n$-dimensional]] [[Definition:Vector|vectors]].
+Let $\mathbf y$ be such that $\map{\mathbf y} a=A$ and $\map{\mathbf y} b=B$.
+Let $J$ be a [[Definition:Real Functional|functional]] such that:
+:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
+{{TFAE|def = Weierstrass E-Function}}
+\end{theorem}
+
+\begin{proof}
+=== Definition 1 implies Definition 2 ===
+By Definition 1:
+:$\map E {x, \mathbf y, \mathbf z, \mathbf w} = \map F {x, \mathbf y, \mathbf w} - \map F {x, \mathbf y, \mathbf z} + \paren {\mathbf w - \mathbf z} \map {F_{\mathbf y'} } {x, \mathbf y, \mathbf z}$
+By [[Taylor's Theorem]], where expansion is done around $\mathbf w = \mathbf z$ and [[Definition:Remainder of Taylor Series|Lagrange form of remainder]] is used:
+:$\ds \map F {x, \mathbf y, \mathbf w} = \map F {x, \mathbf y, \mathbf z} + \frac {\partial \map F {x, \mathbf y, \mathbf z} } {\partial \mathbf y'} \paren {\mathbf w - \mathbf z} + \frac 1 2 \sum_{i, j \mathop = 1}^n \paren {w_i - z_i} \paren {w_j - z_j} \frac {\partial^2 \map F {x, \mathbf y, \mathbf z + \theta \paren {\mathbf z - \mathbf w} } } {\partial y_i'y_j'}$
+where $\theta \in \R: 0 < \theta < 1$.
+Insertion of this expansion into the definition for [[Definition:Weierstrass E-Function|Weierstrass E-Function]] leads to the desired result.
+{{qed|lemma}}
+=== Definition 2 implies Definition 1 ===
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Conditions for Strong Minimum of Functional}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $\mathbf y$ be an [[Definition:Dimension|$n$-dimensional]] [[Definition:Vector|vector]] such that $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$
+Let $J$ be a [[Definition:Real Functional|functional]] such that:
+:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
+Let $\gamma$ be an [[Definition:Extremum of Functional|extremal]] [[Definition:Curve|curve]] of $J$.
+Let the following be the [[Definition:Field of Directions of Functional|field of the functional]] $J$:
+:$\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$
+Let $R$ be an [[Definition:Open Region|open region]] containing $\gamma$ and have the [[Definition:Field of Directions of Functional|field]] $\boldsymbol \psi$ defined as $\forall \paren {x, \mathbf y} \in R$.
+Let $\mathbf w$ be a finite [[Definition:Vector|vector]].
+Suppose that:
+:$\forall \paren {x, \mathbf y} \in R: \map E {x, \mathbf y, \boldsymbol \psi, \mathbf w}\ge 0$
+where $E$ is [[Definition:Weierstrass E-Function|Weierstrass E-Function]].
+Then $J$ has a strong [[Definition:Minimum Value of Functional|minimum]] for $\gamma$.
+{{explain|What is a strong minimum?}}
+\end{theorem}
+
+\begin{proof}
+By definition, the [[Definition:Increment of Functional|increment]] of $J$ is:
+:$\ds \Delta J = \int_{\gamma^*} \map F {x, \mathbf y, \mathbf y'} \rd x - \int_\gamma \map F {x, \mathbf y, \mathbf y'} \rd x$
+where $\gamma$ and $\gamma^*$ are [[Definition:Curve|curves]] described by $\paren {x, \map {\mathbf y} x}$ and $\paren {x, \paren {\mathbf y^*} x}$ respectively, such that $\map {\mathbf y^*} x - \map {\mathbf y} x = \map {\mathbf h} x$.
+Consider [[Definition:Hilbert's Invariant Integral|Hilbert's invariant integral]]:
+{{begin-eqn}}
+{{eqn | l = \map g {x, \mathbf y}
+ | r = \int_\Gamma \paren {-H \rd x + \mathbf p \rd \mathbf y}
+}}
+{{eqn | r = \int_\Gamma \paren {-\paren {-F + \mathbf y' F_{\mathbf y'} } \rd x + F_{\mathbf y'} \rd \mathbf y}
+ | c = {{Defof|Hamiltonian}} and [[Definition:Canonical Variable|Momentum]]
+}}
+{{eqn | r = \int_\Gamma \paren {F \rd x - \mathbf y' F_{\mathbf y'} \rd x + F_{\mathbf y'} \rd \mathbf y}
+}}
+{{eqn | r = \int_{\Gamma} \paren {F \rd x - \boldsymbol \psi F_{\mathbf y'} \rd x + F_{\mathbf y'} \rd \mathbf y}
+ | c = $\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$
+}}
+{{end-eqn}}
+Since the [[Definition:Integrand|integrand]] is full differential, the [[Definition:Definite Integral|integral]] does not depend on the shape of $ \Gamma $, but only on its [[Definition:Endpoint of Path|endpoints]].
+Therefore, for $\Gamma = \gamma$ and $\Gamma = \gamma^*$ the value of the [[Definition:Definite Integral|integral]] is the same.
+Since $\map {\mathbf y'} x = \map {\boldsymbol \psi} {x, \mathbf y}$ determines [[Definition:Boundary Condition|boundary conditions]] for $\gamma$, $\Gamma = \gamma$ is one of the trajectories of the field $\map {\mathbf y'} x = \map {\boldsymbol \psi} {x, \mathbf y}$.
+Hence, $\d \mathbf y$ is constrained by $\d \mathbf y = \boldsymbol \psi \rd x$:
+:$\ds \map g {x, \mathbf y} = \int_\gamma \map F {x, \mathbf y, \mathbf y'} \rd x$
+Thus, $\map g {x, \mathbf y'}$ can be written in two different ways.
+:$\ds \int_\gamma \map F {x, \mathbf y, \mathbf y'} \rd x = \int_{\gamma^*} \paren {\paren {\map F {x, \mathbf y, \boldsymbol \psi} - \boldsymbol \psi F_{\mathbf y'} \paren {x, \mathbf y, \boldsymbol \psi} } \rd x + F_{\mathbf y'} \paren {x, \mathbf y, \boldsymbol \psi} \rd \mathbf y}$
+Substitute this into the expression for $\Delta J$:
+{{begin-eqn}}
+{{eqn | l = \Delta J
+ | r = \int_{\gamma^*} \map F {x, \mathbf y, \mathbf y'} \rd x - \int_{\gamma^*} \paren {\paren {\map F {x, \mathbf y, \boldsymbol \psi} - \boldsymbol \psi \map {F_{\mathbf y'} } {x, \mathbf y, \boldsymbol \psi} }\rd x + \map {F_{\mathbf y'} } {x, \mathbf y, \boldsymbol \psi} \rd \mathbf y}
+}}
+{{eqn | r = \int_{\gamma^*} \paren {\map F {x, \mathbf y, \mathbf y'} - \map F {x, \mathbf y, \boldsymbol \psi} - \paren {\mathbf y'-\boldsymbol \psi} \map {F_{\mathbf y'} } {x, \mathbf y, \boldsymbol \psi} }\rd x
+ | c = as $\d \mathbf y = \boldsymbol \psi \rd x$
+}}
+{{eqn | r = \int_{\gamma^*} \map E {x, \mathbf y, \boldsymbol\psi, \mathbf y'} \rd x
+ | c = {{Defof|Weierstrass E-Function}}
+}}
+{{end-eqn}}
+By assumption:
+:$\map E {x, \mathbf y, \boldsymbol \psi, \mathbf y'} \ge 0$
+Hence the [[Definition:Integrand|integrand]] is [[Definition:Bounded Below|bounded below]] by $0$ and [[Definition:Bounded Above|above]] by its [[Definition:Maximum Element|maximum]] in the [[Definition:Closed Interval|interval]] of integration.
+Hence, the [[Definition:Integral|integral]] is [[Definition:Bounded Below|bounded below]] by $0$ and [[Definition:Bounded Above|above]] by some [[Definition:Positive Real Number|positive number]].
+Therefore:
+:$\Delta J \ge 0$
+{{qed}}
+{{improve|Improve clarity, revisit linked pages}}
+\end{proof}<|endoftext|>
+\section{Factor of Mersenne Number Mp is of form 2kp + 1}
+Tags: Mersenne Numbers
+
+\begin{theorem}
+:$q = 2 k p + 1$
+for some [[Definition:Integer|integer]] $k$.
+\end{theorem}
+
+\begin{proof}
+Let $q \divides M_p$.
+Then:
+:$2^p \equiv 1 \pmod q$
+From [[Integer to Power of Multiple of Order]], the [[Definition:Multiplicative Order of Integer|multiplicative order]] of $2 \pmod q$ [[Definition:Divisor of Integer|divides]] $p$.
+By [[Fermat's Little Theorem]], the [[Definition:Multiplicative Order of Integer|multiplicative order]] of $2 \pmod q$ also [[Definition:Divisor of Integer|divides]] $q - 1$.
+Hence:
+:$q - 1 = 2 k p$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Factor of Mersenne Number Mp equivalent to 1 mod p}
+Tags: Mersenne Numbers
+
+\begin{theorem}
+:$q \equiv 1 \pmod p$
+\end{theorem}
+
+\begin{proof}
+Let $q \mathrel \backslash M_p$.
+From [[Factor of Mersenne Number Mp is of form 2kp + 1|Factor of Mersenne Number $M_p$ is of form $2 k p + 1$]]:
+:$q = 2 k p + 1$
+and so by definition of [[Definition:Congruence Modulo Integer|congruence modulo an integer]]:
+:$q \equiv 1 \pmod p$
+immediately.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Factor of Mersenne Number equivalent to +-1 mod 8}
+Tags: Mersenne Numbers
+
+\begin{theorem}
+:$q \equiv \pm 1 \pmod 8$
+\end{theorem}
+
+\begin{proof}
+Suppose $q \mathop \backslash M_p$.
+From [[Factor of Mersenne Number Mp is of form 2kp + 1|Factor of Mersenne Number $M_p$ is of form $2 k p + 1$]]:
+:$q - 1 = 2 k p$
+From above:
+:$2^{\left({q - 1}\right) / 2} \equiv 2 k p \equiv 1 \pmod q$
+and so $2$ is a [[Definition:Quadratic Residue|quadratic residue]] $\pmod q$.
+From [[Second Supplement to Law of Quadratic Reciprocity]]:
+:$q \equiv \pm 1 \pmod 8$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers with 7 or more Prime Factors}
+Tags: Prime Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] with $7$ or more [[Definition:Prime Factor|prime factors]] (not necessarily [[Definition:Distinct|distinct]]) begins:
+:$128, 192, 256, 288, 320, 384, 432, 448, 480, 512, \ldots$
+{{OEIS|A046307}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 128
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2
+ | c =
+}}
+{{eqn | l = 192
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3
+ | c =
+}}
+{{eqn | l = 256
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \left({\times 2}\right)
+ | c =
+}}
+{{eqn | l = 288
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3
+ | c =
+}}
+{{eqn | l = 320
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5
+ | c =
+}}
+{{eqn | l = 384
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \left({\times 3}\right)
+ | c =
+}}
+{{eqn | l = 432
+ | r = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3
+ | c =
+}}
+{{eqn | l = 448
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7
+ | c =
+}}
+{{eqn | l = 480
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5
+ | c =
+}}
+{{eqn | l = 512
+ | r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \left({\times 2 \times 2}\right)
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers not Sum of Distinct Squares}
+Tags: Sums of Squares
+
+\begin{theorem}
+The [[Definition:Positive Integer|positive integers]] which are not the [[Definition:Integer Addition|sum]] of $1$ or more [[Definition:Distinct|distinct]] [[Definition:Square Number|squares]] are:
+:$2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128$
+{{OEIS|A001422}}
+\end{theorem}
+
+\begin{proof}
+{{finish|a) Demonstration that these cannot be so expressed, b) demonstration that all others below 324 can be so expressed}}
+It will be proved that the largest [[Definition:Integer|integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Square Number|squares]] is $128$.
+The remaining [[Definition:Integer|integers]] in the [[Definition:Integer Sequence|sequence]] can be identified by inspection.
+We prove this using a variant of [[Second Principle of Mathematical Induction]].
+Let $\map P n$ be the proposition:
+:$n$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Square Number|squares]].
+=== Basis for the induction ===
+We verify the result up to $18^2 = 324$.
+Then we conclude all $n$ with $129 \le n \le 324$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Square Number|squares]].
+So $\map P n$ is true for all $129 \le n \le 324$.
+This is the [[Definition:Basis for the Induction|basis for the induction]].
+=== Induction Hypothesis ===
+Suppose for some $k > 324$, $\map P j$ is true for all $129 \le j < k$.
+That is, every [[Definition:Integer|integer]] between $129$ and $k - 1$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Square Number|squares]].
+This is the [[Definition:Induction Hypothesis|induction hypothesis]].
+=== Induction Step ===
+This is the [[Definition:Induction Step|induction step]]:
+We find the largest [[Definition:Integer|integer]] $i$ such that $i^2 < k \le \paren {i + 1}^2$.
+We show that $k - \paren {i - 4}^2$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Square Number|squares]], and the [[Definition:Integer Addition|sum]] does not involve $\paren {i - 4}^2$.
+Then $k$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Square Number|squares]].
+Since $k > 324 = 18^2$, we must have $i \ge 18$.
+Hence $k > k - \paren {i - 4}^2 > i^2 - \paren {i - 4}^2 = 4 \paren {2 i - 4} \ge 4 \times 32 = 128$.
+Thus $k - \paren {i - 4}^2$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Square Number|squares]] by Induction Hypothesis.
+A sufficient condition such that the [[Definition:Integer Addition|sum]] does not involve $\paren {i - 4}^2$ is $\paren {i - 4}^2 > k - \paren {i - 4}^2$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \paren {i - 4}^2 - \paren {k - \paren {i - 4}^2 }
+ | o = \ge
+ | r = 2 \paren {i - 4}^2 - \paren {i + 1}^2
+ | c = From $k \le T_{i + 1}$
+}}
+{{eqn | r = 2 i^2 - 16 i + 32 - i^2 - 2 i - 1
+}}
+{{eqn | r = i^2 - 18 i + 31
+}}
+{{eqn | r = i \paren {i - 18} + 31
+}}
+{{eqn | o = \ge
+ | r = 31
+ | c = From $i \ge 18$
+}}
+{{eqn | o = >
+ | r = 0
+}}
+{{end-eqn}}
+Therefore we have $\paren {i - 4}^2 > k - \paren {i - 4}^2$.
+Hence $\map P k$ is true.
+By the [[Second Principle of Mathematical Induction]], $\map P n$ is true for all $n \ge 128$.
+Thus every [[Definition:Integer|integer]] greater than $128$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Square Number|square numbers]].
+{{Proofread}}
+\end{proof}<|endoftext|>
+\section{132 is Sum of all 2-Digit Numbers formed from its Digits}
+Tags: 132
+
+\begin{theorem}
+$132$ is the smallest [[Definition:Integer Addition|sum]] of all the [[Definition:Digit|$2$-digit]] [[Definition:Positive Integer|(positive) integers]] formed from its own [[Definition:Digit|digits]].
+\end{theorem}
+
+\begin{proof}
+It is necessary to postulate that such [[Definition:Positive Integer|(positive) integers]] have $3$ [[Definition:Digit|digits]] or more, as the $2$ [[Definition:Digit|digit]] solution is trivial.
+Let $n = \sqbrk {abc}$ be a $3$-[[Definition:Digit|digit]] number.
+Let $\map s n$ denote the [[Definition:Integer Addition|sum]] of all the [[Definition:Digit|$2$-digit]] [[Definition:Positive Integer|(positive) integers]] formed from the [[Definition:Digit|digits]] of $n$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \map s n
+ | r = \sqbrk {ab} + \sqbrk {ac} + \sqbrk {bc} + \sqbrk {ba} + \sqbrk {ca} + \sqbrk {cb}
+ | c =
+}}
+{{eqn | r = \paren {10 a + b} + \paren {10 a + c} + \paren {10 b + c} + \paren {10 b + a} + \paren {10 c + a} + \paren {10 c + b}
+ | c =
+}}
+{{eqn | r = 2 \times \paren {10 a + a} + 2 \times \paren {10 b + b} + 2 \times \paren {10 c + c}
+ | c =
+}}
+{{eqn | r = 22 \times \paren {a + b + c}
+ | c =
+}}
+{{end-eqn}}
+So for $n = \map s n$, $n = \sqbrk {abc}$ needs to be [[Definition:Divisor of Integer|divisible]] by both $22$ and $a + b + c$.
+{{begin-eqn}}
+{{eqn | l = 22 \times 4
+ | r = 88
+ | c = too small
+}}
+{{eqn | l = 22 \times 5
+ | r = 110
+ | c = but $\map s {110} = 11 + 10 + 01 = 22$
+}}
+{{eqn | l = 22 \times 6
+ | r = 132
+ | c = and $\map s {132} = 12 + 13 + 21 + 23 + 31 + 32 = 132$
+}}
+{{end-eqn}}
+and the result is apparent.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers which are Sum of Increasing Powers of Digits}
+Tags: Powers, Recreational Mathematics
+
+\begin{theorem}
+The following [[Definition:Integer|integers]] are the [[Definition:Integer Addition|sum]] of the increasing [[Definition:Integer Power|powers]] of their [[Definition:Digit|digits]] taken in order begin as follows:
+:$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2 \, 646 \, 798, 12 \, 157 \, 692 \, 622 \, 039 \, 623 \, 539$
+{{OEIS|A032799}}
+\end{theorem}
+
+\begin{proof}
+Single [[Definition:Digit|digit]] [[Definition:Integer|integers]] are trivial:
+:$d^1 = d$
+for all $d \in \Z$.
+Then we have:
+{{begin-eqn}}
+{{eqn | l = 8^1 + 9^2
+ | r = 8 + 81
+ | rr= = 89
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 1^1 + 3^2 + 5^3
+ | r = 1 + 9 + 125
+ | rr= = 135
+ | c =
+}}
+{{eqn | l = 1^1 + 7^2 + 5^3
+ | r = 1 + 49 + 125
+ | rr= = 175
+ | c =
+}}
+{{eqn | l = 5^1 + 1^2 + 8^3
+ | r = 5 + 1 + 512
+ | rr= = 518
+ | c =
+}}
+{{eqn | l = 5^1 + 9^2 + 8^3
+ | r = 5 + 81 + 512
+ | rr= = 598
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 1^1 + 3^2 + 0^3 + 6^4
+ | r = 1 + 9 + 0 + 1296
+ | rr= = 1306
+ | c =
+}}
+{{eqn | l = 1^1 + 6^2 + 7^3 + 6^4
+ | r = 1 + 36 + 343 + 1296
+ | rr= = 1676
+ | c =
+}}
+{{eqn | l = 2^1 + 4^2 + 2^3 + 7^4
+ | r = 2 + 16 + 8 + 2401
+ | rr= = 2427
+ | c =
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | r = 2^1 + 6^2 + 4^3 + 6^4 + 7^5 + 9^6 + 8^7
+ | o =
+ | c =
+}}
+{{eqn | r = 2 + 36 + 64 + 1296 + 16 \, 807 + 531 \, 441 + 1 \, 097 \, 152
+ | c =
+}}
+{{eqn | r = 2 \, 646 \, 798
+ | c =
+}}
+{{end-eqn}}
+and finally:
+{{begin-eqn}}
+{{eqn | r = 1 + 2^2 + 1^3 + 5^4 + 7^5 + 6^6 + 9^7 + 2^8 + 6^9 + 2^{10} + 2^{11} + 0^{12} + 3^{13} + 9^{14} + 6^{15} + 2^{16} + 3^{17} + 5^{18} + 3^{19} + 9^{20}
+ | o =
+ | c =
+}}
+{{eqn | r = 1 + 4 + 1 + 625 + 16 \, 807 + 46 \, 656 + 4 \, 782 \, 969 + 256 + 10 \, 077 \, 696 + 1024 + 2048 + 0 + 1 \, 594 \, 323
+ | c =
+}}
+{{eqn | o =
+ | ro= +
+ | r = 22 \, 876 \, 792 \, 454 \, 961 + 470 \, 184 \, 984 \, 576 + 65 \, 536 + 129 \, 140 \, 163 + 3 \, 814 \, 697 \, 265 \, 625
+ | c =
+}}
+{{eqn | o =
+ | ro= +
+ | r = 1 \, 162 \, 261 \, 467 + 12 \, 157 \, 665 \, 459 \, 056 \, 928 \, 801
+ | c =
+}}
+{{eqn | r = 12 \, 157 \, 692 \, 622 \, 039 \, 623 \, 539
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be proved that there are no more like this.}}
+\end{proof}<|endoftext|>
+\section{Prime Gaps of 10}
+Tags: Prime Gaps
+
+\begin{theorem}
+The following [[Definition:Ordered Pair|pairs]] of consecutive [[Definition:Prime Number|prime numbers]] are those whose [[Definition:Integer Subtraction|difference]] is $10$:
+:$\tuple {139, 149}, \tuple {181, 191}, \tuple {241, 251}, \tuple {283, 293}, \ldots$
+{{OEIS|A031928|order = lower}}
+\end{theorem}
+
+\begin{proof}
+Demonstrated by listing the [[Definition:Prime Gap|prime gaps]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Harmonic Mean of Divisors in terms of Tau and Sigma}
+Tags: Harmonic Mean, Sigma Function, Tau Function
+
+\begin{theorem}
+Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]].
+The [[Definition:Harmonic Mean|harmonic mean]] of the [[Definition:Divisor of Integer|divisors]] of $n$ is given by:
+:$H \left({n}\right) = \dfrac {n \, \tau \left({n}\right)} {\sigma \left({n}\right)}$
+where:
+:$\tau \left({n}\right)$ denotes the [[Definition:Tau Function|$\tau$ (tau) function]]: the number of [[Definition:Divisor of Integer|divisors]] of $n$
+:$\sigma \left({n}\right)$ denotes the [[Definition:Sigma Function|$\sigma$ (sigma) function]]: the [[Definition:Integer Addition|sum]] of the [[Definition:Divisor of Integer|divisors]] of $n$.
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Harmonic Mean|harmonic mean]]:
+:$\dfrac 1 {H \left({n}\right)} = \dfrac 1 {\tau \left({n}\right)} \left({\displaystyle \sum_{d \mathop \backslash n} \dfrac 1 d}\right)$
+From [[Sum of Reciprocals of Divisors equals Abundancy Index]]:
+: $\displaystyle \sum_{d \mathop \backslash n} \frac 1 d = \frac {\sigma \left({n}\right)} n$
+and so:
+:$\dfrac 1 {H \left({n}\right)} = \dfrac {\sigma \left({n}\right)} {n \tau \left({n}\right)}$
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Perfect Number is Ore Number}
+Tags: Ore Numbers
+
+\begin{theorem}
+Let $n \in \Z_{>0}$ be a [[Definition:Perfect Number|perfect number]].
+Then $n$ is an [[Definition:Ore Number|Ore number]].
+\end{theorem}
+
+\begin{proof}
+From [[Harmonic Mean of Divisors in terms of Tau and Sigma]], the [[Definition:Harmonic Mean|harmonic mean]] of the [[Definition:Divisor of Integer|divisors]] of $n$ is given by:
+:$H \left({n}\right) = \dfrac {n \tau \left({n}\right)} {\sigma \left({n}\right)}$
+where:
+:$\tau \left({n}\right)$ denotes the [[Definition:Tau Function|$\tau$ (tau) function]]: the number of [[Definition:Divisor of Integer|divisors]] of $n$
+:$\sigma \left({n}\right)$ denotes the [[Definition:Sigma Function|$\sigma$ (sigma) function]]: the [[Definition:Integer Addition|sum]] of the [[Definition:Divisor of Integer|divisors]] of $n$.
+Let $n$ be a [[Definition:Perfect Number|perfect number]].
+By definition of [[Definition:Perfect Number|perfect number]]:
+:$\dfrac {\sigma \left({n}\right)} n = 2$
+From [[Tau Function Odd Iff Argument is Square]]:
+:$\tau \left({n}\right) = 2 k$
+for some $k \in \Z$.
+Hence:
+:$H \left({n}\right) = \dfrac {2 k} 2 = k$
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors}
+Tags: Ore Numbers
+
+\begin{theorem}
+The following [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] have the property that both the [[Definition:Harmonic Mean|harmonic mean]] and [[Definition:Arithmetic Mean|arithmetic mean]] of their [[Definition:Divisor of Integer|divisors]] are [[Definition:Integer|integers]]:
+:$1, 6, 140, 270, 672, \ldots$
+{{OEIS|A007340}}
+\end{theorem}
+
+\begin{proof}
+Note the [[Definition:Integer|integers]] whose [[Definition:Harmonic Mean|harmonic mean]] of their [[Definition:Divisor of Integer|divisors]] are [[Definition:Integer|integers]] are the [[Definition:Ore Number|Ore numbers]]:
+:$1, 6, 28, 140, 270, 496, 672, \ldots$
+{{OEIS|A001599}}
+It remains to calculate the [[Definition:Arithmetic Mean|arithmetic mean]] of their [[Definition:Divisor of Integer|divisors]].
+Let $A \left({n}\right)$ denote the [[Definition:Arithmetic Mean|arithmetic mean]] of the [[Definition:Divisor of Integer|divisors]] of $n$.
+Then we have:
+:$A \left({n}\right) = \dfrac {\sigma \left({n}\right)} {\tau \left({n}\right)}$
+where:
+:$\tau \left({n}\right)$ denotes the [[Definition:Tau Function|$\tau$ (tau) function]]: the number of [[Definition:Divisor of Integer|divisors]] of $n$
+:$\sigma \left({n}\right)$ denotes the [[Definition:Sigma Function|$\sigma$ (sigma) function]]: the [[Definition:Integer Addition|sum]] of the [[Definition:Divisor of Integer|divisors]] of $n$.
+{{begin-eqn}}
+{{eqn | l = \sigma \left({6}\right)
+ | r = 12
+ | c = {{SigmaLink|6}}
+}}
+{{eqn | l = \tau \left({6}\right)
+ | r = 4
+ | c = {{TauLink|6}}
+}}
+{{eqn | ll= \leadsto
+ | l = A \left({6}\right)
+ | r = \dfrac {12} 4 = 3
+ | c = and so the [[Definition:Arithmetic Mean|arithmetic mean]] is an [[Definition:Integer|integer]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \sigma \left({28}\right)
+ | r = 56
+ | c = {{SigmaLink|28}}
+}}
+{{eqn | l = \tau \left({28}\right)
+ | r = 4
+ | c = {{TauLink|28}}
+}}
+{{eqn | ll= \leadsto
+ | l = A \left({28}\right)
+ | r = \dfrac {56} 6 = 9 \cdotp \dot 3
+ | c = and so the [[Definition:Arithmetic Mean|arithmetic mean]] is not an [[Definition:Integer|integer]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \sigma \left({140}\right)
+ | r = 336
+ | c = {{SigmaLink|140}}
+}}
+{{eqn | l = \tau \left({140}\right)
+ | r = 12
+ | c = {{TauLink|140}}
+}}
+{{eqn | ll= \leadsto
+ | l = A \left({140}\right)
+ | r = \dfrac {336} {12} = 28
+ | c = and so the [[Definition:Arithmetic Mean|arithmetic mean]] is an [[Definition:Integer|integer]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \sigma \left({270}\right)
+ | r = 720
+ | c = {{SigmaLink|270}}
+}}
+{{eqn | l = \tau \left({270}\right)
+ | r = 16
+ | c = {{TauLink|270}}
+}}
+{{eqn | ll= \leadsto
+ | l = A \left({270}\right)
+ | r = \dfrac {720} {16} = 45
+ | c = and so the [[Definition:Arithmetic Mean|arithmetic mean]] is an [[Definition:Integer|integer]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \sigma \left({496}\right)
+ | r = 992
+ | c = {{SigmaLink|496}}
+}}
+{{eqn | l = \tau \left({496}\right)
+ | r = 10
+ | c = {{TauLink|496}}
+}}
+{{eqn | ll= \leadsto
+ | l = A \left({496}\right)
+ | r = \dfrac {992} {10} = 9 \cdotp 92
+ | c = and so the [[Definition:Arithmetic Mean|arithmetic mean]] is not an [[Definition:Integer|integer]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \sigma \left({672}\right)
+ | r = 2016
+ | c = {{SigmaLink|672}}
+}}
+{{eqn | l = \tau \left({672}\right)
+ | r = 24
+ | c = {{TauLink|672}}
+}}
+{{eqn | ll= \leadsto
+ | l = A \left({672}\right)
+ | r = \dfrac {2016} {24} = 84
+ | c = and so the [[Definition:Arithmetic Mean|arithmetic mean]] is an [[Definition:Integer|integer]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \sigma \left({1638}\right)
+ | r = 4368
+ | c = {{SigmaLink|1638}}
+}}
+{{eqn | l = \tau \left({1638}\right)
+ | r = 24
+ | c = {{TauLink|1638}}
+}}
+{{eqn | ll= \leadsto
+ | l = A \left({1638}\right)
+ | r = \dfrac {4368} {24} = 182
+ | c = and so the [[Definition:Arithmetic Mean|arithmetic mean]] is an [[Definition:Integer|integer]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Square Fibonacci Number}
+Tags: Fibonacci Numbers, Square Numbers
+
+\begin{theorem}
+After $1$, there exists exactly one [[Definition:Fibonacci Number|Fibonacci number]] which is also [[Definition:Square Number|square]]:
+:$F_{12} = 144 = 12^2$
+which is also coincidentally the [[Definition:Square (Algebra)|square]] of its index.
+\end{theorem}<|endoftext|>
+\section{Carmichael's Theorem}
+Tags: Fibonacci Numbers, Primitive Prime Factors
+
+\begin{theorem}
+Let $n \in \Z$ such that $n > 12$.
+Then the $n$th [[Definition:Fibonacci Numbers|Fibonacci number]] $F_n$ has at least one [[Definition:Prime Factor|prime factor]] which does not [[Definition:Divisor of Integer|divide]] any smaller [[Definition:Fibonacci Numbers|Fibonacci number]].
+The exceptions for $n \le 12$ are:
+:$F_1 = 1, F_2 = 1$: neither have any [[Definition:Prime Factor|prime factors]]
+:$F_6 = 8$ whose only [[Definition:Prime Factor|prime factor]] is $2$ which is $F_3$
+:$F_{12} = 144$ whose only [[Definition:Prime Factor|prime factors]] are $2$ (which is $F_3$) and $3$ (which is $F_4$).
+\end{theorem}
+
+\begin{proof}
+We have that:
+:$1$ has no [[Definition:Prime Factor|prime factors]].
+Hence, [[Definition:Vacuous Truth|vacuously]], $1$ has no [[Definition:Primitive Prime Factor|primitive prime factors]].
+:$8 = 2^3$
+and $2 \divides 2 = F_3$
+:$144 = 2^4 3^2$
+and:
+:$2 \divides 8 = F_6$
+:$3 \divides 21 = F_8$
+for example.
+{{ProofWanted|It remains to be shown that these are the only examples.}}
+\end{proof}<|endoftext|>
+\section{Weierstrass's Necessary Condition}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $\mathbf y: \R \to \R^n$ be an [[Definition:Dimension|$n$-dimensional]] [[Definition:Vector-Valued Function|vector-valued function]] such that $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$.
+Let $J$ be a [[Definition:Real Functional|functional]] such that:
+:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
+Let $\mathbf w$ be an [[Definition:Dimension|$n$-dimensional]] [[Definition:Vector|vector]] such that $\mathbf w \in \R^n$.
+Let $\gamma$ be a strong [[Definition:Minimum Value of Functional|minimum]] of $J$.
+{{explain|exactly what a ''strong'' minimum is, by means of a link to a definition page}}
+Then along $\gamma$ and for every $\mathbf w$:
+:$\map E {x, \mathbf y, \mathbf y', \mathbf w} \ge 0$
+{{explain|exactly what ''along'' $\gamma$ means, by means of a link to a definition page}}
+where $E$ stands for the [[Definition:Weierstrass E-Function|Weierstrass E-Function]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+{{Namedfor|Karl Theodor Wilhelm Weierstrass|cat = Weierstrass}}
+\end{proof}<|endoftext|>
+\section{Squares Ending in Repeated Digits}
+Tags: Square Numbers
+
+\begin{theorem}
+A [[Definition:Square Number|square number]] $n^2$ can end in a repeated [[Definition:Digit|digit]] {{iff}} either:
+:$(1): \quad n^2$ is a [[Definition:Integer Multiple|multiple]] of $100$, in which case $n$ is a [[Definition:Integer Multiple|multiple]] of $10$
+:$(2): \quad n^2$ ends in $44$ and $n$ ends in $12, 38, 62$ or $88$.
+\end{theorem}
+
+\begin{proof}
+Let $n \in \Z_{>0}$ end in $a b$.
+By the [[Basis Representation Theorem]], $n$ can be expressed as:
+:$n = 100 k + 10 a + b$
+for some $k \in \Z_{>0}$ and for $0 \le a < 10, 0 \le b < 10$.
+Then:
+{{begin-eqn}}
+{{eqn | l = n^2
+ | r = \paren {100k + 10 a + b}^2
+ | c =
+}}
+{{eqn | r = 100^2 k^2 + 2000 k a + 200 k b + 100 b^2 + 20 a b + b^2
+ | c =
+}}
+{{eqn | o = \equiv
+ | r = 20 a b + b^2
+ | rr = \pmod {100}
+}}
+{{end-eqn}}
+Thus the nature of the last $2$ [[Definition:Digit|digits]] of $n^2$ are not dependent upon $k$.
+So we can ignore all [[Definition:Digit|digits]] of $n$ except the last $2$.
+Note that if $b = 0$ we have that $b^2 = 0$ and $20 a b = 0$.
+So the last $2$ [[Definition:Digit|digits]] of the [[Definition:Square (Algebra)|square]] of a [[Definition:Integer Multiple|multiple]] of $10$ are both $0$, and $n^2$ is a [[Definition:Integer Multiple|multiple]] of $100$.
+Let $a = 5 + c$ where $0 \le c < 5$.
+We have that:
+{{begin-eqn}}
+{{eqn | l = \paren {10 a + b}^2
+ | r = \paren {10 \paren {5 + c} + b}^2
+ | c =
+}}
+{{eqn | r = \paren {10 c + 50 + b}^2
+ | c =
+}}
+{{eqn | r = 100 c^2 + 2 \times 500 c + 2 \times 10 b c + 2 \times 50 b + b^2
+ | c =
+}}
+{{eqn | r = 100 c^2 + 1000 c + 100 b + 20 b c + b^2
+ | c =
+}}
+{{eqn | r = 1000 c + 100 b + \paren {10 c + b}^2
+ | c =
+}}
+{{eqn | o = \equiv
+ | r = \paren {10 c + b}^2
+ | rr = \pmod {100}
+}}
+{{end-eqn}}
+So the [[Definition:Square (Algebra)|square]] of a number ending in $c b$, where $0 \le c < 5$, ends in the same $2$ [[Definition:Digit|digits]] as the [[Definition:Square (Algebra)|square]] a number ending in $a b$ where $a = c + 5$.
+It remains to list the [[Definition:Integer|integers]] from $1$ to $49$, generating their [[Definition:Square (Algebra)|squares]] and investigating their last $2$ [[Definition:Digit|digits]]:
+{{begin-eqn}}
+{{eqn | l = 01^2
+ | r = 01
+}}
+{{eqn | l = 02^2
+ | r = 04
+}}
+{{eqn | l = 03^2
+ | r = 09
+}}
+{{eqn | l = 04^2
+ | r = 16
+}}
+{{eqn | l = 05^2
+ | r = 25
+}}
+{{eqn | l = 06^2
+ | r = 36
+}}
+{{eqn | l = 07^2
+ | r = 49
+}}
+{{eqn | l = 08^2
+ | r = 64
+}}
+{{eqn | l = 09^2
+ | r = 81
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 11^2
+ | r = 121
+}}
+{{eqn | l = 12^2
+ | r = 144
+ | c = $n^2$ ends in $44$ and $n$ ends in $12$
+}}
+{{eqn | l = 13^2
+ | r = 169
+}}
+{{eqn | l = 14^2
+ | r = 196
+}}
+{{eqn | l = 15^2
+ | r = 225
+}}
+{{eqn | l = 16^2
+ | r = 256
+}}
+{{eqn | l = 17^2
+ | r = 289
+}}
+{{eqn | l = 18^2
+ | r = 324
+}}
+{{eqn | l = 19^2
+ | r = 361
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 21^2
+ | r = 441
+}}
+{{eqn | l = 22^2
+ | r = 484
+}}
+{{eqn | l = 23^2
+ | r = 529
+}}
+{{eqn | l = 24^2
+ | r = 576
+}}
+{{eqn | l = 25^2
+ | r = 225
+}}
+{{eqn | l = 26^2
+ | r = 676
+}}
+{{eqn | l = 27^2
+ | r = 729
+}}
+{{eqn | l = 28^2
+ | r = 784
+}}
+{{eqn | l = 29^2
+ | r = 841
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 31^2
+ | r = 961
+}}
+{{eqn | l = 32^2
+ | r = 1024
+}}
+{{eqn | l = 33^2
+ | r = 1089
+}}
+{{eqn | l = 34^2
+ | r = 1156
+}}
+{{eqn | l = 35^2
+ | r = 1225
+}}
+{{eqn | l = 36^2
+ | r = 1296
+}}
+{{eqn | l = 37^2
+ | r = 1369
+}}
+{{eqn | l = 38^2
+ | r = 1444
+ | c = $n^2$ ends in $44$ and $n$ ends in $38$
+}}
+{{eqn | l = 39^2
+ | r = 1521
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 41^2
+ | r = 1681
+}}
+{{eqn | l = 42^2
+ | r = 1764
+}}
+{{eqn | l = 43^2
+ | r = 1849
+}}
+{{eqn | l = 44^2
+ | r = 1936
+}}
+{{eqn | l = 45^2
+ | r = 2025
+}}
+{{eqn | l = 46^2
+ | r = 2116
+}}
+{{eqn | l = 47^2
+ | r = 2209
+}}
+{{eqn | l = 48^2
+ | r = 2304
+}}
+{{eqn | l = 49^2
+ | r = 2401
+}}
+{{end-eqn}}
+It is seen that the only $n^2$ ending in a repeated [[Definition:Digit|digit]] end in $44$.
+The corresponding $n$ are seen to be $12$ and $38$.
+Adding $5$ to the $10$s [[Definition:Digit|digit]] of each gives us $62$ and $88$ as further such:
+{{begin-eqn}}
+{{eqn | l = 62^2
+ | r = 3844
+}}
+{{eqn | l = 88^2
+ | r = 7744
+}}
+{{end-eqn}}
+The result follows.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Magic Constant of Smallest Prime Magic Square with Consecutive Primes}
+Tags: Prime Magic Squares, 4,440,084,513
+
+\begin{theorem}
+The [[Definition:Magic Constant|magic constant]] of the smallest [[Definition:Prime Magic Square|prime magic square]] whose [[Definition:Element of Array|elements]] are consecutive [[Definition:Odd Prime|odd primes]] is $4 \, 440 \, 084 \, 513$.
+\end{theorem}
+
+\begin{proof}
+The smallest [[Definition:Prime Magic Square|prime magic square]] whose [[Definition:Element of Array|elements]] are consecutive [[Definition:Odd Prime|odd primes]] is:
+{{:Prime Magic Square/Examples/Order 3/Smallest with Consecutive Primes}}
+As can be seen by inspection, the [[Definition:Integer Addition|sums]] of the [[Definition:Element of Array|elements]] in the [[Definition:Row of Array|rows]], [[Definition:Column of Array|columns]] and [[Definition:Diagonal of Array|Diagonal of Array]] is $4 \, 440 \, 084 \, 513$:
+{{begin-eqn}}
+{{eqn | l = 1 \, 480 \, 028 \, 159 + 1 \, 480 \, 028 \, 153 + 1 \, 480 \, 028 \, 201
+ | r = 4 \, 440 \, 084 \, 513
+}}
+{{eqn | l = 1 \, 480 \, 028 \, 213 + 1 \, 480 \, 028 \, 171 + 1 \, 480 \, 028 \, 129
+ | r = 4 \, 440 \, 084 \, 513
+}}
+{{eqn | l = 1 \, 480 \, 028 \, 141 + 1 \, 480 \, 028 \, 189 + 1 \, 480 \, 028 \, 183
+ | r = 4 \, 440 \, 084 \, 513
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 1 \, 480 \, 028 \, 159 + 1 \, 480 \, 028 \, 213 + 1 \, 480 \, 028 \, 141
+ | r = 4 \, 440 \, 084 \, 513
+}}
+{{eqn | l = 1 \, 480 \, 028 \, 153 + 1 \, 480 \, 028 \, 171 + 1 \, 480 \, 028 \, 189
+ | r = 4 \, 440 \, 084 \, 513
+}}
+{{eqn | l = 1 \, 480 \, 028 \, 201 + 1 \, 480 \, 028 \, 129 + 1 \, 480 \, 028 \, 183
+ | r = 4 \, 440 \, 084 \, 513
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 1 \, 480 \, 028 \, 159 + 1 \, 480 \, 028 \, 171 + 1 \, 480 \, 028 \, 183
+ | r = 4 \, 440 \, 084 \, 513
+}}
+{{eqn | l = 1 \, 480 \, 028 \, 141 + 1 \, 480 \, 028 \, 171 + 1 \, 480 \, 028 \, 201
+ | r = 4 \, 440 \, 084 \, 513
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be demonstrated that there are no prime magic squares that are any smaller than this.}}
+[[Category:Prime Magic Squares]]
+[[Category:4,440,084,513]]
+h04gygpinc4wf0ihrubo1lf8vghj185
+\end{proof}<|endoftext|>
+\section{Magic Constant of Smallest Prime Magic Square with Consecutive Primes from 3}
+Tags: Prime Magic Squares, 4514
+
+\begin{theorem}
+The [[Definition:Magic Constant|magic constant]] of the smallest [[Definition:Prime Magic Square|prime magic square]] whose [[Definition:Element of Array|elements]] are consecutive [[Definition:Odd Prime|odd primes]] from $3$ upwards is $4514$.
+\end{theorem}
+
+\begin{proof}
+The smallest [[Definition:Prime Magic Square|prime magic square]] whose [[Definition:Element of Array|elements]] are the first consecutive [[Definition:Odd Prime|odd primes]] is:
+{{:Prime Magic Square/Examples/Order 12/Smallest with Consecutive Primes from 3}}
+The fact that this is the smallest is proven [[Prime Magic Square/Examples/Order 12/Smallest with Consecutive Primes from 3|here]].
+The [[Definition:Integer Addition|sum]] of the first $144$ [[Definition:Prime Number|prime numbers]] can either be calculated or looked up: it is $54 \, 169$.
+We see that $2$ is not included in this [[Definition:Prime Magic Square|prime magic square]], but instead $1$ is used.
+So the total of all the [[Definition:Element of Array|elements]] of this [[Definition:Prime Magic Square|prime magic square]] is $54 \, 169 - 2 + 1 = 54 \, 168$.
+There are $12$ [[Definition:Row of Array|rows]], and $12$ [[Definition:Column of Array|columns]], each with the same [[Definition:Magic Constant|magic constant]].
+This [[Definition:Magic Constant|magic constant]] must therefore be $\dfrac {54 \, 168} {12} = 4514$.
+As can be seen by inspection, the [[Definition:Integer Addition|sums]] of the [[Definition:Element of Array|elements]] in the [[Definition:Row of Array|rows]], [[Definition:Column of Array|columns]] and [[Definition:Diagonal of Array|diagonals]] is $4514$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Factorions Base 10}
+Tags: Factorions
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integers]] are the only [[Definition:Factorion|factorions base $10$]]:
+:$1, 2, 145, 40 \, 585$
+\end{theorem}
+
+\begin{proof}
+From [[Factorial/Examples|examples of factorials]]:
+{{begin-eqn}}
+{{eqn | l = 1
+ | r = 1!
+ | c =
+}}
+{{eqn | l = 2
+ | r = 2!
+ | c =
+}}
+{{eqn|| l = 145
+ | r = 1 + 24 + 120
+ | c =
+}}
+{{eqn | r = 1! + 4! + 5!
+ | c =
+}}
+{{eqn|| l = 40 \, 585
+ | r = 24 + 1 + 120 + 40 \, 320 + 120
+ | c =
+}}
+{{eqn | r = 4! + 0! + 5! + 8! + 5!
+ | c =
+}}
+{{end-eqn}}
+A computer search can verify solutions under $2540160 = 9! \times 7$ in seconds.
+Let $n$ be a $7$-[[Definition:Digit|digit]] number with $n > 2540160$.
+Then the [[Definition:Integer Addition|sum]] of the [[Definition:Factorial|factorials]] of its [[Definition:Digit|digits]] is not more than $9! \times 7 = 2540160$.
+So $n$ cannot be a [[Definition:Factorion|factorion base $10$]].
+Now let $n$ be a $k$-[[Definition:Digit|digit]] number, for $k \ge 8$.
+Then the [[Definition:Integer Addition|sum]] of the [[Definition:Factorial|factorials]] of its [[Definition:Digit|digits]] is not more than $9! \times k$.
+But we have:
+{{begin-eqn}}
+{{eqn | l = n
+ | o = \ge
+ | r = 10^{k - 1}
+}}
+{{eqn | r = 10^7 \times 10^{k - 8}
+}}
+{{eqn | o = \ge
+ | r = 10^7 \times \paren {1 + 9 \paren {k - 8} }
+ | c = [[Bernoulli's Inequality]]
+}}
+{{eqn | o = >
+ | r = 8 \times 9! \times \paren {9 k - 71}
+ | c = $8 \times 9! = 2903040$
+}}
+{{eqn | r = 9! \paren {72 k - 71 \times 8}
+}}
+{{eqn | o = >
+ | r = 9! \times k
+ | c = $k \ge 8$
+}}
+{{eqn | o = \ge
+ | r = \text{sum of the factorials of digits of } n
+}}
+{{end-eqn}}
+So there are no more [[Definition:Factorion|factorions base $10$]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum of 2 Squares in 2 Distinct Ways/Examples/145}
+Tags: Sum of 2 Squares in 2 Distinct Ways, 145
+
+\begin{theorem}
+$145$ can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|square numbers]] in two [[Definition:Distinct|distinct]] ways:
+{{begin-eqn}}
+{{eqn | l = 145
+ | r = 12^2 + 1^2
+}}
+{{eqn | r = 9^2 + 8^2
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+We have that:
+:$145 = 5 \times 29$
+Both $5$ and $29$ can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Distinct|distinct]] [[Definition:Square Number|square numbers]]:
+{{begin-eqn}}
+{{eqn | l = 5
+ | r = 1^2 + 2^2
+}}
+{{eqn | l = 29
+ | r = 2^2 + 5^2
+}}
+{{end-eqn}}
+Thus:
+{{begin-eqn}}
+{{eqn | r = \paren {1^2 + 2^2} \paren {2^2 + 5^2}
+ | c =
+}}
+{{eqn | r = \paren {1 \times 2 + 2 \times 5}^2 + \paren {1 \times 5 - 2 \times 2}^2
+ | c = [[Brahmagupta-Fibonacci Identity]]
+}}
+{{eqn | r = \paren {2 + 10}^2 + \paren {5 - 4}^2
+ | c =
+}}
+{{eqn | r = 12^2 + 1^2
+ | c =
+}}
+{{eqn | r = 144 + 1
+ | c =
+}}
+{{eqn | r = 145
+ | c =
+}}
+{{end-eqn}}
+and:
+{{begin-eqn}}
+{{eqn | r = \paren {1^2 + 2^2} \paren {2^2 + 5^2}
+ | c =
+}}
+{{eqn | r = \paren {1 \times 2 - 2 \times 5}^2 + \paren {1 \times 5 + 2 \times 2}^2
+ | c = [[Brahmagupta-Fibonacci Identity/Corollary|Brahmagupta-Fibonacci Identity/Corollary]]
+}}
+{{eqn | r = \paren {2 - 10}^2 + \paren {5 + 4}^2
+ | c =
+}}
+{{eqn | r = \paren {8 - 2}^2 + \paren {5 + 4}^2
+ | c =
+}}
+{{eqn | r = 8^2 + 9^2
+ | c =
+}}
+{{eqn | r = 64 + 81
+ | c =
+}}
+{{eqn | r = 145
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Representation of 1 as Sum of n Unit Fractions}
+Tags: Unit Fractions
+
+\begin{theorem}
+Let $U \left({n}\right)$ denote the number of different ways of representing $1$ as the [[Definition:Rational Addition|sum]] of $n$ [[Definition:Unit Fraction|unit fractions]].
+Then for various $n$, $U \left({n}\right)$ is given by the following table:
+:{| border="1"
+|-
+! align="right" style = "padding: 2px 10px" | $n$
+! align="right" style = "padding: 2px 10px" | $U \left({n}\right)$
+|-
+| align="right" style = "padding: 2px 10px" | $1$
+| align="right" style = "padding: 2px 10px" | $1$
+|-
+| align="right" style = "padding: 2px 10px" | $2$
+| align="right" style = "padding: 2px 10px" | $1$
+|-
+| align="right" style = "padding: 2px 10px" | $3$
+| align="right" style = "padding: 2px 10px" | $3$
+|-
+| align="right" style = "padding: 2px 10px" | $4$
+| align="right" style = "padding: 2px 10px" | $14$
+|-
+| align="right" style = "padding: 2px 10px" | $5$
+| align="right" style = "padding: 2px 10px" | $147$
+|-
+| align="right" style = "padding: 2px 10px" | $6$
+| align="right" style = "padding: 2px 10px" | $3462$
+|}
+{{OEIS|A002966}}
+\end{theorem}
+
+\begin{proof}
+Trivially:
+:$1 = \dfrac 1 1$
+and it follows that:
+$U \left({1}\right) = 1$
+Also trivially:
+:$1 = \dfrac 1 2 + \dfrac 1 2$
+and it follows that:
+$U \left({2}\right) = 1$
+From [[Sum of 3 Unit Fractions that equals 1]]:
+:$U \left({3}\right) = 3$
+From [[Sum of 4 Unit Fractions that equals 1]]:
+:$U \left({4}\right) = 14$
+From [[Sum of 5 Unit Fractions that equals 1]]:
+:$U \left({5}\right) = 147$
+{{ProofWanted|and for the rest}}
+\end{proof}<|endoftext|>
+\section{Sum of 3 Unit Fractions that equals 1}
+Tags: Unit Fractions, Recreational Mathematics
+
+\begin{theorem}
+There are $3$ ways to represent $1$ as the sum of exactly $3$ [[Definition:Unit Fraction|unit fractions]].
+\end{theorem}
+
+\begin{proof}
+Let:
+:$1 = \dfrac 1 a + \dfrac 1 b + \dfrac 1 c$
+where:
+:$0 < a \le b \le c$
+and:
+{{AimForCont}} $a = 1$.
+Then:
+:$1 = \dfrac 1 1 + \dfrac 1 b + \dfrac 1 c$
+and so:
+:$\dfrac 1 b + \dfrac 1 c = 0$
+which [[Definition:Contradiction|contradicts]] the stipulation that $b, c > 0$.
+So there is no solution possible when $a = 1$.
+Therefore $a \ge 2$.
+=== $a = 2$ ===
+Let $a = 2$.
+==== $b = 2$ ====
+Let $b = 2$.
+Then:
+:$\dfrac 1 a + \dfrac 1 b = 1$
+leaving no room for $c$.
+Hence there are no solutions where $a = 2$ and $b = 2$.
+{{qed|lemma}}
+==== $b = 3$ ====
+Let $b = 3$.
+{{begin-eqn}}
+{{eqn | l = \dfrac 1 a + \dfrac 1 b
+ | r = \dfrac 1 2 + \dfrac 1 3
+ | c =
+}}
+{{eqn | r = \dfrac {3 + 2} 6
+ | c =
+}}
+{{eqn | r = \dfrac 5 6
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \dfrac 1 c
+ | r = 1 - \dfrac 5 6
+ | c =
+}}
+{{eqn | r = \dfrac 1 6
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \dfrac 1 c
+ | r = \dfrac 1 6
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = c
+ | r = 6
+ | c =
+}}
+{{end-eqn}}
+Thus we have:
+:$(1): \quad 1 = \dfrac 1 2 + \dfrac 1 3 + \dfrac 1 6$
+{{qed|lemma}}
+==== $b = 4$ ====
+Let $b = 4$.
+{{begin-eqn}}
+{{eqn | l = \dfrac 1 a + \dfrac 1 b
+ | r = \dfrac 1 2 + \dfrac 1 4
+ | c =
+}}
+{{eqn | r = \dfrac {2 + 1} 4
+ | c =
+}}
+{{eqn | r = \dfrac 3 4
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \dfrac 1 c
+ | r = 1 - \dfrac 3 4
+ | c =
+}}
+{{eqn | r = \dfrac 1 4
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = c
+ | r = 4
+ | c =
+}}
+{{end-eqn}}
+Thus we have:
+:$(2): \quad 1 = \dfrac 1 2 + \dfrac 1 4 + \dfrac 1 4$
+==== $b > 4$ ====
+Let $b > 4$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \dfrac 1 a + \dfrac 1 b + \dfrac 1 c
+ | o = <
+ | r = \dfrac 1 2 + \dfrac 1 4 + \dfrac 1 4
+ | c =
+}}
+{{eqn | r = 1
+ | c =
+}}
+{{end-eqn}}
+Hence there are no solutions such that $a = 2, b > 4$.
+{{qed|lemma}}
+=== $a = 3$ ===
+Let $a = 3$.
+==== $b = 3$ ====
+Let $b = 3$.
+{{begin-eqn}}
+{{eqn | l = \dfrac 1 a + \dfrac 1 b
+ | r = \dfrac 1 3 + \dfrac 1 3
+ | c =
+}}
+{{eqn | r = \dfrac 2 3
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \dfrac 1 c
+ | r = 1 - \dfrac 2 3
+ | c =
+}}
+{{eqn | r = \dfrac 1 3
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = c
+ | r = 3
+ | c =
+}}
+{{end-eqn}}
+Thus we have:
+:$(3): \quad 1 = \dfrac 1 3 + \dfrac 1 3 + \dfrac 1 3$
+{{qed|lemma}}
+=== $a > 3$ ===
+Let $a > 3$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \dfrac 1 a + \dfrac 1 b + \dfrac 1 c
+ | o = \le
+ | r = \dfrac 1 a + \dfrac 1 a + \dfrac 1 a
+ | c =
+}}
+{{eqn | o = <
+ | r = 1
+ | c =
+}}
+{{end-eqn}}
+Hence there are no solutions for $a>3$.
+{{qed|lemma}}
+=== Summary ===
+Hence our $3$ solutions:
+{{begin-eqn}}
+{{eqn | n = 1
+ | r = \dfrac 1 2 + \dfrac 1 3 + \dfrac 1 6
+ | o =
+}}
+{{eqn | n = 2
+ | r = \dfrac 1 2 + \dfrac 1 4 + \dfrac 1 4
+ | o =
+}}
+{{eqn | n = 3
+ | r = \dfrac 1 3 + \dfrac 1 3 + \dfrac 1 3
+ | o =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Necessary Condition for Integral Functional to have Extremum for given function/Dependent on n Variables}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $\mathbf x$ be an $n$-dimensional vector.
+Let $\map u {\mathbf x}$ be a real function.
+Let $R$ be a fixed [[Definition:Region|region]].
+Let $J$ be a [[Definition:Real Functional|functional]] such that
+:$\ds J \sqbrk u = \idotsint_R \map F {\mathbf x, u, u_{\mathbf x} } \rd x_1 \cdots \rd x_n$
+Then a [[Definition:Necessary Condition|necessary condition]] for $J \sqbrk u$ to have an [[Definition:Extremum of Functional|extremum]] (strong or weak) for a given [[Definition:Mapping|mapping]] $\map u {\mathbf x}$ is that $\map u {\mathbf x}$ satisfies [[Definition:Euler's Equation for Vanishing Variation|Euler's equation]]:
+:$F_u - \dfrac {\partial} {\partial \mathbf x} F_{u_{\mathbf x} } = 0$
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Increment of Functional|increment of the functional]]:
+:$\ds J \sqbrk {u + h} - J \sqbrk u = \idotsint_R \paren {F \sqbrk {x, u + h, u_{\mathbf x} + h_{\mathbf x} } - F \sqbrk {x, u, u_{\mathbf x} } } \rd x_1 \cdots \rd x_n$
+Use multivariate [[Taylor's Theorem]] on $F$ around the point $\tuple {\mathbf x, u, u_{\mathbf x} }$:
+:$F \sqbrk {\mathbf x, u + h, u_{\mathbf x} + h_{\mathbf x} } = F \sqbrk {\mathbf x, u, u_{\mathbf x} } + \dfrac {\partial F \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial u} h + \dfrac {\partial F \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial u_{\mathbf x} } h_{\mathbf x} + \map \OO {h^2, h h_{\mathbf x}, h_{\mathbf x}^2}$
+where $\OO$ stands for [[Definition:Big-O Notation|Big-O]].
+Then:
+:$\ds \Delta J \sqbrk {u, h} = \idotsint_R \paren {\dfrac {\partial F \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial u} h + \dfrac {\partial F \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial u_{\mathbf x} } h_{\mathbf x} + \map \OO {h^2, h h_{\mathbf x}, h_{\mathbf x}^2} } \rd x_1 \cdots \rd x_n$
+By definition of [[Definition:Differentiable Functional|variation of the functional]]:
+{{begin-eqn}}
+{{eqn | l = \delta J
+ | r = \idotsint_R \paren {\dfrac {\partial F \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial u} + \dfrac {\partial F \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial u_{\mathbf x} } h_{\mathbf x} } \rd x_1 \cdots \rd x_n
+}}
+{{eqn | r = \idotsint_R \paren {\dfrac {\partial F \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial u} - \dfrac {\partial F_{u_{\mathbf x} } \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial \mathbf x} } \map h {\mathbf x} \rd x_1 \cdots \rd x_n + \idotsint_R \dfrac {\partial} {\partial \mathbf x} \sqbrk {\dfrac {\partial F \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial u_{\mathbf x} } \map h {\mathbf x} } \rd x_1 \cdots \rd x_n
+ | c = as $F_{u_{\mathbf x} } h_{\mathbf x} = \dfrac {\partial} {\partial \mathbf x} \sqbrk {F_{u_{\mathbf x} } h} - \dfrac {\partial F_{u_{\mathbf x} } } {\partial \mathbf x} h$
+}}
+{{end-eqn}}
+By [[Green's Theorem|Green's theorem]]:
+:$\ds \idotsint_R \dfrac {\partial} {\partial \mathbf x} \sqbrk {\dfrac {\partial F \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial u_{\mathbf x} } \map h {\mathbf x} } \rd x_1 \cdots \rd x_n = \idotsint_\Gamma \map h {\mathbf x} F_{u_{\mathbf x} } \sqbrk {\mathbf x, u, u_{\mathbf x} } \boldsymbol \nu \rd \sigma$
+where:
+:$\Gamma$ denotes the [[Definition:Boundary (Topology)|boundary]] of $R$
+:$\boldsymbol \nu$ is an outward normal to $\Gamma$.
+Since the [[Definition:Region|region]] is fixed, so are its [[Definition:Boundary (Topology)|boundary]] [[Definition:Point|points]].
+Hence, the difference function $h$ has to vanish at the [[Definition:Boundary (Topology)|boundary]].
+In other words:
+:$\forall \mathbf x \in \Gamma: \map h {\mathbf x} = 0$
+This leaves only the first [[Definition:Definite Integral|integral]].
+:$\ds \delta J = \idotsint_R \paren {\dfrac {\partial F \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial u} - \dfrac{\partial F_{u_{ \mathbf x } } \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial \mathbf x} } \map h {\mathbf x} \rd x_1 \cdots \rd x_n$
+For arbitrary $h$ the [[Definition:Differential of Functional|first variation]] vanishes if the term in brackets vanishes:
+:$\dfrac {\partial F \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial u} - \dfrac {\partial F_{u_{\mathbf x} } \sqbrk {\mathbf x, u, u_{\mathbf x} } } {\partial \mathbf x} = 0$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Repeated Sum of Cubes of Digits of Multiple of 3}
+Tags: Recreational Mathematics, Pluperfect Digital Invariants, Proofs by Induction
+
+\begin{theorem}
+Let $k \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]].
+Let $f: \Z_{>0} \to \Z_{>0}$ be the [[Definition:Mapping|mapping]] defined as:
+:$\forall m \in \Z_{>0}: \map f m = $ the [[Definition:Integer Addition|sum]] of the [[Definition:Cube (Algebra)|cubes]] of the [[Definition:Digit|digits]] of $n$.
+Let $n_0 \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]] which is a [[Definition:Multiple of Integer|multiple]] of $3$.
+Consider the sequence:
+:$s_n = \begin{cases} n_0 & : n = 0 \\
+\map f {s_{n - 1} } & : n > 0 \end{cases}$
+Then:
+:$\exists r \in \N_{>0}: s_r = 153$
+That is, by performing $f$ repeatedly on a [[Definition:Multiple of Integer|multiple]] of $3$ eventually results in the [[Definition:Pluperfect Digital Invariant|pluperfect digital invariant]] $153$.
+\end{theorem}
+
+\begin{proof}
+We verify by brute force:
+:Starting on $n_0 \le 2916 = 4 \times 9^3$, we will end on $153$.
+{{qed|lemma}}
+First we prove that if $3 \divides n_0$, then $3 \divides \map f {n_0}$.
+From [[Divisibility by 3]]:
+The [[Definition:Integer Addition|sum]] of [[Definition:Digit|digits]] of $n_0$ is [[Definition:Divisor of Integer|divisible]] by $3$.
+By [[Fermat's Little Theorem]]:
+:$\forall x \in \Z: x^3 \equiv x \pmod 3$
+Therefore [[Definition:Integer Addition|sum]] of the [[Definition:Cube|cubes]] of the [[Definition:Digit|digits]] of $n_0$ is also [[Definition:Divisor of Integer|divisible]] by $3$.
+That is the definition of $\map f {n_0}$.
+Thus $3 \divides \map f {n_0}$.
+{{qed|lemma}}
+It remains to be shown that for every $n_0 > 2916$, $\map f {n_0} < n_0$.
+For $n_0 \le 9999$:
+:$\map f {n_0} \le 9^3 + 9^3 + 9^3 + 9^3 = 2916 < n_0$
+Now suppose $n_0$ is a $k$-[[Definition:Digit|digit]] number, where $k \ge 5$.
+Then:
+{{begin-eqn}}
+{{eqn | l = n_0
+ | o = >
+ | r = 10^{k - 1}
+ | c = the smallest $k$-[[Definition:Digit|digit]] number
+}}
+{{eqn | r = 10^4 \times 10^{k - 5}
+}}
+{{eqn | o = \ge
+ | r = 10^4 \paren {1 + 9 \paren {k - 5} }
+ | c = [[Bernoulli's Inequality]]
+}}
+{{eqn | o = >
+ | r = 10 \times 9^3 \paren {9 k - 44}
+ | c = $10 > 9$
+}}
+{{eqn | o = >
+ | r = 9^3 \paren {90 k - 440}
+}}
+{{eqn | o = \ge
+ | r = 9^3 \paren {k + 445 - 440}
+ | c = $k \ge 5$
+}}
+{{eqn | o = >
+ | r = 9^3 \times k
+}}
+{{eqn | o = \ge
+ | r = \map f {n_0}
+}}
+{{end-eqn}}
+This shows that $\map f {n_0} < n_0$ for all $n_0 > 2916$.
+Thus for any $n_0 > 2916$, $s_n$ is eventually less than $2916$.
+Therefore we will eventually reach $153$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers for which Euler Phi Function of 2n + 1 is less than that of 2n}
+Tags: Euler Phi Function
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] for which:
+:$\map \phi {2 n + 1} < \map \phi {2 n}$
+begins:
+:$157, 262, 367, 412, \ldots$
+{{OEIS|A001837}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 2 \times 157
+ | r = 314
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 157}
+ | r = 156
+ | c = {{EulerPhiLink|314}}
+}}
+{{eqn | l = 2 \times 157 + 1
+ | r = 315
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 157 + 1}
+ | r = 144
+ | c = {{EulerPhiLink|315}}
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 157 + 1}
+ | o = <
+ | r = \map \phi {2 \times 157}
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 2 \times 262
+ | r = 524
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 262}
+ | r = 260
+ | c = {{EulerPhiLink|524}}
+}}
+{{eqn | l = 2 \times 262 + 1
+ | r = 525
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 262 + 1}
+ | r = 240
+ | c = {{EulerPhiLink|525}}
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 262 + 1}
+ | o = <
+ | r = \map \phi {2 \times 262}
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 2 \times 367
+ | r = 734
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 367}
+ | r = 366
+ | c = {{EulerPhiLink|734}}
+}}
+{{eqn | l = 2 \times 367 + 1
+ | r = 735
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 367 + 1}
+ | r = 336
+ | c = {{EulerPhiLink|735}}
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 367 + 1}
+ | o = <
+ | r = \map \phi {2 \times 367}
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 2 \times 412
+ | r = 824
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 412}
+ | r = 408
+ | c = {{EulerPhiLink|824}}
+}}
+{{eqn | l = 2 \times 412 + 1
+ | r = 825
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 412 + 1}
+ | r = 400
+ | c = {{EulerPhiLink|825}}
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 412 + 1}
+ | o = <
+ | r = \map \phi {2 \times 412}
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 2 \times 472
+ | r = 944
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 472}
+ | r = 464
+ | c = {{EulerPhiLink|944}}
+}}
+{{eqn | l = 2 \times 472 + 1
+ | r = 945
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 472 + 1}
+ | r = 432
+ | c = {{EulerPhiLink|945}}
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 472 + 1}
+ | o = <
+ | r = \map \phi {2 \times 472}
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 2 \times 487
+ | r = 974
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 487}
+ | r = 486
+ | c = {{EulerPhiLink|974}}
+}}
+{{eqn | l = 2 \times 487 + 1
+ | r = 975
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 487 + 1}
+ | r = 480
+ | c = {{EulerPhiLink|975}}
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 487 + 1}
+ | o = <
+ | r = \map \phi {2 \times 487}
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 2 \times 577
+ | r = 1154
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 577}
+ | r = 576
+ | c = {{EulerPhiLink|1154}}
+}}
+{{eqn | l = 2 \times 577 + 1
+ | r = 1155
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 577 + 1}
+ | r = 480
+ | c = {{EulerPhiLink|1155}}
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 577 + 1}
+ | o = <
+ | r = \map \phi {2 \times 577}
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 2 \times 682
+ | r = 1364
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 682}
+ | r = 600
+ | c = {{EulerPhiLink|1364}}
+}}
+{{eqn | l = 2 \times 682 + 1
+ | r = 1365
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 682 + 1}
+ | r = 576
+ | c = {{EulerPhiLink|1365}}
+}}
+{{eqn | ll= \leadsto
+ | l = \map \phi {2 \times 682 + 1}
+ | o = <
+ | r = \map \phi {2 \times 682}
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|that these are the smallest such. The philosophical significance of this result is not immediately clear.}}
+\end{proof}<|endoftext|>
+\section{Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares}
+Tags: Sums of Squares, 157
+
+\begin{theorem}
+The largest [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive integer]] that cannot be expressed as the [[Definition:Integer Addition|sum]] of exactly $4$ [[Definition:Distinct Elements|distinct]] non-zero [[Definition:Square Number|square numbers]] with [[Definition:Greatest Common Divisor/Integers/General Definition|greatest common divisor]] $1$ is $157$.
+The full [[Definition:Integer Sequence|sequence]] of such [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive integers]] which cannot be so expressed is:
+:$1, \ldots, 37, 41, 43, 45, 47, 49, 53, 55, 59, 61, 67, 69, 73, 77, 83, 89, 97, 101, 103, 115, 157$
+where the [[Definition:Integer Sequence|sequence]] contains all [[Definition:Odd Integer|odd integers]] between $1$ and $37$.
+This [[Definition:Integer Sequence|sequence]] appears not to be documented on the {{OEISLink}}.
+\end{theorem}
+
+\begin{proof}
+The statement of the result was taken from the paper cited below by {{AuthorRef|Paul T. Bateman}} and others:
+:''The largest [[Definition:Odd Integer|odd integer]] not expressible as a [[Definition:Integer Addition|sum]] of $4$ [[Definition:Distinct Elements|distinct]] non-zero [[Definition:Square Number|squares]] with [[Definition:Greatest Common Divisor/Integers/General Definition|greatest common divisor]] $1$ is $157$.''
+By a brute force exercise, we assemble all the [[Definition:Set|sets]] of $4$ [[Definition:Distinct Elements|distinct]] non-zero [[Definition:Integer|integers]] and calculate the [[Definition:Integer Addition|sum]] of their [[Definition:Square Number|squares]] such that those [[Definition:Integer Addition|sums]] are less than $157$.
+It is noted in passing that all such [[Definition:Set|sets]] have a [[Definition:Greatest Common Divisor/Integers/General Definition|greatest common divisor]] of $1$.
+Thus we have:
+{{begin-eqn}}
+{{eqn | l = 39
+ | r = 1 + 4 + 9 + 25
+ | rr= = 1^2 + 2^2 + 3^2 + 5^2
+ | c =
+}}
+{{eqn | l = 51
+ | r = 1 + 9 + 16 + 25
+ | rr= = 1^2 + 3^2 + 4^2 + 5^2
+ | c =
+}}
+{{eqn | l = 57
+ | r = 1 + 4 + 16 + 36
+ | rr= = 1^2 + 2^2 + 4^2 + 6^2
+ | c =
+}}
+{{eqn | l = 63
+ | r = 1 + 4 + 9 + 49
+ | rr= = 1^2 + 2^2 + 3^2 + 7^2
+ | c =
+}}
+{{eqn | l = 65
+ | r = 4 + 9 + 16 + 36
+ | rr= = 2^2 + 3^2 + 4^2 + 6^2
+ | c =
+}}
+{{eqn | l = 71
+ | r = 1 + 9 + 25 + 36
+ | rr= = 1^2 + 3^2 + 5^2 + 6^2
+ | c =
+}}
+{{eqn | l = 75
+ | r = 1 + 9 + 16 + 49
+ | rr= = 1^2 + 3^2 + 4^2 + 7^2
+ | c =
+}}
+{{eqn | l = 79
+ | r = 1 + 2 + 25 + 49
+ | rr= = 1^2 + 2^2 + 5^2 + 7^2
+ | c =
+}}
+{{eqn | l = 81
+ | r = 4 + 16 + 25 + 36
+ | rr= = 2^2 + 4^2 + 5^2 + 6^2
+ | c =
+}}
+{{eqn | l = 85
+ | r = 1 + 4 + 16 + 64
+ | rr= = 1^2 + 2^2 + 4^2 + 8^2
+ | c =
+}}
+{{eqn | l = 87
+ | r = 4 + 9 + 25 + 49
+ | rr= = 2^2 + 3^2 + 5^2 + 7^2
+ | c =
+}}
+{{eqn | l = 91
+ | r = 1 + 16 + 25 + 49
+ | rr= = 1^2 + 4^2 + 5^2 + 7^2
+ | c =
+}}
+{{eqn | l = 93
+ | r = 4 + 9 + 16 + 64
+ | rr= = 2^2 + 3^2 + 4^2 + 8^2
+ | c =
+}}
+{{eqn | l = 95
+ | r = 1 + 9 + 36 + 49
+ | rr= = 1^2 + 3^2 + 6^2 + 7^2
+ | c =
+}}
+{{eqn | l = 99
+ | r = 1 + 9 + 25 + 64
+ | rr= = 1^2 + 3^2 + 5^2 + 8^2
+ | c =
+}}
+{{eqn | r = 9 + 16 + 25 + 49
+ | rr= = 3^2 + 4^2 + 5^2 + 7^2
+ | c =
+}}
+{{eqn | l = 105
+ | r = 4 + 16 + 36 + 47
+ | rr= = 2^2 + 4^2 + 6^2 + 7^2
+ | c =
+}}
+{{eqn | l = 107
+ | r = 1 + 9 + 16 + 81
+ | rr= = 1^2 + 3^2 + 4^2 + 9^2
+ | c =
+}}
+{{eqn | l = 109
+ | r = 4 + 16 + 25 + 64
+ | rr= = 2^2 + 4^2 + 5^2 + 8^2
+ | c =
+}}
+{{eqn | l = 111
+ | r = 1 + 25 + 36 + 49
+ | rr= = 1^2 + 5^2 + 6^2 + 7^2
+ | c =
+}}
+{{eqn | r = 1 + 4 + 25 + 81
+ | rr= = 1^2 + 2^2 + 5^2 + 9^2
+ | c =
+}}
+{{eqn | l = 113
+ | r = 4 + 9 + 36 + 64
+ | rr= = 2^2 + 3^2 + 6^2 + 8^2
+ | c =
+}}
+{{eqn | l = 117
+ | r = 1 + 16 + 36 + 64
+ | rr= = 1^2 + 4^2 + 6^2 + 8^2
+ | c =
+}}
+{{eqn | l = 119
+ | r = 4 + 9 + 25 + 81
+ | rr= = 2^2 + 3^2 + 5^2 + 9^2
+ | c =
+}}
+{{eqn | r = 9 + 25 + 36 + 49
+ | rr= = 3^2 + 5^2 + 6^2 + 7^2
+ | c =
+}}
+{{eqn | l = 121
+ | r = 1 + 4 + 16 + 100
+ | rr= = 1^2 + 2^2 + 4^2 + 10^2
+ | c =
+}}
+{{eqn | l = 123
+ | r = 1 + 9 + 49 + 64
+ | rr= = 1^2 + 3^2 + 7^2 + 8^2
+ | c =
+}}
+{{eqn | r = 1 + 16 + 25 + 81
+ | rr= = 1^2 + 4^2 + 5^2 + 9^2
+ | c =
+}}
+{{eqn | l = 125
+ | r = 9 + 16 + 36 + 64
+ | rr= = 3^2 + 4^2 + 6^2 + 8^2
+ | c =
+}}
+{{eqn | l = 127
+ | r = 1 + 9 + 36 + 81
+ | rr= = 1^2 + 3^2 + 6^2 + 9^2
+ | c =
+}}
+{{eqn | l = 129
+ | r = 4 + 9 + 16 + 100
+ | rr= = 2^2 + 3^2 + 4^2 + 10^2
+ | c =
+}}
+{{eqn | r = 4 + 25 + 36 + 64
+ | rr= = 2^2 + 5^2 + 6^2 + 8^2
+ | c =
+}}
+{{eqn | l = 131
+ | r = 9 + 16 + 25 + 81
+ | rr= = 3^2 + 4^2 + 5^2 + 9^2
+ | c =
+}}
+{{eqn | l = 133
+ | r = 4 + 16 + 49 + 64
+ | rr= = 2^2 + 4^2 + 7^2 + 8^2
+ | c =
+}}
+{{eqn | l = 135
+ | r = 1 + 4 + 9 + 121
+ | rr= = 1^2 + 2^2 + 3^2 + 11^2
+ | c =
+}}
+{{eqn | r = 1 + 4 + 49 + 81
+ | rr= = 1^2 + 2^2 + 7^2 + 9^2
+ | c =
+}}
+{{eqn | r = 1 + 9 + 25 + 100
+ | rr= = 1^2 + 3^2 + 5^2 + 10^2
+ | c =
+}}
+{{eqn | l = 137
+ | r = 4 + 16 + 36 + 81
+ | rr= = 2^2 + 4^2 + 6^2 + 9^2
+ | c =
+}}
+{{eqn | l = 139
+ | r = 1 + 25 + 49 + 64
+ | rr= = 1^2 + 5^2 + 7^2 + 8^2
+ | c =
+}}
+{{eqn | l = 141
+ | r = 16 + 25 + 36 + 64
+ | rr= = 4^2 + 5^2 + 6^2 + 8^2
+ | c =
+}}
+{{eqn | l = 143
+ | r = 1 + 25 + 36 + 81
+ | rr= = 1^2 + 5^2 + 6^2 + 9^2
+ | c =
+}}
+{{eqn | r = 4 + 9 + 49 + 81
+ | rr= = 2^2 + 3^2 + 7^2 + 9^2
+ | c =
+}}
+{{eqn | l = 145
+ | r = 4 + 16 + 25 + 100
+ | rr= = 2^2 + 4^2 + 5^2 + 10^2
+ | c =
+}}
+{{eqn | l = 147
+ | r = 1 + 16 + 49 + 81
+ | rr= = 1^2 + 4^2 + 7^2 + 9^2
+ | c =
+}}
+{{eqn | r = 1 + 9 + 16 + 121
+ | rr= = 1^2 + 3^2 + 4^2 + 11^2
+ | c =
+}}
+{{eqn | r = 9 + 25 + 49 + 64
+ | rr= = 3^2 + 5^2 + 7^2 + 8^2
+ | c =
+}}
+{{eqn | l = 149
+ | r = 1 + 4 + 25 + 121
+ | rr= = 1^2 + 2^2 + 5^2 + 11^2
+ | c =
+}}
+{{eqn | r = 4 + 9 + 36 + 100
+ | rr= = 2^2 + 3^2 + 6^2 + 10^2
+ | c =
+}}
+{{eqn | l = 151
+ | r = 9 + 25 + 36 + 81
+ | rr= = 3^2 + 5^2 + 6^2 + 9^2
+ | c =
+}}
+{{eqn | l = 153
+ | r = 4 + 36 + 49 + 64
+ | rr= = 2^2 + 6^2 + 7^2 + 8^2
+ | c =
+}}
+{{eqn | l = 155
+ | r = 9 + 16 + 49 + 81
+ | rr= = 3^2 + 4^2 + 7^2 + 9^2
+ | c =
+}}
+{{end-eqn}}
+It remains to be demonstrated that the above [[Definition:Integer Sequence|sequence]] is indeed complete.
+The original article by {{AuthorRef|Franz Halter-Koch}} states:
+:{{SmallCaps|Satz}} $3$. ''Genau dann is $n \in N$ nicht Summe von vier verschiedenen positiven (im Falle $n \not \equiv 0 \bmod 8$ auch teilerfremden) Quadraten, wenn entwieder''
+::''$n = 4^h a \quad$ mit $h \ge 0$ und $a \in \set {1, 2, \ldots, 19, 22, 23, 25, 26, 27, 31, 33, 34, 37, 38, 42, 43, 47, 55, 58, 67, 73, 82, 97, 103}$''
+:''oder''
+::''$n \in \set {21, 29, 35, 41, 45, 49, 53, 59, 61, 69, 77, 83, 89, 101, 115, 157}$
+That is, in English:
+:''{{SmallCaps|Statement}} $3$. $n \in N$ is not the sum of four different positive squares (in the case where $n \not \equiv 0 \bmod 8$ also non-prime) if and only if:
+::''$n = 4^h a \quad$ where $h \ge 0$ and $a \in \set {1, 2, \ldots, 19, 22, 23, 25, 26, 27, 31, 33, 34, 37, 38, 42, 43, 47, 55, 58, 67, 73, 82, 97, 103}$
+:''or''
+::''$n \in \set {21, 29, 35, 41, 45, 49, 53, 59, 61, 69, 77, 83, 89, 101, 115, 157}$
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Odd Numbers Not Expressible as Sum of 5 Distinct Non-Zero Coprime Squares}
+Tags: Sums of Squares, 245
+
+\begin{theorem}
+The largest [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive integer]] that cannot be expressed as the [[Definition:Integer Addition|sum]] of exactly $5$ non-zero [[Definition:Square Number|square numbers]] all of which are [[Definition:Coprime Integers|coprime]] is $245$.
+{{expand|The full list of those numbers is to be investigated.}}
+\end{theorem}<|endoftext|>
+\section{159 is not Expressible as Sum of Fewer than 19 Fourth Powers}
+Tags: Fourth Powers, Hilbert-Waring Theorem, 159
+
+\begin{theorem}
+:$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$
+\end{theorem}
+
+\begin{proof}
+We have:
+:$4^4 = 256 > 159$
+:$3^4 = 81$
+:$2^4 = 16$
+:$1^4 = 1$
+Let us attempt to construct an expression of $159$ as the [[Definition:Integer Addition|sum]] of fewer than $19$ [[Definition:Fourth Power|fourth powers]]:
+If no $3^4$ is used in our [[Definition:Integer Addition|sum]], the [[Definition:Integer Addition|sum]] consists only of $2^4$ and $1^4$.
+Using $2^4$ is more efficient than using $1^4$, since $2^4$ can replace $16 \times 1^4$.
+So we have:
+:$159 = 9 \times 2^4 + 15 \times 1^4$
+which uses $24$ [[Definition:Fourth Power|fourth powers]].
+If $3^4$ is used in our [[Definition:Integer Addition|sum]], the most efficient way is demonstrated above:
+:$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$
+which uses $19$ [[Definition:Fourth Power|fourth powers]].
+We cannot use more than $2$ instances of $3^4$, as $2 \times 3^4 = 162 > 159$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers not Expressible as Sum of Distinct Pentagonal Numbers}
+Tags: Pentagonal Numbers
+
+\begin{theorem}
+The [[Definition:Positive Integer|positive integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct|distinct]] [[Definition:Pentagonal Number|pentagonal numbers]] are:
+:$2, 3, 4, 7, 8, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30,$
+:$31, 32, 33, 37, 38, 42, 43, 44, 45, 46, 49, 50, 54, 55, 59, 60, 61, 65,$
+:$66, 67, 72, 77, 80, 81, 84, 89, 94, 95, 96, 100, 101, 102, 107, 112, 116,$
+:$124, 136, 137, 141, 142, 147, 159$
+\end{theorem}
+
+\begin{proof}
+It will be proved that the largest [[Definition:Integer|integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Pentagonal Number|pentagonal numbers]] is $159$.
+The remaining [[Definition:Integer|integers]] in the [[Definition:Integer Sequence|sequence]] can be identified by inspection.
+We prove this using a variant of [[Second Principle of Mathematical Induction]].
+Let $\map P n$ be the proposition:
+:$n$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Pentagonal Number|pentagonal numbers]].
+=== Basis for the induction ===
+We verify the result up to $P_{19} = 532$.
+Then we conclude all $n$ with $160 \le n \le 532$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Pentagonal Number|pentagonal numbers]].
+So $\map P n$ is true for all $160 \le n \le 532$.
+This is the [[Definition:Basis for the Induction|basis for the induction]].
+=== Induction Hypothesis ===
+Suppose for some $k > 532$, $\map P j$ is true for all $160 \le j < k$.
+That is, every [[Definition:Integer|integer]] between $160$ and $k - 1$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Pentagonal Number|pentagonal numbers]].
+This is the [[Definition:Induction Hypothesis|induction hypothesis]].
+=== Induction Step ===
+This is the [[Definition:Induction Step|induction step]]:
+We find the largest [[Definition:Integer|integer]] $i$ such that $P_i < k \le P_{i + 1}$.
+We show that $k - P_{i - 4}$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Pentagonal Number|pentagonal numbers]], and the [[Definition:Integer Addition|sum]] does not involve $P_{i - 4}$.
+Then $k$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Pentagonal Number|pentagonal numbers]].
+Since $k > 160 = P_{19}$, we must have $i \ge 19$.
+Hence $k > k - P_{i - 4} \ge P_i - P_{i - 4} = \paren {3 i - 2} + \paren {3 i - 5} + \paren {3 i - 8} + \paren {3 i - 11} \ge 55 + 52 + 49 + 47 = 202 > 160$.
+Thus $k - P_{i - 4}$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Pentagonal Number|pentagonal numbers]] by Induction Hypothesis.
+A sufficient condition such that the [[Definition:Integer Addition|sum]] does not involve $P_{i - 4}$ is $P_{i - 4} > k - P_{i - 4}$.
+We have:
+{{begin-eqn}}
+{{eqn | l = P_{i - 4} - \paren {k - P_{i - 4} }
+ | o = \ge
+ | r = 2 P_{i - 4} - P_{i + 1}
+ | c = From $k \le P_{i + 1}$
+}}
+{{eqn | r = 2 \paren {\frac {\paren {i - 4} \paren {3 i - 13} } 2} - \frac {\paren {i + 1} \paren {3 i + 2} } 2
+ | c = [[Closed Form for Pentagonal Numbers]]
+}}
+{{eqn | r = \frac 1 2 \paren {6 i^2 - 50 i + 104 - 3 i^2 - 5 i - 2}
+}}
+{{eqn | r = \frac 1 2 \paren {3 i^2 - 55 i + 102}
+}}
+{{eqn | r = \frac 3 2 \paren {i \paren {i - \frac {55} 3} + 34}
+}}
+{{eqn | o = \ge
+ | r = \frac 3 2 \paren {34}
+ | c = From $i \ge 19$
+}}
+{{eqn | r = 51
+}}
+{{eqn | o = >
+ | r = 0
+}}
+{{end-eqn}}
+Therefore we have $P_{i - 4} > k - P_{i - 4}$.
+Hence $\map P k$ is true.
+By the [[Second Principle of Mathematical Induction]], $\map P n$ is true for all $n \ge 160$.
+Thus every [[Definition:Integer|integer]] greater than $159$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Pentagonal Number|pentagonal numbers]].
+{{Proofread}}
+\end{proof}<|endoftext|>
+\section{Sum of Distinct Primes of form 6n-1}
+Tags: Prime Numbers, 161
+
+\begin{theorem}
+$161$ is the largest [[Definition:Integer|integer]] that cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct|distinct]] [[Definition:Prime Number|primes]] of the form $6 n - 1$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|All the numbers that cannot be so expressed need to be determined.}}
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Polynomial Ring in One Variable}
+Tags: Polynomial Theory
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+The following definitions of [[Definition:Polynomial Ring|polynomial ring]] are equivalent in the following sense:
+: For every two constructions, there exists an [[Definition:R-isomorphism|$R$-isomorphism]] which sends [[Definition:Indeterminate|indeterminates]] to indeterminates.
+{{explain|this statement has to be made more precise}}
+=== [[Definition:Polynomial Ring/Sequences|Definition 1: As a Ring of Sequences]] ===
+{{:Definition:Polynomial Ring/Sequences}}
+=== [[Definition:Polynomial Ring/Monoid Ring on Natural Numbers|Definition 2: As a Monoid Ring on the Natural Numbers]] ===
+{{:Definition:Polynomial Ring/Monoid Ring on Natural Numbers}}
+\end{theorem}
+
+\begin{proof}
+Use [[Polynomial Ring of Sequences Satisfies Universal Property]]
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Smallest Triplet of Integers whose Product with Tau Value are Equal}
+Tags: Tau Function, 168, 192, 224
+
+\begin{theorem}
+Let $\map \tau n$ denote the [[Definition:Tau Function|$\tau$ function]]: the number of [[Definition:Divisor of Integer|divisors]] of $n$.
+The smallest [[Definition:Set|set]] of $3$ [[Definition:Positive Integer|integers]] $T$ such that $m \, \map \tau m$ is equal for each $m \in T$ is:
+:$\set {168, 192, 224}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 168 \times \map \tau {168}
+ | r = 168 \times 16
+ | c = {{TauLink|168}}
+}}
+{{eqn | r = \paren {2^3 \times 3 \times 7} \times 2^4
+ | c =
+}}
+{{eqn | r = 2^7 \times 3 \times 7
+ | c =
+}}
+{{eqn | r = 2688
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 192 \times \map \tau {192}
+ | r = 192 \times 14
+ | c = {{TauLink|192}}
+}}
+{{eqn | r = \paren {2^6 \times 3} \times \paren {2 \times 7}
+ | c =
+}}
+{{eqn | r = 2^7 \times 3 \times 7
+ | c =
+}}
+{{eqn | r = 2688
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 224 \times \map \tau {224}
+ | r = 224 \times 12
+ | c = {{TauLink|224}}
+}}
+{{eqn | r = \paren {2^5 \times 7} \times \paren {2^2 \times 3}
+ | c =
+}}
+{{eqn | r = 2^7 \times 3 \times 7
+ | c =
+}}
+{{eqn | r = 2688
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|This needs to be identified as the smallest such.}}
+\end{proof}<|endoftext|>
+\section{General Variation of Integral Functional/Dependent on n Variables}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $\mathbf x$ be an [[Definition:Dimension|$n$-dimensional]] [[Definition:Vector|vector]].
+Let $u = \map u {\mathbf x}$ be a [[Definition:Real-Valued Function|real-valued function]].
+Let $J$ be a [[Definition:Real Functional|functional]] such that:
+:$\ds J \sqbrk u = \int_R \map F {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } \rd x_1 \dotsm \rd x_n$
+Let $\mathbf x^*, u^*$ be defined by the following transformations $\paren \star$:
+{{begin-eqn}}
+{{eqn | l = \mathbf x^*
+ | r = \map {\boldsymbol \Phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; \epsilon} = \mathbf x + \epsilon \map {\boldsymbol \phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } + \map \OO {\epsilon^2}
+}}
+{{eqn | l = u^*
+ | r = \map \Psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; \epsilon} = \mathbf x + \epsilon \map \psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x } } + \map \OO {\epsilon^2}
+}}
+{{end-eqn}}
+where:
+:$\boldsymbol \Phi$, $\Psi$ are [[Definition:Differentiable Mapping|differentiable]] {{WRT|Differentiation}} $\epsilon$
+and:
+{{begin-eqn}}
+{{eqn | l = \map {\boldsymbol \Phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; 0}
+ | r = \mathbf x
+}}
+{{eqn | l = \map \Psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; 0}
+ | r = u
+}}
+{{eqn | l = \map {\boldsymbol \phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} }
+ | r = \valueat {\dfrac {\partial \boldsymbol \Phi} {\partial \epsilon} } {\epsilon \mathop = 0}
+}}
+{{eqn | l = \map \psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} }
+ | r = \valueat {\dfrac {\partial \Psi} {\partial \epsilon} } {\epsilon \mathop = 0}
+}}
+{{end-eqn}}
+Then the [[Definition:Differential of Functional|variation]] of the [[Definition:Real Functional|functional]] $J$ due to the original [[Definition:Mapping|mapping]] being transformed by $\paren \star$ reads:
+{{explain|not clear what "aforementioned transformation" refers to. Best to identify it with a label and refer to that label.}}
+:$\ds \delta J = \epsilon \int_R \paren {F_u - \dfrac {\partial F_{u_{\mathbf x} } } {\partial \mathbf x} } \overline \psi \rd x_1 \dotsm \rd x_n + \epsilon \int_R \map {\dfrac {\partial} {\partial \mathbf x} } {F_{u_x} \overline {\boldsymbol \psi} + F \boldsymbol \phi} \rd x_1 \dotsm \rd x_n$
+where:
+:$\overline \psi = \psi - u_{\mathbf x} \boldsymbol \phi$
+\end{theorem}
+
+\begin{proof}
+By definition:
+{{begin-eqn}}
+{{eqn | l = \Delta J
+ | r = J \sqbrk {\map {u^*} {x^*} } - J \sqbrk {\map u x}
+}}
+{{eqn | r = \int_{R^*} \map F {\mathbf x^*, u^*, \frac {\partial u^*} {\partial \mathbf x^*} } \rd x_1^* \dotsm \rd x_n^* - \int_R \map F {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } \rd x_1 \dotsm \rd x_n
+}}
+{{eqn | r = \int_R \paren {\map F {\mathbf x^*, u^*, \dfrac {\partial u^*} {\partial \mathbf x^*} } \det \paren {J_{\mathbf x^*} } - \map F {\mathbf x, u, \frac {\partial u} {\partial \mathbf x} } } \rd x_1 \dotsm \rd x_n
+ | c = Volume element transformation
+}}
+{{end-eqn}}
+From the definition of $\mathbf x^*$ a [[Definition:Jacobian Matrix|Jacobian matrix]] can be constructed:
+:$\ds \dfrac {\partial x_i^*} {\partial x_j} = \delta_i^j + \epsilon \dfrac {\partial \phi_i} {\partial x_j} + \map \OO {\epsilon^2}$
+By [[Taylor's Theorem|Taylor's theorem]], The corresponding [[Definition:Jacobian Determinant|Jacobian determinant]] can be expanded {{WRT}} $\epsilon$.
+The $\epsilon^0$ term is obtained from the [[Definition:Diagonal|diagonal]] [[Definition:Product (Algebra)|product]] by opening brackets and collecting $\epsilon$-free terms.
+The $\epsilon^1$ term is obtained from the same [[Definition:Diagonal|diagonal]] [[Definition:Product (Algebra)|product]] by choosing non-constant term once.
+:$\ds \det \paren {J_{\mathbf x^*} } = 1 + \epsilon \dfrac {\partial \boldsymbol \phi} {\partial \mathbf x} + \map \OO {\epsilon^2}$
+{{explain|make explanation more precise}}
+:$\ds \Delta J = \int_R \paren {\map F {\mathbf x^*, u^* \dfrac {\partial u^*} {\partial \mathbf x^*} } \paren {1 + \epsilon \dfrac {\partial \boldsymbol \phi} {\partial \mathbf x} } - \map F {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } } \rd x_1 \dotsm \rd x_n + \map \OO {\epsilon^2}$
+By definition, the [[Definition:Differentiable Functional|principal part]] is:
+:$\ds \delta J = \int_R \paren {F_{\mathbf x} \delta \mathbf x + F_u \delta u + F_{u_{\mathbf x} } \delta u_{\mathbf x} + \epsilon F \dfrac {\partial \boldsymbol \phi} {\partial \mathbf x} } \rd x_1 \dotsm \rd x_n$
+{{explain|If use of square brackets has a particular significance and meaning here, explain it. If it is just a variant on conventional parenthesis, then best if $\paren \cdot$ is used instead. It's ultimately more clear to use one style of parenthesis than mix and match various different types.}}
+Introduce the following [[Definition:Difference|differences]]:
+{{begin-eqn}}
+{{eqn | l = \Delta u
+ | r = \map {u^*} {x^*} - \map u x
+}}
+{{eqn | l = \overline {\Delta u}
+ | r = \map {u^*} x - \map u x
+}}
+{{eqn | l = \Delta x
+ | r = x^*-x
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \overline {\Delta u}
+ | r = \epsilon \overline \psi + \map \OO {\epsilon^2}
+}}
+{{eqn | l = \overline {\delta u}
+ | r = \epsilon \overline \psi
+}}
+{{eqn | l = \Delta u
+ | r = \dfrac {\partial u} {\partial \mathbf x} \delta \mathbf x + \overline {\delta u} + \map \OO {\epsilon^2}
+}}
+{{eqn | l = \delta u
+ | r = \dfrac {\partial u} {\partial \mathbf x} \delta \mathbf x + \overline {\delta u}
+}}
+{{eqn | l = \paren {\delta u}_{x_i}
+ | r = \paren {\overline {\delta u} }_{x_i} + u_{\mathbf x x_i} {\delta \mathbf x}_{x_i}
+}}
+{{eqn | l = \Delta \mathbf x
+ | r = \epsilon \boldsymbol \phi + \map \OO {\epsilon^2}
+}}
+{{eqn | l = \delta \mathbf x
+ | r = \epsilon \boldsymbol \phi
+}}
+{{end-eqn}}
+Here $\delta$ [[Definition:Difference|difference]] stands for principal part, which is [[Definition:Constant|constant]] or linear {{WRT}} $\epsilon$.
+Thus, the [[Definition:Differentiable Functional|variation]] of $J$ reads:
+:$\ds \delta J = \int_R \paren {F_{\mathbf x} \delta \mathbf x + F_u \overline {\delta u} + F_u u_{\mathbf x} \delta \mathbf x + F_{u_{\mathbf x} } {\overline {\delta u} }_{\mathbf x} + \sum_{i, j \mathop = 1}^n F_{u_{x_i} } u_{x_i x_j} \delta x_j + \map F {\delta \mathbf x}_{\mathbf x} } \rd x_1 \dotsm \rd x_n$
+A few terms can be rewritten using:
+:$\ds \dfrac {\partial} {\partial \mathbf x} \paren {F \delta \mathbf x} = F_{\mathbf x} \delta \mathbf x + \map F {\overline {\delta \mathbf x} }_{\mathbf x} + F_u u_{\mathbf x} \delta \mathbf x + \sum_{i, j\mathop = 1}^n F_{u_{x_i} } u_{x_i x_j} \delta x_j$
+and:
+:$\ds \map {F_{u_{\mathbf x} } } {\overline{\delta u} }_{\mathbf x} = \dfrac {\partial} {\partial \mathbf x} {F_{u_{\mathbf x} } \overline {\delta u} } - \dfrac {\partial F_{u_{\mathbf x} } } {\partial \mathbf x} \delta u$
+Substitution of previous results into [[Definition:Differentiable Functional|variation]] of $J$ leads to:
+:$\ds \delta J = \int_R \paren {F_u - \dfrac {\partial F_{u_{\mathbf x} } } {\partial \mathbf x} } \overline {\delta u} \rd x_1 \dotsm \rd x_n + \int_R \map {\dfrac \partial {\partial \mathbf x} } {F_{u_{\mathbf x} } \overline {\delta u} + F \delta \mathbf x} \rd x_1 \dotsm \rd x_n$
+Substitute expressions for $\overline {\delta u}$ and $\delta\mathbf x$ to obtain the desired result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Pairs of Integers whose Product with Tau Value are Equal}
+Tags: Tau Function
+
+\begin{theorem}
+Let $\tau \left({n}\right)$ denote the [[Definition:Tau Function|$\tau$ function]]: the number of [[Definition:Divisor of Integer|divisors]] of $n$.
+The following [[Definition:Doubleton|pairs]] of [[Definition:Positive Integer|integers]] $T$ have the property that $m \tau \left({m}\right)$ is equal for each $m \in T$:
+:$\left\{ {18, 27}\right\}$
+:$\left\{ {24, 32}\right\}$
+:$\left\{ {56, 64}\right\}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 18 \times \tau \left({18}\right)
+ | r = 18 \times 6
+ | c = [[Tau Function of 18]]
+}}
+{{eqn | r = \left({2 \times 3^2}\right) \times \left({2 \times 3}\right)
+ | c =
+}}
+{{eqn | r = 2^2 \times 3^3
+ | c =
+}}
+{{eqn | r = 108
+ | c =
+}}
+{{eqn | l = 27 \times \tau \left({27}\right)
+ | r = 27 \times 4
+ | c = [[Tau Function of 27]]
+}}
+{{eqn | r = 2^2 \times 3^3
+ | c =
+}}
+{{eqn | r = 108
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 24 \times \tau \left({24}\right)
+ | r = 24 \times 8
+ | c = [[Tau Function of 24]]
+}}
+{{eqn | r = \left({2^3 \times 3}\right) \times 2^3
+ | c =
+}}
+{{eqn | r = 2^6 \times 3
+ | c =
+}}
+{{eqn | r = 192
+ | c =
+}}
+{{eqn | l = 32 \times \tau \left({32}\right)
+ | r = 32 \times 6
+ | c = [[Tau Function of 32]]
+}}
+{{eqn | r = 2^5 \times \left({2 \times 3}\right)
+ | c =
+}}
+{{eqn | r = 192
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 56 \times \tau \left({56}\right)
+ | r = 56 \times 8
+ | c = [[Tau Function of 56]]
+}}
+{{eqn | r = \left({2^3 \times 7}\right) \times 2^3
+ | c =
+}}
+{{eqn | r = 2^6 \times 7
+ | c =
+}}
+{{eqn | r = 448
+ | c =
+}}
+{{eqn | l = 64 \times \tau \left({64}\right)
+ | r = 64 \times 7
+ | c = [[Tau Function of 64]]
+}}
+{{eqn | r = 2^6 \times 7
+ | c =
+}}
+{{eqn | r = 448
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sequence of Record Peaks in Values of Sigma}
+Tags: Highly Abundant Numbers
+
+\begin{theorem}
+Let $d: \Z_{>0} \to \Z_{>0}$ be the [[Definition:Mapping|mapping]] defined as:
+:$d \left({n}\right) = \sigma \left({n}\right) - \sigma \left({n'}\right)$
+where:
+:$n$ denotes a [[Definition:Highly Abundant Number|highly abundant number]]
+:$n'$ denotes the previous [[Definition:Highly Abundant Number|highly abundant number]].
+The following $n \in \Z_{>0}$ have the property that they have a higher value of $d \left({n}\right)$ than any smaller $n$:
+:$\forall m \in \Z_{>0}: m < n \implies \sigma \left({m}\right) < \sigma \left({n}\right)$
+That is, they are the peak [[Definition:Sigma Function|$\sigma$ values]] which exceed the previous peak by a higher number than any previous peak.
+That is, they are [[Definition:Highly Abundant Number|highly abundant number]] which have [[Definition:Sigma Function|$\sigma$ values]] whose difference with that of the [[Definition:Sigma Function|$\sigma$ (sigma) value]] of the previous [[Definition:Highly Abundant Number|highly abundant numbers]] is greater than that with the previous record difference.
+:{| border="1"
+|-
+! align="right" style = "padding: 2px 10px" | $n$
+! align="right" style = "padding: 2px 10px" | $n'$
+! align="right" style = "padding: 2px 10px" | $\sigma \left({n}\right)$
+! align="right" style = "padding: 2px 10px" | $\sigma \left({n'}\right)$
+! style = "padding: 2px 10px" | $d \left({n}\right)$
+|-
+| align="right" style = "padding: 2px 10px" | $2$
+| align="right" style = "padding: 2px 10px" | $1$
+| align="right" style = "padding: 2px 10px" | $3$
+| align="right" style = "padding: 2px 10px" | $1$
+| align="right" style = "padding: 2px 10px" | $2$
+|-
+| align="right" style = "padding: 2px 10px" | $4$
+| align="right" style = "padding: 2px 10px" | $3$
+| align="right" style = "padding: 2px 10px" | $7$
+| align="right" style = "padding: 2px 10px" | $4$
+| align="right" style = "padding: 2px 10px" | $3$
+|-
+| align="right" style = "padding: 2px 10px" | $6$
+| align="right" style = "padding: 2px 10px" | $4$
+| align="right" style = "padding: 2px 10px" | $12$
+| align="right" style = "padding: 2px 10px" | $7$
+| align="right" style = "padding: 2px 10px" | $5$
+|-
+| align="right" style = "padding: 2px 10px" | $12$
+| align="right" style = "padding: 2px 10px" | $10$
+| align="right" style = "padding: 2px 10px" | $28$
+| align="right" style = "padding: 2px 10px" | $18$
+| align="right" style = "padding: 2px 10px" | $10$
+|-
+| align="right" style = "padding: 2px 10px" | $24$
+| align="right" style = "padding: 2px 10px" | $20$
+| align="right" style = "padding: 2px 10px" | $60$
+| align="right" style = "padding: 2px 10px" | $42$
+| align="right" style = "padding: 2px 10px" | $18$
+|-
+| align="right" style = "padding: 2px 10px" | $36$
+| align="right" style = "padding: 2px 10px" | $30$
+| align="right" style = "padding: 2px 10px" | $91$
+| align="right" style = "padding: 2px 10px" | $72$
+| align="right" style = "padding: 2px 10px" | $19$
+|-
+| align="right" style = "padding: 2px 10px" | $48$
+| align="right" style = "padding: 2px 10px" | $42$
+| align="right" style = "padding: 2px 10px" | $124$
+| align="right" style = "padding: 2px 10px" | $96$
+| align="right" style = "padding: 2px 10px" | $28$
+|-
+| align="right" style = "padding: 2px 10px" | $60$
+| align="right" style = "padding: 2px 10px" | $48$
+| align="right" style = "padding: 2px 10px" | $168$
+| align="right" style = "padding: 2px 10px" | $124$
+| align="right" style = "padding: 2px 10px" | $44$
+|-
+| align="right" style = "padding: 2px 10px" | $120$
+| align="right" style = "padding: 2px 10px" | $108$
+| align="right" style = "padding: 2px 10px" | $360$
+| align="right" style = "padding: 2px 10px" | $280$
+| align="right" style = "padding: 2px 10px" | $80$
+|}
+\end{theorem}<|endoftext|>
+\section{Order Isomorphism forms Galois Connection}
+Tags: Galois Connections, Order Isomorphisms
+
+\begin{theorem}
+Let $L_1 = \left({S_1, \preceq_1}\right)$, $L_2 = \left({S_2, \preceq_2}\right)$ be [[Definition:Ordered Set|ordered sets]].
+Let $f:S_1 \to S_2$ be an [[Definition:Order Isomorphism|order isomorphism]] between $L_1$ and $L_2$.
+Then $\left({f, f^{-1} }\right)$ is a [[Definition:Galois Connection|Galois connection]].
+\end{theorem}
+
+\begin{proof}
+Let $t \in S_2$, $s \in S_1$.
+We will prove that
+:$t \preceq_2 f\left({s}\right) \implies f^{-1}\left({t}\right) \preceq_1 s$
+Assume that
+:$t \preceq_2 f\left({s}\right)$
+By [[Inverse of Order Isomorphism is Order Isomorphism]]:
+:$f^{-1}$ is an [[Definition:Order Isomorphism|order isomorphism]].
+By definition of [[Definition:Order Isomorphism|order isomorphism]]:
+:$f^{-1}$ is an [[Definition:Order Embedding|order embedding]].
+By definition of [[Definition:Order Embedding|order embedding]]:
+:$f^{-1}\left({t}\right) \preceq_1 f^{-1}\left({f\left({s}\right)}\right)$
+Thus by definition of [[Definition:Bijection|bijection]]:
+:$f^{-1}\left({t}\right) \preceq_1 s$
+{{qed|lemma}}
+We will prove that
+:$f^{-1}\left({t}\right) \preceq_1 s \implies t \preceq_2 f\left({s}\right)$
+Assume that
+:$f^{-1}\left({t}\right) \preceq_1 s$
+By definition of [[Definition:Order Embedding|order embedding]]:
+:$f\left({f^{-1}\left({t}\right)}\right) \preceq_2 f\left({s}\right)$
+Thus by definition of [[Definition:Bijection|bijection]]:
+:$t \preceq_2 f\left({s}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers of Primes with at most n Digits}
+Tags: Prime Numbers
+
+\begin{theorem}
+Let $p: \Z_{>0} \to \Z_{>0}$ be the [[Definition:Mapping|mapping]] defined as:
+:$\forall n \in \Z_{>0}: p \left({n}\right) = $ the number of [[Definition:Prime Number|prime numbers]] with no more than $n$ [[Definition:Digit|digits]]
+Then the value of $p$ for the first few numbers is given below:
+:{| border="1"
+|-
+! align="right" style = "padding: 2px 10px" | $n$
+! align="right" style = "padding: 2px 10px" | $p \left({n}\right)$
+|-
+| align="right" style = "padding: 2px 10px" | $1$
+| align="right" style = "padding: 2px 10px" | $4$
+|-
+| align="right" style = "padding: 2px 10px" | $2$
+| align="right" style = "padding: 2px 10px" | $25$
+|-
+| align="right" style = "padding: 2px 10px" | $3$
+| align="right" style = "padding: 2px 10px" | $168$
+|-
+| align="right" style = "padding: 2px 10px" | $4$
+| align="right" style = "padding: 2px 10px" | $1229$
+|-
+| align="right" style = "padding: 2px 10px" | $5$
+| align="right" style = "padding: 2px 10px" | $9592$
+|-
+| align="right" style = "padding: 2px 10px" | $6$
+| align="right" style = "padding: 2px 10px" | $78 \, 498$
+|-
+| align="right" style = "padding: 2px 10px" | $7$
+| align="right" style = "padding: 2px 10px" | $664 \, 579$
+|-
+| align="right" style = "padding: 2px 10px" | $8$
+| align="right" style = "padding: 2px 10px" | $5 \, 761 \, 455$
+|-
+| align="right" style = "padding: 2px 10px" | $9$
+| align="right" style = "padding: 2px 10px" | $50 \, 847 \, 534$
+|-
+| align="right" style = "padding: 2px 10px" | $10$
+| align="right" style = "padding: 2px 10px" | $455 \, 052 \, 511$
+|}
+{{OEIS|A006880}}
+\end{theorem}<|endoftext|>
+\section{Sequence of Square Centered Hexagonal Numbers}
+Tags: Centered Hexagonal Numbers, Square Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Centered Hexagonal Number|centered hexagonal numbers]] which are also [[Definition:Square Number|square]] begins:
+:$1, 169, 32 \, 761, 6 \, 355 \, 441, \ldots$
+{{OEIS|A006051}}
+\end{theorem}
+
+\begin{proof}
+{{:Definition:Centered Hexagonal Number/Sequence}}
+We have that:
+{{begin-eqn}}
+{{eqn | l = 1
+ | r = 1^2
+ | c = {{Defof|Square Number}}
+}}
+{{eqn | r = 3 \times 1 \times \paren {1 - 1} + 1
+ | c = [[Closed Form for Centered Hexagonal Numbers]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 169
+ | r = 13^2
+ | c = {{Defof|Square Number}}
+}}
+{{eqn | r = 3 \times 8 \times \paren {8 - 1} + 1
+ | c = [[Closed Form for Centered Hexagonal Numbers]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 32 \, 761
+ | r = 181^2
+ | c = {{Defof|Square Number}}
+}}
+{{eqn | r = 3 \times 105 \times \paren {105 - 1} + 1
+ | c = [[Closed Form for Centered Hexagonal Numbers]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \, 355 \, 441
+ | r = 2521^2
+ | c = {{Defof|Square Number}}
+}}
+{{eqn | r = 3 \times 1456 \times \paren {1456 - 1} + 1
+ | c = [[Closed Form for Centered Hexagonal Numbers]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{169 as Sum of up to 155 Squares}
+Tags: Sums of Squares, 169
+
+\begin{theorem}
+$169$ can be expressed as the [[Definition:Integer Addition|sum]] of $n$ non-zero [[Definition:Square Number|squares]] for all $n$ from $1$ to $155$.
+\end{theorem}
+
+\begin{proof}
+We note the following:
+{{begin-eqn}}
+{{eqn | l = 169
+ | r = 13^2
+ | c =
+}}
+{{eqn | r = 12^2 + 5^2
+ | c =
+}}
+{{eqn | r = 12^2 + 4^2 + 3^2
+ | c =
+}}
+{{eqn | r = 8^2 + 8^2 + 5^2 + 4^2
+ | c =
+}}
+{{eqn | r = 8^2 + 8^2 + 4^2 + 4^2 + 3^2
+ | c =
+}}
+{{end-eqn}}
+Let $41 \le n \le 155$.
+Let $n \equiv 0 \pmod 3$.
+Then $169$ can be represented as:
+:$169 = 2 \times 3^2 + \left({37 - \dfrac {n - 42} 3}\right) \times 2^2 + \left({\dfrac {4 \left({n - 42}\right)} 3 + 3}\right) \times 1^2$
+Let $n \equiv 1 \pmod 3$.
+Then $169$ can be represented as:
+:$169 = \left({42 - \dfrac {n - 43} 3}\right) \times 2^2 + \left({\dfrac {4 \left({n - 43}\right)} 3 + 1}\right) \times 1^2$
+Let $n \equiv 2 \pmod 3$.
+Then $169$ can be represented as:
+:$169 = 3^2 + \left({40 - \dfrac {n - 41} 3}\right) \times 2^2 + \left({\dfrac {4 \left({n - 41}\right)} 3}\right) \times 1^2$
+Thus in each case $169$ has been expressed in terms of $n$ squares: a combination of varying numbers of $1^2$, $2^2$ and $3^2$.
+For $n > 155$ this construction does not work, because it results in a negative number of instances of $2^2$.
+Similarly, for $n < 41$ this construction does not work, because it results in a negative number of instances of $1^2$.
+For $n < 41$, we have:
+:$169 = 25 + k$
+where $k$ equals one of:
+:$1 \times 12^2$
+:$9 \times 4^2$
+:$16 \times 3^2$
+:$14 \times 3^2 + 3 \times 2^2 + 6 \times 1^2$
+:$10 \times 3^2 + 11 \times 2^2 + 10 \times 1^2$
+That is, $169$ equals $25$ plus $1$, $9$, $16$, $23$ or $31$ other [[Definition:Square Number|squares]].
+But from [[25 as Sum of 4 to 11 Squares]], $25$ can also be written as a [[Definition:Integer Addition|sum]] of $n$ [[Definition:Square Number|squares]] for $4 \le n \le 11$.
+Thus, by using judicious combinations of:
+: [[Definition:Integer Addition|sums]] of [[Definition:Square Number|squares]] totalling $25$
+: [[Definition:Integer Addition|sums]] of [[Definition:Square Number|squares]] totalling $169 - 25$
+we can obtain a representation of $169$ as the [[Definition:Integer Addition|sum]] of $n$ [[Definition:Square Number|squares]] where $5 \le n \le 40$.
+In summary, therefore, there exists a representation of $169$ as the [[Definition:Integer Addition|sum]] of $n$ [[Definition:Square Number|squares]] for $1 \le n \le 155$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Lattice of Power Set is Arithmetic}
+Tags: Continuous Lattices
+
+\begin{theorem}
+Let $X$ be a [[Definition:Set|set]].
+Let $P = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be a [[Definition:Lattice|lattice]] of [[Definition:Power Set|power set]].
+Then $P$ is [[Definition:Arithmetic Ordered Set|arithmetic]].
+\end{theorem}
+
+\begin{proof}
+Define $C = \left({K\left({P}\right), \preceq}\right)$ as an [[Definition:Ordered Subset|ordered subset]] of $P$
+where $K\left({P}\right)$ denotes the [[Definition:Compact Subset of Lattice|compact subset]] of $P$.
+Thus by [[Lattice of Power Set is Algebraic]]:
+:$P$ is [[Definition:Algebraic Ordered Set|algebraic]].
+It remains to prove that
+:$K\left({P}\right)$ is [[Definition:Meet Closed|meet closed]].
+Let $x, y \in K\left({P}\right)$.
+By definition of [[Definition:Compact Subset of Lattice|compact subset]]:
+:$x$ is [[Definition:Compact Element|compact]].
+By [[Element is Finite iff Element is Compact in Lattice of Power Set]]
+:$x$ is [[Definition:Finite Set|finite]].
+By [[Intersection is Subset]]:
+:$x \cap y \subseteq x$
+By [[Subset of Finite Set is Finite]]:
+:$x \cap y$ is [[Definition:Finite Set|finite]].
+By [[Element is Finite iff Element is Compact in Lattice of Power Set]]
+:$x \cap y$ is [[Definition:Compact Element|compact]].
+Thus by definition of [[Definition:Compact Subset of Lattice|compact subset]]:
+:$x \cap y \in K\left({P}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Nth Derivative of Natural Logarithm}
+Tags: Derivatives, Natural Logarithms, Proofs by Induction
+
+\begin{theorem}
+The [[Definition:Higher Derivative|$n$th derivative]] of $\map \ln x$ for $n \ge 1$ is:
+:$\dfrac {\d^n} {\d x^n} \ln x = \dfrac {\paren {n - 1}! \paren {-1}^{n - 1} } {x^n}$
+\end{theorem}
+
+\begin{proof}
+Proof by [[Principle of Mathematical Induction|induction]]:
+For all $n \in \N_{>1}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
+:$\dfrac {\d^n} {\d x^n} \ln x = \dfrac {\paren {n - 1}! \paren {-1}^{n - 1} } {x^n}$
+=== Basis for the Induction ===
+$\map P 1$ is true, as this just says:
+:$\dfrac \d {\d x} \ln x = \dfrac 1 x$
+This follows by [[Derivative of Natural Logarithm Function]]
+This is our [[Definition:Basis for the Induction|basis for the induction]].
+=== Induction Hypothesis ===
+Now we need to show that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
+So this is our [[Definition:Induction Hypothesis|induction hypothesis]]:
+:$\dfrac {\d^k} {\d x^k} \ln x = \dfrac {\paren {k - 1}! \paren {-1}^{k - 1} } {x^k}$
+Then we need to show:
+:$\dfrac {\d^{k + 1} } {\d x^{k + 1} } \ln x = \dfrac {k! \paren {-1}^k} {x^{k + 1} }$
+=== Induction Step ===
+This is our [[Definition:Induction Step|induction step]]:
+{{begin-eqn}}
+{{eqn | l = \frac {\d^{k + 1} } {\d x^{k + 1} } \ln x
+ | r = \map {\frac \d {\d x} } {\frac {\d^k} {\d x^k} \ln x}
+ | c =
+}}
+{{eqn | r = \map {\frac \d {\d x} } {\paren {k - 1}! \paren {-1}^{k - 1} x^{-k} }
+ | c = [[Nth Derivative of Natural Logarithm#Induction Hypothesis|Induction Hypothesis]]
+}}
+{{eqn | r = \paren {k - 1}! \paren {-1}^{k - 1} \map {\frac \d {\d x} } {x^{-k} }
+ | c = [[Derivative of Constant Multiple]]
+}}
+{{eqn | r = \paren {k - 1}! \paren {-1}^{k - 1} \paren {k x^{-k - 1} }
+ | c = [[Power Rule for Derivatives]]
+}}
+{{eqn | r = k! \paren {-1}^k x^{-\paren {k + 1} }
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = \frac {k! \paren {-1}^k} {x^{k + 1} }
+ | c =
+}}
+{{end-eqn}}
+So $\map P k \implies \map P {k + 1}$ and the result follows by the [[Principle of Mathematical Induction]].
+Therefore:
+:$\dfrac {\d^n} {\d x^n} \ln x = \dfrac {\paren {n - 1}! \paren {-1}^{n - 1} } {x^n}$
+{{Qed}}
+[[Category:Derivatives]]
+[[Category:Natural Logarithms]]
+[[Category:Proofs by Induction]]
+4kn1vgarampqz7xu9rh0ek5g1vjhroe
+\end{proof}<|endoftext|>
+\section{Sum of Factorials of Digits of 169}
+Tags: 169, Factorials
+
+\begin{theorem}
+Let the [[Definition:Factorial|factorials]] of the [[Definition:Digit|digits]] of $169$ be [[Definition:Integer Addition|added]].
+Let the same process be done on the result.
+Repeat.
+After $3$ iterations, the result will be $169$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1! + 6! + 9!
+ | r = 1 + 720 + 362 \, 880
+ | c =
+}}
+{{eqn | r = 363 \, 601
+ | c =
+}}
+{{eqn | l = 3! + 6! + 3! + 6! + 0! + 1!
+ | r = 6 + 720 + 6 + 720 + 1 + 1
+ | c =
+}}
+{{eqn | r = 1454
+ | c =
+}}
+{{eqn | l = 1! + 4! + 5! + 4!
+ | r = 1 + 24 + 120 + 24
+ | c =
+}}
+{{eqn | r = 169
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Squares which are Difference between Two Cubes}
+Tags: 169, Square Numbers, Cube Numbers
+
+\begin{theorem}
+$169$ is the smallest [[Definition:Square Number|square number]] which is the [[Definition:Integer Subtraction|difference]] between two [[Definition:Cube Number|cubes]]:
+:$169 = 8^3 - 7^3$
+{{expand|Add the rest of the sequence, having found out what they are.}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 8^3 - 7^3
+ | r = 512 - 343
+ | c =
+}}
+{{eqn | r = 169
+ | c =
+}}
+{{eqn | r = 13^2
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|Establish that this is the smallest such.}}
+\end{proof}<|endoftext|>
+\section{Cube of 180 is Sum of Sequence of Consecutive Cubes}
+Tags: Cube Numbers, 180
+
+\begin{theorem}
+:$180^3 = \displaystyle \sum_{k \mathop = 6}^{69} k^3$
+That is:
+:$180^3 = 6^3 + 7^3 + \cdots + 67^3 + 68^3 + 69^3$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sum_{k \mathop = 1}^{69} k^3
+ | r = \paren {\dfrac {69 \paren {69 + 1} } 2}^2
+ | c = [[Sum of Sequence of Cubes]]
+}}
+{{eqn | r = 5 \, 832 \, 225
+ | c =
+}}
+{{eqn | l = \sum_{k \mathop = 1}^5 k^3
+ | r = \paren {\dfrac {5 \paren {5 + 1} } 2}^2
+ | c = [[Sum of Sequence of Cubes]]
+}}
+{{eqn | r = 225
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \sum_{k \mathop = 6}^{69} k^3
+ | r = \sum_{k \mathop = 1}^{69} k^3 - \sum_{k \mathop = 1}^5 k^3
+ | c =
+}}
+{{eqn | r = 5 \, 832 \, 225 - 225
+ | c =
+}}
+{{eqn | r = 5 \, 832 \, 000
+ | c =
+}}
+{{eqn | r = 180^3
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{3-Digit Numbers forming Longest Reverse-and-Add Sequence}
+Tags: Reverse-and-Add
+
+\begin{theorem}
+Let $m \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]] expressed in [[Definition:Decimal Notation|decimal notation]].
+Let $r \left({m}\right)$ be the [[Definition:Reverse-and-Add|reverse-and-add process]] on $m$.
+Let $r$ be applied iteratively to $m$.
+The $3$-[[Definition:Digit|digit]] [[Definition:Positive Integer|integers]] $m$ which need the largest number of iterations before reaching a [[Definition:Palindromic Number|palindromic number]] are:
+:$187, 286, 385, 583, 682, 781, 880$
+all of which need $23$ iterations.
+:$869$ and $968$ are the result of the first iteration for each of these, and so take $22$ iterations to reach a [[Definition:Palindromic Number|palindromic number]].
+\end{theorem}
+
+\begin{proof}
+The [[Definition:Integer Sequence|sequence]] obtained by iterating $r$ on $187$ is:
+:$187, 968, 1837, 9218, 17347, 91718, 173437, 907808, 1716517, 8872688,$
+:$17735476, 85189247, 159487405, 664272356, 1317544822, 3602001953, 7193004016, 13297007933,$
+:$47267087164, 93445163438, 176881317877, 955594506548, 170120002107, 8713200023178$
+{{OEIS|A033670}}
+Note that $r \left({187}\right) = r \left({781}\right) = 968$, so the [[Definition:Integer Sequence|sequence]] obtained by iterating $r$ on $781$ is the same.
+The rest of the numbers converge on the same sequence immediately:
+:$286, 968, \ldots$
+:$385, 968, \ldots$
+:$880, 88, \ldots$
+:$869, 1837, \ldots$
+as do their reversals.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers Expressible as Sum of Five Distinct Squares}
+Tags: Specific Numbers, 188
+
+\begin{theorem}
+The largest [[Definition:Positive Integer|positive integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of no more than $5$ [[Definition:Distinct|distinct]] [[Definition:Square Number|squares]] is $188$.
+Both $188$ and $124$ require as many as $6$ [[Definition:Distinct|distinct]] [[Definition:Square Number|squares]] to represent them:
+{{begin-eqn}}
+{{eqn | l = 124
+ | r = 1 + 4 + 9 + 25 + 36 + 49
+ | c =
+}}
+{{eqn | r = 1^2 + 2^2 + 3^2 + 5^2 + 6^2 + 7^2
+ | c =
+}}
+{{eqn | l = 188
+ | r = 1 + 4 + 9 + 25 + 49 + 100
+ | c =
+}}
+{{eqn | r = 1^2 + 2^2 + 3^2 + 5^2 + 7^2 + 10^2
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+From [[Numbers not Sum of Distinct Squares]], the following [[Definition:Positive Integer|positive integers]] cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct|distinct]] [[Definition:Square Number|squares]] at all:
+:$2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128$
+{{OEIS|A001422}}
+{{ProofWanted|It remains to be shown that all other [[Definition:Positive Integer|positive integers]] less than $188$ can be expressed as the sum of no more than 5 distinct squares, and all numbers over $188$ can also be so expressed.}}
+\end{proof}<|endoftext|>
+\section{Pandigital Sum whose Components are Multiples}
+Tags: Recreational Mathematics
+
+\begin{theorem}
+The following [[Definition:Integer Addition|sums]] use each of the [[Definition:Digit|digits]] $1$ to $9$ exactly once each, while one [[Definition:Summand|summand]] equals $2$ times the other [[Definition:Summand|summand]]:
+{{begin-eqn}}
+{{eqn | l = 192 + 384
+ | r = 576
+ | c = where $2 \times 192 = 384$ and so $3 \times 192 = 576$
+}}
+{{eqn | l = 219 + 438
+ | r = 657
+ | c = where $2 \times 219 = 438$ and so $3 \times 219 = 657$
+}}
+{{eqn | l = 273 + 546
+ | r = 819
+ | c = where $2 \times 273 = 546$ and so $3 \times 273 = 819$
+}}
+{{eqn | l = 327 + 654
+ | r = 981
+ | c = where $2 \times 327 = 654$ and so $3 \times 327 = 981$
+}}
+{{end-eqn}}
+{{OEIS|A039667}}
+It is interesting to note how the $4$ [[Definition:Integer Addition|sums]] can be divided into two pairs in which each element of the sum is an [[Definition:Anagram|anagram]] of its corresponding element in the other.
+\end{theorem}
+
+\begin{proof}
+We establish some parameters for the number $x$ such that the equation $x + 2 x = 3 x$ is pandigital.
+Firstly, among the three numbers, there cannot be repeating [[Definition:Digit|digits]] or $0$.
+If $5 \divides x$:
+:$10 \divides 2 x$ and $2 x$ ends in $0$.
+By [[Divisibility by 5]], $x$ cannot end in $5$.
+Secondly, $x$ must be [[Definition:Divisor of Integer|divisible]] by $3$.
+To see this, consider the concatenation of the three numbers, $\sqbrk {x || 2 x || 3 x}$.
+Since the [[Definition:Digit|digits]] of this number is a permutation of $\set {1, \dots, 9}$, they [[Definition:Integer Addition|sum]] to $45$.
+Since $9 \divides 45$, by [[Divisibility by 9]]:
+:$9 \divides \sqbrk {x || 2 x || 3 x}$
+We have:
+{{begin-eqn}}
+{{eqn | l = \sqbrk {x \vert \vert 2 x \vert \vert 3 x}
+ | r = 10^6 x + 10^3 \paren {2 x} + 3 x
+}}
+{{eqn | o = \equiv
+ | r = x + 2 x + 3 x
+ | rr = \pmod 9
+ | c = [[Congruence of Powers]]
+}}
+{{eqn | o = \equiv
+ | r = 6 x
+ | rr = \pmod 9
+}}
+{{end-eqn}}
+Hence $9 \divides 6 x$.
+This gives $3 \divides 2 x$.
+By [[Euclid's Lemma]]:
+:$3 \divides x$
+Finally, as $3 x < 1000$, we must have $100 \le x \le 333$.
+Putting repeating [[Definition:Digit|digits]] and zeroes into consideration, we list and check all possible candidates from $123$ to $327$:
+{{begin-eqn}}
+{{eqn | ll = 12X:
+ | l = 12X \times 2
+ | r = 2XX
+ | c = $2$ is repeated
+}}
+{{eqn | ll = 132:
+ | l = 132 \times 2
+ | r = 264
+ | c = $2$ is repeated
+}}
+{{eqn | ll = 138:
+ | l = 138 \times 3
+ | r = 414
+ | c = $4$ is repeated
+}}
+{{eqn | ll = 147:
+ | l = 147 \times 2
+ | r = 294
+ | c = $4$ is repeated
+}}
+{{eqn | ll = 153:
+ | l = 153 \times 2
+ | r = 306
+ | c = $3$ is repeated
+}}
+{{eqn | ll = 156:
+ | l = 156 \times 2
+ | r = 312
+ | c = $1$ is repeated
+}}
+{{eqn | ll = 159:
+ | l = 159 \times 2
+ | r = 318
+ | c = $1$ is repeated
+}}
+{{eqn | ll = 162:
+ | l = 162 \times 2
+ | r = 324
+ | c = $2$ is repeated
+}}
+{{eqn | ll = 168:
+ | l = 168 \times 2
+ | r = 336
+ | c = $3, 6$ are repeated
+}}
+{{eqn | ll = 174:
+ | l = 174 \times 2
+ | r = 348
+ | c = $4$ is repeated
+}}
+{{eqn | ll = 183:
+ | l = 183 \times 2
+ | r = 366
+ | c = $3, 6$ are repeated
+}}
+{{eqn | ll = 186:
+ | l = 186 \times 3
+ | r = 558
+ | c = $5, 8$ are repeated
+}}
+{{eqn | ll = 189:
+ | l = 189 \times 2
+ | r = 378
+ | c = $8$ is repeated
+}}
+{{eqn | ll = 192:
+ | l = 192 + 384
+ | r = 576
+ | c = is the $1$st solution
+}}
+{{eqn | ll = 198:
+ | l = 198 \times 2
+ | r = 396
+ | c = $9$ is repeated
+}}
+{{eqn | ll = 213:
+ | l = 213 \times 2
+ | r = 426
+ | c = $2$ is repeated
+}}
+{{eqn | ll = 216:
+ | l = 216 \times 2
+ | r = 432
+ | c = $2$ is repeated
+}}
+{{eqn | ll = 219:
+ | l = 219 + 438
+ | r = 657
+ | c = is the $2$nd solution
+}}
+{{eqn | ll = 231:
+ | l = 231 \times 2
+ | r = 462
+ | c = $2$ is repeated
+}}
+{{eqn | ll = 234:
+ | l = 234 \times 2
+ | r = 468
+ | c = $4$ is repeated
+}}
+{{eqn | ll = 237:
+ | l = 237 \times 2
+ | r = 474
+ | c = $4, 7$ are repeated
+}}
+{{eqn | ll = 24X:
+ | l = 24X \times 2
+ | r = 4XX
+ | c = $4$ is repeated
+}}
+{{eqn | ll = 258:
+ | l = 258 \times 2
+ | r = 516
+ | c = $5$ is repeated
+}}
+{{eqn | ll = 261:
+ | l = 261 \times 2
+ | r = 522
+ | c = $2$ is repeated
+}}
+{{eqn | ll = 264:
+ | l = 264 \times 2
+ | r = 528
+ | c = $2$ is repeated
+}}
+{{eqn | ll = 267:
+ | l = 267 \times 3
+ | r = 801
+ | c = $0$ is used
+}}
+{{eqn | ll = 273:
+ | l = 273 + 546
+ | r = 819
+ | c = is the $3$rd solution
+}}
+{{eqn | ll = 276:
+ | l = 276 \times 2
+ | r = 552
+ | c = $2, 5$ are repeated
+}}
+{{eqn | ll = 279:
+ | l = 279 \times 3
+ | r = 837
+ | c = $7$ is repeated
+}}
+{{eqn | ll = 291:
+ | l = 291 \times 2
+ | r = 582
+ | c = $2$ is repeated
+}}
+{{eqn | ll = 294:
+ | l = 294 \times 2
+ | r = 588
+ | c = $8$ is repeated
+}}
+{{eqn | ll = 297:
+ | l = 297 \times 2
+ | r = 594
+ | c = $9$ is repeated
+}}
+{{eqn | ll = 312:
+ | l = 312 \times 2
+ | r = 624
+ | c = $2$ is repeated
+}}
+{{eqn | ll = 318:
+ | l = 318 \times 2
+ | r = 636
+ | c = $3, 6$ are repeated
+}}
+{{eqn | ll = 321:
+ | l = 321 \times 2
+ | r = 642
+ | c = $2$ is repeated
+}}
+{{eqn | ll = 324:
+ | l = 324 \times 2
+ | r = 648
+ | c = $4$ is repeated
+}}
+{{eqn | ll = 327:
+ | l = 327 + 654
+ | r = 981
+ | c = is the $4$th solution
+}}
+{{end-eqn}}
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Closed Form for Heptagonal Pyramidal Numbers}
+Tags: Closed Forms, Pyramidal Numbers
+
+\begin{theorem}
+The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Heptagonal Pyramidal Number|heptagonal pyramidal number]] is:
+:$Q_n = \dfrac {n \paren {n + 1} \paren {5 n - 2} } 6$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = Q_n
+ | r = \sum_{k \mathop = 1}^n H_n
+ | c = {{Defof|Heptagonal Pyramidal Number}}
+}}
+{{eqn | r = \sum_{k \mathop = 1}^n \dfrac {k \paren {5 k - 3} } 2
+ | c = [[Closed Form for Heptagonal Numbers]]
+}}
+{{eqn | r = \dfrac 1 2 \paren {5 \sum_{k \mathop = 1}^n k^2 - 3 \sum_{k \mathop = 1}^n k}
+ | c =
+}}
+{{eqn | r = \dfrac 1 2 \paren {5 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 - 3 \sum_{k \mathop = 1}^n k}
+ | c = [[Sum of Sequence of Squares]]
+}}
+{{eqn | r = \dfrac 1 2 \paren {5 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 - 3 \dfrac {n \paren {n + 1} } 2}
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{eqn | r = \dfrac 1 2 \paren {\frac {5 n \paren {n + 1} \paren {2 n + 1} - 9 n \paren {n + 1} } 6}
+ | c =
+}}
+{{eqn | r = \dfrac 1 2 \paren {\frac {n \paren {n + 1} \paren {5 \paren {2 n + 1} - 9} } 6}
+ | c =
+}}
+{{eqn | r = \dfrac 1 2 \paren {\frac {n \paren {n + 1} \paren {10 n + 5 - 9} } 6}
+ | c =
+}}
+{{eqn | r = \dfrac 1 2 \paren {\frac {n \paren {n + 1} \paren {10 n - 4} } 6}
+ | c =
+}}
+{{eqn | r = \frac {n \paren {n + 1} \paren {5 n - 2} } 6
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Heptagonal Pyramidal Numbers which are Square}
+Tags: Pyramidal Numbers, Square Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Heptagonal Pyramidal Number|heptagonal pyramidal numbers]] which also have the property of being [[Definition:Square Number|square]] begins:
+:$0, 1, 196, \ldots$
+{{expand|Further research needed}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | r = \dfrac {0 \left({0 + 1}\right) \left({5 \times 0 - 2}\right)} 6
+ | o =
+ | c = [[Closed Form for Heptagonal Pyramidal Numbers]]
+}}
+{{eqn | r = \dfrac {0 \times 1 \times \left({- 2}\right)} 6
+ | c =
+}}
+{{eqn | r = 0
+ | c =
+}}
+{{eqn | r = 0^2
+ | c = {{Defof|Square Number}}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | r = \dfrac {1 \left({1 + 1}\right) \left({5 \times 1 - 2}\right)} 6
+ | o =
+ | c = [[Closed Form for Heptagonal Pyramidal Numbers]]
+}}
+{{eqn | r = \dfrac {1 \times 2 \times 3} 6
+ | c =
+}}
+{{eqn | r = 1
+ | c =
+}}
+{{eqn | r = 1^2
+ | c = {{Defof|Square Number}}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | r = \dfrac {6 \left({6 + 1}\right) \left({5 \times 6 - 2}\right)} 6
+ | o =
+ | c = [[Closed Form for Heptagonal Pyramidal Numbers]]
+}}
+{{eqn | r = \dfrac {6 \times 7 \times 28} 6
+ | c =
+}}
+{{eqn | r = 7 \times \left({2^2 \times 7}\right)
+ | c =
+}}
+{{eqn | r = \left({2 \times 7}\right)^2
+ | c = {{Defof|Square Number}}
+}}
+{{eqn | r = 196
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Conditions for Limit Function to be Limit Minimizing Function of Functional}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $y$ be a [[Definition:Real Function|real function]].
+Let $J \sqbrk y$ be a [[Definition:Real Functional|functional]].
+Let $\sequence {y_n}$ be a [[Definition:Minimizing Sequence of Functional|minimizing sequence]] of $J$.
+Let:
+:$\ds \lim_{n \mathop \to \infty} y_n = \hat y$
+Suppose $J$ is [[Definition:Lower Semicontinuous|lower semicontinuous]] at $y = \hat y$.
+Then:
+:$\ds J \sqbrk {\hat y} = \lim_{n \mathop \to \infty} J \sqbrk {y_n}$
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Minimizing Sequence of Functional|minimizing sequence]]:
+:$\ds \inf_y J \sqbrk y = \lim_{n \mathop \to \infty} J \sqbrk {y_n}$
+Any [[Definition:Mapping|mapping]] from this [[Definition:Sequence|sequence]] either minimises the [[Definition:Real Functional|functional]] or not.
+This is true for the limit [[Definition:Mapping|mapping]] as well:
+:$\ds J \sqbrk {\hat y} \ge \inf_y J \sqbrk y$
+By [[Definition:Assumption|assumption]], $J$ is [[Definition:Lower Semicontinuous|lower semicontinuous]] at $\hat y$:
+:$\ds \forall \epsilon_1 > 0: \forall n \in \N: \exists N \in \N: \paren {n > N} \implies \paren {J \sqbrk {\hat y} - J \sqbrk {y_n} < \epsilon_1}$
+Hence, for [[Definition:Sufficiently Large|sufficiently large]] $n$:
+:$\ds \inf_y J \sqbrk y \le J \sqbrk {\hat y} < J \sqbrk {y_n} + \epsilon_1$
+By the definition of [[Definition:Minimizing Sequence of Functional|minimizing sequence]], where the label $n$ has been replaced with $m$:
+:$\ds \forall \epsilon_2: \forall m \in \N: \exists M \in \N: \paren {m > M} \implies \paren {\size {J \sqbrk {y_m} - \inf_y J \sqbrk y} < \epsilon_2}$
+From this, for [[Definition:Sufficiently Large|sufficiently large]] $m$:
+:$\ds J \sqbrk {y_m} - \epsilon_2 < \inf_y J \sqbrk y$
+
+Then:
+:$J \sqbrk {y_m} - \epsilon_2 < J \sqbrk {\hat y} < J \sqbrk {y_n} + \epsilon_1$
+Here $\epsilon_1$, $n$ have similar properties like $\epsilon_2$, $m$, but are otherwise arbitrary and independent.
+Let $\epsilon_1 = \epsilon_2 = \epsilon$, $n = m$, $M = N$.
+Arbitrariness is still not affected.
+Then:
+:$J \sqbrk {y_n} - \epsilon < J \sqbrk {\hat y} < J \sqbrk {y_n} + \epsilon$
+Subtract $J \sqbrk {y_n}$ from all the terms.
+Hence:
+:$-\epsilon < J \sqbrk {\hat y} - J \sqbrk {y_n} < \epsilon$
+or:
+:$\size {J \sqbrk {y_n} - J \sqbrk {\hat y} } < \epsilon$
+Therefore this relation inherits all the constraints on its values $n$, $N$, $\epsilon$, and by definition is a [[Definition:Limit of Sequence|limit]]:
+:$\ds \lim_{n \mathop \to \infty} J \sqbrk {y_n} = J \sqbrk {\hat y}$
+{{qed}}
+{{Proofread|may contain mistakes, since original was not satisfactory enough}}
+\end{proof}<|endoftext|>
+\section{Smallest 10 Primes in Arithmetic Sequence}
+Tags: Prime Numbers, Arithmetic Sequences
+
+\begin{theorem}
+The smallest $10$ [[Definition:Prime Number|primes]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] are:
+:$199 + 210 n$
+for $n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9$.
+These are also the smallest $8$ and $9$ [[Definition:Prime Number|primes]] in [[Definition:Arithmetic Sequence|arithmetic sequence]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 199 + 0 \times 210
+ | r = 199
+ | c = which is the $46$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 199 + 1 \times 210
+ | r = 409
+ | c = which is the $80$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 199 + 2 \times 210
+ | r = 619
+ | c = which is the $114$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 199 + 3 \times 210
+ | r = 829
+ | c = which is the $145$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 199 + 4 \times 210
+ | r = 1039
+ | c = which is the $175$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 199 + 5 \times 210
+ | r = 1249
+ | c = which is the $204$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 199 + 6 \times 210
+ | r = 1459
+ | c = which is the $232$nd [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 199 + 7 \times 210
+ | r = 1669
+ | c = which is the $263$rd [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 199 + 8 \times 210
+ | r = 1879
+ | c = which is the $289$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 199 + 9 \times 210
+ | r = 2089
+ | c = which is the $316$th [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{OEIS|A033168}}
+But note that $199 + 10 \times 210 = 2299 = 11^2 \times 19$ and so is not [[Definition:Prime Number|prime]].
+{{ProofWanted|It remains to be shown that there are no smaller such APs}}
+\end{proof}<|endoftext|>
+\section{Numbers that cannot be made Prime by changing 1 Digit}
+Tags: Prime Numbers, Recreational Mathematics, Numbers that cannot be made Prime by changing 1 Digit
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integers]] cannot be made into [[Definition:Prime Number|prime numbers]] by changing just one [[Definition:Digit|digit]]:
+:$200, 202, 204, 205, 206, 208, \ldots$
+{{OEIS|A192545}}
+\end{theorem}
+
+\begin{proof}
+In order to make any one of these [[Definition:Positive Integer|positive integers]] into a [[Definition:Prime Number|prime number]] one would have to change the last [[Definition:Digit|digit]].
+Otherwise the number it was changed into would be either [[Definition:Even Integer|even]], or [[Definition:Divisor of Integer|divisible]] by $5$, and so not [[Definition:Prime Number|prime]].
+But all the other [[Definition:Positive Integer|integers]] between $200$ and $209$ are [[Definition:Composite Number|composite]]:
+{{begin-eqn}}
+{{eqn | l = 201
+ | r = 3 \times 67
+ | c =
+}}
+{{eqn | l = 203
+ | r = 7 \times 29
+ | c =
+}}
+{{eqn | l = 207
+ | r = 3^2 \times 23
+ | c =
+}}
+{{eqn | l = 209
+ | r = 11 \times 19
+ | c =
+}}
+{{end-eqn}}
+In order for there to be a smaller number with this property, a [[Definition:Prime Gap|prime gap]] would need to be found which spans an entire decade.
+There are none such.
+On the other hand, each of $201, 203, 207, 209$ can themselves be turned into a [[Definition:Prime Number|prime number]] by changing the initial $2$ into a $1$:
+:$101, 103, 107,109$
+are all [[Definition:Prime Number|prime]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum of Cubes of 3 Consecutive Integers which is Square}
+Tags: Sums of Cubes
+
+\begin{theorem}
+The following [[Definition:Integer Sequence|sequence]] of $3$ consecutive [[Definition:Positive Integer|positive integers]] have [[Definition:Cube Number|cubes]] that [[Definition:Integer Addition|sum]] to a [[Definition:Square Number|square]]:
+:$23, 24, 25$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 23^3 + 24^3 + 25^3
+ | r = 12 \, 167 + 13 \, 824 + 15 \, 625
+ | c =
+}}
+{{eqn | r = 41 \, 616
+ | c =
+}}
+{{eqn | r = 204^2
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integer Greater than 205 is Sum of Distinct Primes of form 6 n + 1}
+Tags: 205, Prime Numbers
+
+\begin{theorem}
+Let $n$ be an [[Definition:Integer|integer]] greater than $205$.
+Then $n$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct|distinct]] [[Definition:Prime Number|prime numbers]] of the form $6 n + 1$.
+\end{theorem}<|endoftext|>
+\section{Triangular Numbers which are also Pentagonal}
+Tags: Triangular Numbers, Pentagonal Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Triangular Number|triangular numbers]] which are also [[Definition:Pentagonal Number|pentagonal]] begins:
+:$1, 210, 40 \, 755, 7 \, 906 \, 276, 1 \, 533 \, 776 \, 805, 297 \, 544 \, 793 \, 910, \ldots$
+{{OEIS|A014979}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1
+ | r = \dfrac {1 \paren {3 \times 1 - 1} } 2
+ | c = [[Closed Form for Pentagonal Numbers]]
+}}
+{{eqn | r = \dfrac {1 \times \paren {1 + 1} } 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 210
+ | r = \dfrac {12 \paren {3 \times 12 - 1} } 2
+ | c = [[Closed Form for Pentagonal Numbers]]
+}}
+{{eqn | r = \dfrac {20 \times \paren {20 + 1} } 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 40 \, 755
+ | r = \dfrac {165 \paren {3 \times 165 - 1} } 2
+ | c = [[Closed Form for Pentagonal Numbers]]
+}}
+{{eqn | r = \dfrac {285 \times \paren {285 + 1} } 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that these are the only such instances.}}
+\end{proof}<|endoftext|>
+\section{Numbers such that Tau divides Phi divides Sigma}
+Tags: Tau Function, Euler Phi Function, Sigma Function, Numbers such that Tau divides Phi divides Sigma
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] $n$ with the property that:
+:$\map \tau n \divides \map \phi n \divides \map \sigma n$
+where:
+:$\divides$ denotes [[Definition:Divisor of Integer|divisibility]]
+:$\tau$ denotes the [[Definition:Tau Function|$\tau$ (tau) function]]: the count of [[Definition:Divisor of Integer|divisors]] of $n$
+:$\phi$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ (phi) function]]: the count of smaller [[Definition:Integer|integers]] [[Definition:Coprime Integers|coprime]] to $n$
+:$\sigma$ denotes the [[Definition:Sigma Function|$\sigma$ (sigma) function]]: the [[Definition:Integer Addition|sum]] of [[Definition:Divisor of Integer|divisors]] of $n$
+begins:
+:$1, 3, 15, 30, 35, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, \ldots$
+{{OEIS|A020493}}
+\end{theorem}
+
+\begin{proof}
+By inspection and investigation.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Number of Representations as Sum of Two Primes}
+Tags: Prime Numbers
+
+\begin{theorem}
+The number of ways an [[Definition:Integer|integer]] $n$ can be represented as the [[Definition:Integer Addition|sum]] of two [[Definition:Prime Number|primes]] is no greater than the number of [[Definition:Prime Number|primes]] in the [[Definition:Closed Real Interval|interval]] $\closedint {\dfrac n 2} {n - 2}$.
+\end{theorem}
+
+\begin{proof}
+Let $n = p + q$ where $p \le q$ and both $p$ and $q$ are [[Definition:Prime Number|primes]].
+There can be no more different representations of $n$ as $p + q$ than there are the number of possible options for $q$.
+As $q \ge p$, it follows that $q \ge \dfrac n 2$.
+Note that as $p \ge 2$, it follows that $q \le n - 2$.
+The number of possible values of $q$ is therefore equal to the number of [[Definition:Prime Number|primes]] between $\dfrac n 2$ and $n - 2$ inclusive.
+That is, the number of [[Definition:Prime Number|primes]] in the [[Definition:Closed Real Interval|interval]] $\closedint {\dfrac n 2} {n - 2}$.
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integers whose Number of Representations as Sum of Two Primes is Maximum}
+Tags: Prime Numbers
+
+\begin{theorem}
+$210$ is the largest [[Definition:Integer|integer]] which can be represented as the [[Definition:Integer Addition|sum]] of two [[Definition:Prime Number|primes]] in the maximum number of ways.
+The full list of such numbers is as follows:
+:$1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 24, 30, 36, 42, 48, 60, 90, 210$
+{{OEIS|A141340}}
+The list contains:
+:$n \le 8$
+:$n \le 18$ where $2 \divides n$
+:$n \le 48$ where $2 \times 3 \divides n$
+:$n \le 90$ where $2 \times 3 \times 5 \divides n$
+:$210 = 2 \times 3 \times 5 \times 7$
+{{WIP|Add this sequence to the number pages}}
+\end{theorem}
+
+\begin{proof}
+From [[Number of Representations as Sum of Two Primes]], the number of ways an [[Definition:Integer|integer]] $n$ can be represented as the [[Definition:Integer Addition|sum]] of two [[Definition:Prime Number|primes]] is no greater than the number of [[Definition:Prime Number|primes]] in the [[Definition:Closed Real Interval|interval]] $\closedint {\dfrac n 2} {n - 2}$.
+The [[Definition:Closed Real Interval|interval]] $\closedint {\dfrac {210} 2} {210 - 2}$ is $\closedint {105} {208}$.
+The [[Definition:Prime Number|primes]] in this [[Definition:Closed Real Interval|interval]] can be enumerated:
+:$107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199$
+It can be seen there are exactly $19$ of them.
+We have:
+{{begin-eqn}}
+{{eqn | l = 11 + 199
+ | r = 210
+}}
+{{eqn | l = 13 + 197
+ | r = 210
+}}
+{{eqn | l = 17 + 193
+ | r = 210
+}}
+{{eqn | l = 19 + 191
+ | r = 210
+}}
+{{eqn | l = 29 + 181
+ | r = 210
+}}
+{{eqn | l = 31 + 179
+ | r = 210
+}}
+{{eqn | l = 37 + 173
+ | r = 210
+}}
+{{eqn | l = 43 + 167
+ | r = 210
+}}
+{{eqn | l = 47 + 163
+ | r = 210
+}}
+{{eqn | l = 53 + 157
+ | r = 210
+}}
+{{eqn | l = 59 + 151
+ | r = 210
+}}
+{{eqn | l = 61 + 149
+ | r = 210
+}}
+{{eqn | l = 71 + 139
+ | r = 210
+}}
+{{eqn | l = 73 + 137
+ | r = 210
+}}
+{{eqn | l = 79 + 131
+ | r = 210
+}}
+{{eqn | l = 83 + 127
+ | r = 210
+}}
+{{eqn | l = 97 + 113
+ | r = 210
+}}
+{{eqn | l = 101 + 109
+ | r = 210
+}}
+{{eqn | l = 103 + 107
+ | r = 210
+}}
+{{end-eqn}}
+and as can be seen, there are $19$ such representations, one for each [[Definition:Prime Number|prime]] in $\closedint {105} {208}$.
+{{ProofWanted|It remains to be shown that $210$ is the largest number that can be represented by the maximum of these.}}
+\end{proof}<|endoftext|>
+\section{25 as Sum of 4 to 11 Squares}
+Tags: Sums of Squares, 25
+
+\begin{theorem}
+$25$ can be expressed as the [[Definition:Integer Addition|sum]] of $n$ non-zero [[Definition:Square Number|squares]] for all $n$ from $4$ to $11$.
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = 25
+ | r = 4^2 + \left({2 \times 2^2}\right) + 1^2
+ | c = $4$ [[Definition:Square Number|squares]]
+}}
+{{eqn | r = 3^2 + \left({4 \times 2^2}\right)
+ | c = $5$ [[Definition:Square Number|squares]]
+}}
+{{eqn | r = \left({2 \times 3^2}\right) + 2^2 + \left({3 \times 1^2}\right)
+ | c = $6$ [[Definition:Square Number|squares]]
+}}
+{{eqn | r = 4^2 + 2^2 + \left({5 \times 1^2}\right)
+ | c = $7$ [[Definition:Square Number|squares]]
+}}
+{{eqn | r = 3^2 + \left({3 \times 2^2}\right) + \left({4 \times 1^2}\right)
+ | c = $8$ [[Definition:Square Number|squares]]
+}}
+{{eqn | r = \left({2 \times 3^2}\right) + \left({7 \times 1^2}\right)
+ | c = $9$ [[Definition:Square Number|squares]]
+}}
+{{eqn | r = 4^2 + \left({9 \times 1^2}\right)
+ | c = $10$ [[Definition:Square Number|squares]]
+}}
+{{eqn | r = 3^2 + \left({2 \times 2^2}\right) + \left({8 \times 1^2}\right)
+ | c = $11$ [[Definition:Square Number|squares]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cubes which are Sum of Three Cubes}
+Tags: Sums of Cubes
+
+\begin{theorem}
+The following [[Definition:Cube Number|cube numbers]] can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Positive Integer|positive]] [[Definition:Cube Number|cube numbers]]:
+:$6^3, 9^3, 12^3, 18^3, 19^3, 20^3, 24^3, 25^3, \ldots$
+{{OEIS|A066890}}
+with [[Definition:Cube Root|cube roots]]:
+:$6, 9, 12, 18, 19, 20, 24, 25, \ldots$
+{{OEIS|A023042}}
+{{WIP|Add these sequences to the separate pages for the numbers themselves}}
+\end{theorem}<|endoftext|>
+\section{Ritz Method implies Not Worse Approximation with Increased Number of Functions}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Consider the [[Definition:Ritz Method|Ritz method]].
+Let $\eta_n = \boldsymbol \alpha \boldsymbol \phi$.
+Let $J \sqbrk {\eta_n} = \mu_n$.
+Then:
+:$\mu_n \ge \mu_{n + 1}$
+\end{theorem}
+
+\begin{proof}
+Denote $\eta_{n + 1} = \eta_n + \alpha_{n + 1} \phi_{n + 1}$.
+For $\alpha_{n + 1} = 0$ we have:
+:$\eta_n = \eta_{n + 1}$
+Suppose that $J \sqbrk {\eta_n}$ has been [[Definition:Minimum Value|minimised]] {{WRT}} $\boldsymbol \alpha$.
+If:
+:$\exists \boldsymbol \alpha \in \R^n: \nexists \alpha_{n + 1} \ne 0: J \sqbrk {\eta_n} > J \sqbrk {\eta_{n + 1} }$
+then $J \sqbrk {\eta_{n + 1} }$ is [[Definition:Minimum Value|minimised]] for previously determined $\boldsymbol \alpha$ and $\alpha_{n + 1} = 0$:
+:$J \sqbrk {\eta_{n + 1} } = J \sqbrk {\eta_n}$
+where $\eta_{n + 1} = \eta_n$.
+Suppose:
+:$\exists \boldsymbol \alpha \in \R^n: \exists \alpha_{n + 1} \ne 0: J \sqbrk {\eta_n} > J \sqbrk {\eta_{n + 1} }$
+Then $\eta_{n + 1} \ne \eta_n$, because their respective $\boldsymbol \alpha$ differ by at least one value: $\alpha_{n + 1}$.
+Hence, for this supposition:
+:$J \sqbrk {\eta_{n + 1} } > J \sqbrk {\eta_n}$
+Finally, both cases together imply that:
+:$J \sqbrk {\eta_{n + 1} } \ge J \sqbrk {\eta_n}$
+or equivalently:
+:$\mu_n \ge \mu_{n + 1}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Dissection of Cube into 3 Cubes using 8 Pieces}
+Tags: Dissections
+
+\begin{theorem}
+A [[Definition:Cube (Geometry)|cube]] can be [[Definition:Dissection|dissected]] into $3$ smaller [[Definition:Cube (Geometry)|cubes]] by cutting it into $8$ pieces and reassembling them.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Apparently it can be done using the result [[Cubes which are Sum of Three Cubes]].}}
+\end{proof}<|endoftext|>
+\section{Smallest Magic Constant of Order 3 Multiplicative Magic Square}
+Tags: Multiplicative Magic Squares
+
+\begin{theorem}
+The [[Definition:Multiplicative Magic Constant|magic constant]] of the smallest possible [[Definition:Multiplicative Magic Square|multiplicative magic square]] with the smallest [[Definition:Multiplicative Magic Constant|magic constant]] is as follows.
+{{:Multiplicative Magic Square/Examples/Order 3/Smallest}}
+
+\end{theorem}
+
+\begin{proof}
+From [[Smallest Multiplicative Magic Square is of Order 3]], the smallest possible [[Definition:Multiplicative Magic Square|multiplicative magic square]] is of [[Definition:Order of Multiplicative Magic Square|order $3$]].
+Let $C$ be a [[Definition:Multiplicative Magic Constant|magic constant]] for an [[Definition:Order of Multiplicative Magic Square|order $3$]] [[Definition:Multiplicative Magic Square|multiplicative magic square]].
+We prove that $C \ge 216$ is smallest in $2$ steps:
+:Step $1$: If $C$ is a [[Definition:Prime Power|prime power]] $p^k$, then $k \ge 8$.
+:Step $2$: For each [[Definition:Prime Number|prime]] $p$ [[Definition:Divisor of Integer|dividing]] $C$, we must have $p^3$ [[Definition:Divisor of Integer|dividing]] $C$ as well.
+Then $216 = 2^3 \cdot 3^3$ is the smallest such constant.
+=== Step $1$ ===
+Suppose that $C = p^k$ for some [[Definition:Prime Number|prime]] $p$.
+Since the [[Definition:Multiplicative Magic Square|multiplicative magic square]] has distinct entries, and each entry of the [[Definition:Multiplicative Magic Square|magic square]] is a [[Definition:Divisor of Integer|factor]] of $C$, $C$ must have at least $9$ distinct [[Definition:Divisor of Integer|factors]].
+From [[Tau of Power of Prime]], $\map \tau {p^k} = k + 1 \ge 9$.
+Therefore we have $k \ge 8$.
+{{qed|lemma}}
+=== Step $2$ ===
+Suppose on the contrary, for some [[Definition:Prime Number|prime]] $p$ [[Definition:Divisor of Integer|dividing]] $C$, $p^3$ does not [[Definition:Divisor of Integer|divide]] $C$.
+Then we have either:
+:$p \divides C$ and $p^2 \nmid C$
+or:
+:$p^2 \divides C$ and $p^3 \nmid C$
+==== Case 1: $p \divides C$ and $p^2 \nmid C$ ====
+Since $p \divides C$, each row, column and diagonal contains one entry [[Definition:Divisor of Integer|divisible]] by $p$.
+However there cannot be anymore, since $p^2 \nmid C$.
+These entries can be in the middle, at a corner or at an edge.
+We will show that none of these will form a [[Definition:Multiplicative Magic Square|magic square]].
+We indicate the entries [[Definition:Divisor of Integer|divisible]] by $p$ with $p$, and the entries cannot be [[Definition:Divisor of Integer|divisible]] by $p$ with $*$.
+Suppose an entry [[Definition:Divisor of Integer|divisible]] by $p$ is in the middle.
+Then:
+:$\begin{array}{|c|c|c|}
+\hline * & * & * \\
+\hline * & p & * \\
+\hline * & * & * \\
+\hline
+\end{array}$
+But then the first row is not [[Definition:Divisor of Integer|divisible]] by $p$, so their product is not $C$.
+Suppose an entry [[Definition:Divisor of Integer|divisible]] by $p$ is at an edge. Then:
+:$\begin{array}{|c|c|c|}
+\hline * & & \\
+\hline p & * & * \\
+\hline * & & \\
+\hline
+\end{array}$
+By the diagonal conditions, the top right and bottom right entries must be [[Definition:Divisor of Integer|divisible]] by $p$.
+But then the rightmost column is [[Definition:Divisor of Integer|divisible]] by $p^2$, so their product is not $C$.
+Suppose an entry [[Definition:Divisor of Integer|divisible]] by $p$ is in a corner.
+Then:
+:$\begin{array}{|c|c|c|}
+\hline p & * & * \\
+\hline * & * & \\
+\hline * & & * \\
+\hline
+\end{array}$
+But then the diagonal from bottom left to top right has no entries [[Definition:Divisor of Integer|divisible]] by $p$, so their product is not $C$.
+{{qed|lemma}}
+==== Case $2$: $p^2 \divides C$ and $p^3 \nmid C$ ====
+The cases where an entry [[Definition:Divisor of Integer|divisible]] by $p^2$ exists is similar to Case $1$.
+So we suppose there is no entry [[Definition:Divisor of Integer|divisible]] by $p^2$.
+Similar to Case 1, we indicate the entries [[Definition:Divisor of Integer|divisible]] by $p$ with $p$, and the entries cannot be [[Definition:Divisor of Integer|divisible]] by $p$ with $*$.
+Each row must contain an entry not [[Definition:Divisor of Integer|divisible]] by $p$.
+Then the cases will be the inverse of the cases in Case 1, with $p$ interchanged with $*$.
+For example, the cases where an entry not [[Definition:Divisor of Integer|divisible]] by $p$ is at the corner.
+Then:
+:$\begin{array}{|c|c|c|}
+\hline * & p & p \\
+\hline p & p & \\
+\hline p & & p \\
+\hline
+\end{array}$
+But then every entry on the diagonal from bottom left to top right are [[Definition:Divisor of Integer|divisible]] by $p$, so their product is not $C$.
+{{qed|lemma}}
+Cases $1$ and $2$ above shows that for each [[Definition:Prime Number|prime]] $p$ [[Definition:Divisor of Integer|dividing]] $C$, we must have $p^3$ [[Definition:Divisor of Integer|dividing]] $C$ as well.
+{{qed|lemma}}
+=== Proof that $216$ is the smallest $C$ ===
+Now suppose $C$ is a [[Definition:Prime Power|prime power]].
+Then from our conclusion in Step $1$:
+:$C = p^k \ge 2^k \ge 2^8 = 512$
+Suppose $C$ is not a [[Definition:Prime Power|prime power]].
+Then $C$ must be [[Definition:Divisor of Integer|divisible]] by at least $2$ [[Definition:Prime Number|primes]].
+Then from our conclusion in Step $2$:
+:$C = p_1^{a_1} p_2^{a_2} k \ge 2^{a_1} 3^{a_2} \ge 2^3 3^3 = 216$
+{{qed}}
+{{explain|There are a number of assumptions which appear to made for the purposes of a [[Proof by Contradiction]] -- worth labelling them as such and indicating where those assumptions are refuted, for clarity.}}
+\end{proof}<|endoftext|>
+\section{Smallest Multiplicative Magic Square is of Order 3}
+Tags: Multiplicative Magic Squares
+
+\begin{theorem}
+The [[Definition:Order of Multiplicative Magic Square|order]] of the smallest [[Definition:Multiplicative Magic Square|multiplicative magic square]] is $3$, for example:
+{{:Multiplicative Magic Square/Examples/Order 3/Smallest}}
+\end{theorem}
+
+\begin{proof}
+Suppose there were an [[Definition:Order of Multiplicative Magic Square|order $2$]] [[Definition:Multiplicative Magic Square|multiplicative magic square]] $M$.
+Let $a$ be the [[Definition:Element of Array|element]] of [[Definition:Row of Array|row $1$]] and [[Definition:Column of Array|column $1$]].
+Let $a b$ be the [[Definition:Multiplicative Magic Constant|magic constant]] of $M$.
+Then $b$ is:
+:the [[Definition:Element of Array|element]] of [[Definition:Row of Array|row $1$]] and [[Definition:Column of Array|column $2$]], to make the [[Definition:Multiplication|product]] of [[Definition:Row of Array|row $1$]] equal to $a b$
+:the [[Definition:Element of Array|element]] of [[Definition:Row of Array|row $2$]] and [[Definition:Column of Array|column $1$]], to make the [[Definition:Multiplication|product]] of [[Definition:Column of Array|column $1$]] equal to $a b$
+and so on.
+But it is a principle of [[Definition:Multiplicative Magic Square|multiplicative magic squares]] that the [[Definition:Element of Array|elements]] are all [[Definition:Distinct|distinct]].
+Hence no [[Definition:Order of Multiplicative Magic Square|order $2$]] [[Definition:Multiplicative Magic Square|multiplicative magic square]] can exist.
+{{qed}}
+[[Category:Multiplicative Magic Squares]]
+by42pajlvdcgeda5on1eie60h053ebo
+\end{proof}<|endoftext|>
+\section{For Complete Ritz Sequence Continuous Functional approaches its Minimal Value}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $J$ be a [[Definition:Continuous Functional|continuous functional]].
+Let $\sequence {\phi_n}$ be a [[Definition:Complete Ritz Sequence|complete Ritz sequence]].
+{{explain|The concept of $\sequence {\phi_n}$ does not appear to be related in any way to the statement of the theorem.}}
+Then:
+:$\displaystyle \lim_{n \mathop \to \infty} \mu_n = \mu$
+where $\displaystyle \mu = \inf_y J \sqbrk y$.
+\end{theorem}
+
+\begin{proof}
+Let $y^*: \R \to \R$ be such that:
+:$\forall \epsilon > 0: J \sqbrk {y^*} < \mu + \epsilon$
+By assumption of [[Definition:Continuous Functional|continuity of $J$]]:
+:$\forall \epsilon > 0: \exists \map \delta \epsilon > 0: \paren {\size {y - y^*} < \delta} \implies \paren {\size {J \sqbrk y - J \sqbrk {y^*} } < \epsilon}$
+Let $\eta_n = \boldsymbol \alpha \boldsymbol \phi$, such that:
+:$\exists n \in\ N: \exists N \in \N: \paren {n > N} \implies \paren {\size {\eta_n - y^*} < \epsilon}$
+where $\boldsymbol \alpha$ is an [[Definition:Dimension of Vector|$n$-dimensional]] [[Definition:Real Vector Space|real vector]].
+Let $y_n = \boldsymbol \alpha \boldsymbol \phi$, where $\boldsymbol \alpha$ is such that it minimises $J \sqbrk {y_n}$.
+Hence:
+:$J \sqbrk {y_n} < J \sqbrk {\eta_n} + \epsilon < \mu + 2 \epsilon$
+On the other hand, by the definition of $y_n$:
+:$\mu \le J \sqbrk {y_n} \le J \sqbrk {\eta_n}$
+Thus:
+:$\mu \le J \sqbrk {y_n} < \mu + 2 \epsilon$
+If $J \sqbrk {y_n} = \mu$, then the [[Definition:Inequality|inequality]] is satisfied, and $n$ is [[Definition:Finite|finite]].
+{{explain|"finite" does not appear in the original formulation of GF; either restrict the theorem, or refine definitions used}}
+If $ \mu < J \sqbrk {y_n}$, then subtract $\mu$ from all the terms:
+:$0 < J \sqbrk {y_n} - \mu < 2 \epsilon$
+We have that:
+:$\paren {\epsilon > 0} \implies \paren {-2 \epsilon < 0}$
+Hence:
+:$-2 \epsilon < J \sqbrk {y_n} - \mu < 2 \epsilon$
+or:
+:$\size {J \sqbrk {y_n} - \mu} < 2 \epsilon$
+Since $\epsilon$ and $n$ inherits their constraints from the definitions of $\eta_n$ and the [[Definition:Continuous Functional|continuity of $J$]], these together with the last [[Definition:Inequality|inequality]] imply:
+:$\displaystyle \lim_{n \mathop \to \infty} J \sqbrk {y_n} = \mu$
+where $J \sqbrk {y_n} = \mu_n$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{219 Fedorov Groups}
+Tags: Fedorov Groups, 219
+
+\begin{theorem}
+There are $219$ [[Definition:Fedorov Group|Fedorov groups]], if [[Definition:Chirality|chiral copies]] are considered the same.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|First the concept of [[Definition:Fedorov Group|Fedorov group]] needs to be explored.}}
+\end{proof}<|endoftext|>
+\section{17 Wallpaper Groups}
+Tags: Wallpaper Groups, 17
+
+\begin{theorem}
+There are $17$ [[Definition:Wallpaper Group|wallpaper groups]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|First the concept of [[Definition:Wallpaper Group|wallpaper group]] needs to be explored.}}
+\end{proof}<|endoftext|>
+\section{230 Fedorov Groups where Chiral Pairs are Distinct}
+Tags: Fedorov Groups, 230
+
+\begin{theorem}
+There are $219$ [[Definition:Fedorov Group|Fedorov groups]], if [[Definition:Chirality|chiral copies]] are considered [[Definition:Distinct|distinct]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|First the concept of [[Definition:Fedorov Group|Fedorov group]] needs to be explored.}}
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Amicable Pair}
+Tags: Amicable Pairs
+
+\begin{theorem}
+Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
+{{TFAE|def = Amicable Pair}}
+\end{theorem}
+
+\begin{proof}
+=== Definition 1 is equivalent to Definition 2 ===
+Let $s \left({n}\right)$ denote the [[Definition:Aliquot Sum|aliquot sum]] of [[Definition:Strictly Positive Integer|(strictly) positive integer]] $n$.
+The [[Definition:Integer Addition|sum]] of all the [[Definition:Divisor of Integer|divisors]] of a [[Definition:Strictly Positive Integer|(strictly) positive integer]] $n$ is $\sigma \left({n}\right)$, where $\sigma$ is the [[Definition:Sigma Function|$\sigma$ function]].
+The [[Definition:Aliquot Sum|aliquot sum]] of $n$ is the [[Definition:Integer Addition|sum]] of the [[Definition:Divisor of Integer|divisors]] of $n$ with $n$ excluded.
+Thus:
+:$s \left({n}\right) = \sigma \left({n}\right) - n$
+Suppose:
+:$s \left({n}\right) = m$
+and:
+:$s \left({m}\right) = n$
+Then:
+{{begin-eqn}}
+{{eqn | l = \sigma \left({n}\right) - n
+ | r = m
+ | c = {{Defof|Proper Divisor of Integer}}
+}}
+{{eqn | ll= \leadsto
+ | l = \sigma \left({n}\right)
+ | r = m + n
+ | c =
+}}
+{{end-eqn}}
+Similarly:
+{{begin-eqn}}
+{{eqn | l = \sigma \left({m}\right) - m
+ | r = n
+ | c = {{Defof|Proper Divisor of Integer}}
+}}
+{{eqn | ll= \leadsto
+ | l = \sigma \left({m}\right)
+ | r = m + n
+ | c =
+}}
+{{end-eqn}}
+Thus:
+:$s \left({n}\right) = s \left({m}\right) = m + n$
+The argument reverses.
+{{qed|lemma}}
+=== Definition 1 is equivalent to Definition 3 ===
+From the definition of [[Definition:Amicable Pair/Definition 1|definition 1 of an amicable pair]]:
+{{:Definition:Amicable Pair/Definition 1}}
+From the definition of a [[Definition:Sociable Chain|sociable chain]]:
+{{:Definition:Sociable Chain}}
+Here it is seen that setting $r = 2$ gives that:
+:$s \left({a_0}\right) = a_1$
+:$s \left({a_1}\right) = a_0$
+and the equivalence follows.
+{{qed}}
+[[Category:Amicable Pairs]]
+hj2x6uy537g5p9ne9incdjnvm7vk9x8
+\end{proof}<|endoftext|>
+\section{Thabit's Rule}
+Tags: Amicable Pairs
+
+\begin{theorem}
+Let $n$ be a [[Definition:Positive Integer|positive integer]] such that:
+{{begin-eqn}}
+{{eqn | l = a
+ | r = 3 \times 2^n - 1
+ | c =
+}}
+{{eqn | l = b
+ | r = 3 \times 2^{n - 1} - 1
+ | c =
+}}
+{{eqn | l = c
+ | r = 9 \times 2^{2 n - 1} - 1
+ | c =
+}}
+{{end-eqn}}
+are all [[Definition:Prime Number|prime]].
+Then:
+:$\tuple {2^n a b, 2^n c}$
+forms an [[Definition:Amicable Pair|amicable pair]].
+\end{theorem}
+
+\begin{proof}
+Let $r = 2^n a b, s = 2^n c$.
+Let $\map \sigma k$ denote the [[Definition:Sigma Function|sigma function]] on an [[Definition:Integer|integer]] $k$.
+From [[Sigma Function of Power of 2]]:
+:$\map \sigma {2^n} = 2^{n + 1} - 1$
+From [[Sigma Function of Prime Number]]:
+{{begin-eqn}}
+{{eqn | l = \map \sigma a
+ | r = 3 \times 2^n
+ | c =
+}}
+{{eqn | l = \map \sigma b
+ | r = 3 \times 2^{n - 1}
+ | c =
+}}
+{{eqn | l = \map \sigma c
+ | r = 9 \times 2^{2 n - 1}
+ | c =
+}}
+{{end-eqn}}
+From [[Sigma Function is Multiplicative]]:
+{{begin-eqn}}
+{{eqn | l = \map \sigma r
+ | r = \map \sigma {2^n} \map \sigma a \map \sigma b
+ | c =
+}}
+{{eqn | r = \paren {2^{n + 1} - 1} \paren {3 \times 2^n} \paren {3 \times 2^{n - 1} }
+ | c =
+}}
+{{eqn | r = \paren {2^{n + 1} - 1} \paren {9 \times 2^{2 n - 1} }
+ | c =
+}}
+{{end-eqn}}
+and:
+{{begin-eqn}}
+{{eqn | l = \map \sigma s
+ | r = \map \sigma {2^n} \map \sigma c
+ | c =
+}}
+{{eqn | r = \paren {2^{n + 1} - 1} \paren {9 \times 2^{2 n - 1} }
+ | c =
+}}
+{{eqn | r = \map \sigma r
+ | c =
+}}
+{{end-eqn}}
+Thus it is seen that:
+:$\map \sigma r = \map \sigma s$
+Now we have:
+{{begin-eqn}}
+{{eqn | l = r + s
+ | r = 2^n a b + 2^n c
+ | c =
+}}
+{{eqn | r = 2^n \paren {\paren {3 \times 2^n - 1} \paren {3 \times 2^{n - 1} - 1} + 9 \times 2^{2 n - 1} - 1 }
+ | c =
+}}
+{{eqn | r = 2^n \paren {9 \times 2^{2 n - 1} - 3 \times 2^n - 3 \times 2^{n - 1} + 1 + 9 \times 2^{2 n - 1} - 1}
+ | c =
+}}
+{{eqn | r = 2^n \paren {2 \times 9 \times 2^{2 n - 1} - 3 \times 2 \times 2^{n - 1} - 3 \times 2^{n - 1} }
+ | c =
+}}
+{{eqn | r = 2^{n + 1} \paren {9 \times 2^{2 n - 1} - 9 \times 2^{n - 2} }
+ | c =
+}}
+{{eqn | r = 2^{n + 1} \paren {9 \times 2^{2 n - 1} } - 2^{n + 1} \paren {9 \times 2^{n - 2} }
+ | c =
+}}
+{{eqn | r = 2^{n + 1} \paren {9 \times 2^{2 n - 1} } - \paren {9 \times 2^{2 n - 1} }
+ | c =
+}}
+{{eqn | r = \paren {2^{n + 1} - 1} \paren {9 \times 2^{2 n - 1} }
+ | c =
+}}
+{{end-eqn}}
+and so it is seen that:
+:$r + s = \map \sigma r = \map \sigma s$
+Hence the result, by definition of [[Definition:Amicable Pair|amicable pair]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smaller of Thabit Pair is Tetrahedral}
+Tags: Amicable Pairs, Tetrahedral Numbers
+
+\begin{theorem}
+Let $\tuple {m_1, m_2}$ be a [[Definition:Thabit Pair|Thabit pair]] such that $m_1 < m_2$.
+Then $m_1$ is a [[Definition:Tetrahedral Number|tetrahedral number]].
+\end{theorem}
+
+\begin{proof}
+By [[Thabit's Rule]]:
+{{begin-eqn}}
+{{eqn | l = m_1
+ | r = 2^n \times \paren {3 \times 2^{n - 1} - 1} \times \paren {3 \times 2^n - 1}
+ | c =
+}}
+{{eqn | l = m_2
+ | r = 2^n \times \paren {9 \times 2^{2 n - 1} - 1}
+ | c =
+}}
+{{end-eqn}}
+for some $n \in \Z_{\ge 0}$.
+We have that:
+{{begin-eqn}}
+{{eqn | l = \paren {3 \times 2^{n - 1} - 1} \times \paren {3 \times 2^n - 1}
+ | r = 9 \times 2^{2 n - 1} - 3 \times 2^n - 3 \times 2^{n - 1} + 1
+ | c =
+}}
+{{eqn | o = <
+ | r = 9 \times 2^{2 n - 1} - 1
+ | c =
+}}
+{{end-eqn}}
+and so $m_1 < m_2$.
+Then:
+{{begin-eqn}}
+{{eqn | l = m_1
+ | r = 2^n \times \paren {3 \times 2^{n - 1} - 1} \times \paren {3 \times 2^n - 1}
+ | c =
+}}
+{{eqn | r = \frac {2^n \times \paren {3 \times 2^{n - 1} - 1} \times \paren {3 \times 2^n - 1} \times 6} 6
+ | c =
+}}
+{{eqn | r = \frac {\paren {3 \times 2^n} \times \paren {2 \times 3 \times 2^{n - 1} - 2} \times \paren {3 \times 2^n - 1} } 6
+ | c =
+}}
+{{eqn | r = \frac {\paren {3 \times 2^n} \times \paren {3 \times 2^n - 1} \times \paren {3 \times 2^n - 2} } 6
+ | c =
+}}
+{{eqn | r = \frac {k \times \paren {k + 1} \times \paren {k + 2} } 6
+ | c = where $k = 3 \times 2^n - 2$
+}}
+{{end-eqn}}
+From [[Closed Form for Tetrahedral Numbers]], this is the $k$th [[Definition:Tetrahedral Number|tetrahedral number]] where $k = 3 \times 2^n - 2$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sturm-Liouville Problem}
+Tags:
+
+\begin{theorem}
+Let:
+:$P \in C^\infty \map P x > 0$
+:$Q \in C^0$
+:$-\paren {P y'}' + Q y = \lambda y$
+:$\map y a = \map y b = 0$
+Then the Sturm-Liouville problem has an infinite sequence of eigenvalues $\set {\lambda^{\paren n} }$, and to each $\lambda^{\paren n}$ corresponds an eigenfunction $ y^{ \left ( { n } \right ) } $, unique up to a constant factor.
+\end{theorem}
+
+\begin{proof}
+:$J \sqbrk y = \displaystyle \int_a^b \paren {P y'^2 + Q y^2} \rd x$
+:$\displaystyle \int_a^b y^2 \rd x = 1$
+:$\displaystyle \int_a^b \paren {P y'^2 + Q y^2} \rd x > \int_a^b Q y^2 \rd x \ge M \int_a^b y^2 \rd x = M$
+:$M = \min \limits_{a \mathop \le x \mathop \le b} \map Q x$
+Assume $a = 0$, $b = \pi$.
+Choose $\set {\map {\phi_n} x} = \set {\sin n x}$
+For $k \ne l$ we have:
+:$\displaystyle \int_0^\pi \sin k x \sin l x \rd x = 0$
+:$\displaystyle \int_0^\pi \paren {\sum_{k \mathop = 1}^n \alpha_k \sin k x}^2 \rd x = \dfrac \pi 2 \sum_{k \mathop = 1}^n \alpha_k^2 = 1$
+:$\displaystyle \map {J_n} {\alpha_1, \ldots, \alpha_n} = \int_o^\pi \paren {\map P {\sum_{k \mathop = 1}^n \alpha_k \sin k x}'^2 + \map Q {\sum_{k \mathop = 1}^n \alpha_k \sin k x}^2} \rd x$
+{{explain|The scope of the $'$ and ${}^2$ in $\displaystyle \map P {\sum_{k \mathop {{=}} 1}^n \alpha_k \sin k x}'^2$ needs to be clarified in the above}}
+:$\displaystyle y_n^{\paren 1} \left ( { x } \right ) = \sum_{k \mathop = 1}^n \alpha_k^{\paren 1} \sin k x$
+:$\set {\lambda_n^{\paren 1} }$
+:$\set {y_n^{\paren 1} }$
+:$\map {J_n} {\alpha_1, \ldots, \alpha_n} = \map {J_{n + 1} } {\alpha_1, \ldots, \alpha_n, 0}$
+:$\lambda_{n + 1}^{\paren 1} \le \lambda_n^{\paren 1}$
+:$\lambda^{\paren 1} = \lim_{n \mathop \to \infty} \lambda_n^{\paren 1}$
+:$\displaystyle \lambda_n^{\paren 1} = \int_0^\pi \paren {P y_n'^2 + Q y_n^2} \rd x$
+:$\displaystyle \int_0^\pi \paren {P y_n'^2 + Q y_n^2} \rd x \le M$
+:$\displaystyle \int_0^\pi Py_n'^2 \rd x \le M + \size {\int_0^\pi Q y_n^2 \rd x} \le M + \max \limits_{a \mathop \le x \mathop \le b } \size {\map Q x} = M_1$
+:$\displaystyle \int_0^\pi \map {y_n'^2} x \rd x \le \dfrac {M_1} {\min \limits_{a \mathop \le x \mathop \le b} \map P x} = M_2$
+:$\map {y_n} 0 = 0$
+:$\displaystyle\size {\map {y_n} x}^2 = \size {\int_0^x \map {y_n'} \zeta \rd \zeta}^2 \le \int_0^x \map {y_n'^2} \zeta \rd \zeta \int_0^x \rd \zeta \le M_2 \pi$
+:$\displaystyle\size {\map {y_n} {x_2} - \map {y_n} {x_1} }^2 = \size {\int_{x_1}^{x_2} \map {y_n'} x \rd x}^2 \le \int_{x_1}^{x_2} y_n'^2 \rd x \size {\int_{x_1}^{x_2} \rd x}^2 \le M_2 \size {x_2 - x_1}$
+:$\map {y^{\paren 1} } x = \lim_{m \mathop \to \infty} \map {y_{n_m} } x$
+:$\int_0^\pi \paren {-\paren {P h'}' + Q_1 h} y \rd x = 0$
+:$\map h x \in \map {\mathcal D_2} {0, \pi}$
+:$\map h 0 = \map h \pi = \map {h'} 0 = \map {h'} \pi = 0$
+:$\map y x \in \map {\mathcal D_2} {0, \pi}$
+:$-\paren {P y'}' + Q_1 y = 0$
+:$\displaystyle \int_0^\pi \paren {-\paren {P y'}' + Q_1 y} y \rd x = -\int_0^\pi P h'' y \rd x - \int_0^\pi P' h' y \rd x + \int_0^\pi Q_1 h y \rd x = - \int_0^\pi \paren {- P y + \int_0^x P' y \rd \zeta + \int_0^x \paren {\int_0^\zeta Q_1 y \rd t} \rd \zeta} \rd x = 0$
+:$-\paren {P y}' + P' y + \displaystyle \int_0^x Q_1 y \rd \zeta = c_1$
+:$-P y' + \displaystyle \int_0^x Q_1 y \rd \zeta = c_1$
+:$-\paren {P y'}' + Q_1 y = 0$
+:$-\paren {P {y^{\paren 1} }'}' + Q y^{\paren 1} = \lambda^{\paren 1} y^{\paren 1}$
+:$\dfrac \partial {\partial \alpha_r} \paren {\map {J_n} {\alpha_1, \ldots, \alpha_n} - \lambda_n^{\paren 1} \displaystyle \int_0^\pi \paren {\sum_{k \mathop = 1}^n \alpha_k \sin k x}^2 \rd x} = 0$
+{{ProofWanted}}
+jif4paeyaqadrmzbik9zvfzx008y0dc
+\end{proof}<|endoftext|>
+\section{Numbers in Even-Even Amicable Pair are not Divisible by 3}
+Tags: Amicable Pairs
+
+\begin{theorem}
+Let $\tuple {m_1, m_2}$ be an [[Definition:Amicable Pair|amicable pair]] such that both $m_1$ and $m_2$ are [[Definition:Even Integer|even]].
+Then neither $m_1$ nor $m_2$ is [[Definition:Divisor of Integer|divisible]] by $3$.
+\end{theorem}
+
+\begin{proof}
+An [[Definition:Amicable Pair|amicable pair]] must be formed from a smaller [[Definition:Abundant Number|abundant number]] and a larger [[Definition:Deficient Number|deficient number]].
+Suppose both $m_1, m_2$ are [[Definition:Divisor of Integer|divisible]] by $3$.
+Since both are [[Definition:Even Integer|even]], they must also be [[Definition:Divisor of Integer|divisible]] by $6$.
+However $6$ is a [[Definition:Perfect Number|perfect number]].
+By [[Multiple of Perfect Number is Abundant]], neither can be [[Definition:Deficient Number|deficient]].
+So $m_1, m_2$ cannot form an [[Definition:Amicable Pair|amicable pair]].
+Therefore at most one of them is [[Definition:Divisor of Integer|divisible]] by $3$.
+{{WLOG}} suppose $m_1$ is [[Definition:Divisor of Integer|divisible]] by $3$.
+Write:
+:$m_1 = 2^r 3^t a, m_2 = 2^s b$
+where $a, b$ are not [[Definition:Divisor of Integer|divisible]] by $2$ or $3$.
+Then:
+{{begin-eqn}}
+{{eqn | l = m + n
+ | r = \map \sigma {m_2}
+ | c = {{Defof|Amicable Pair|subdef = Definition 2}}
+}}
+{{eqn | r = \map \sigma {2^s} \map \sigma b
+ | c = [[Sigma Function is Multiplicative]]
+}}
+{{eqn | r = \paren {2^{s + 1} - 1} \map \sigma b
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | o = \equiv
+ | r = \paren {\paren {-1}^{s + 1} - 1} \map \sigma b
+ | rr = \pmod 3
+ | c = [[Congruence of Powers]]
+}}
+{{end-eqn}}
+Since $m + n$ is not [[Definition:Divisor of Integer|divisible]] by $3$, $s$ must be [[Definition:Even Integer|even]].
+Similarly, by [[Sigma Function is Multiplicative]], $t$ must also be [[Definition:Even Integer|even]].
+In particular, both $s, t$ are at least $2$.
+Now write:
+:$m_1 = 2^2 \cdot 3 k, m_2 = 2^2 \cdot l$
+where $k, l$ are some [[Definition:Integer|integers]].
+By [[Multiple of Perfect Number is Abundant]], $m_1$ is [[Definition:Abundant Number|abundant number]].
+Therefore $m_2 > m_1$.
+This leads to $l > 3 k \ge 3$.
+By [[Abundancy Index of Product is greater than Abundancy Index of Proper Factors]]:
+{{begin-eqn}}
+{{eqn | l = \frac {\map \sigma {m_1} } {m_1}
+ | o = \ge
+ | r = \frac {\map \sigma {12} } {12}
+ | c = equality occurs {{iff}} $m_1 = 12$
+}}
+{{eqn | r = \frac 7 3
+}}
+{{eqn | l = \frac {\map \sigma {m_2} } {m_2}
+ | o = >
+ | r = \frac {\map \sigma 4} 4
+}}
+{{eqn | r = \frac 7 4
+}}
+{{end-eqn}}
+But:
+{{begin-eqn}}
+{{eqn | l = 1
+ | r = \frac {m_1 + m_2} {m_1 + m_2}
+}}
+{{eqn | r = \frac {m_1} {\map \sigma {m_1} } + \frac {m_2} {\map \sigma {m_2} }
+ | c = {{Defof|Amicable Pair|subdef = Definition 2}}
+}}
+{{eqn | o = <
+ | r = \frac 3 7 + \frac 4 7
+}}
+{{eqn | r = 1
+}}
+{{end-eqn}}
+which is a [[Definition:Contradiction|contradiction]].
+Therefore neither $m_1$ nor $m_2$ is [[Definition:Divisor of Integer|divisible]] by $3$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Amicable Pair with Smallest Common Prime Factor 5}
+Tags: Amicable Pairs
+
+\begin{theorem}
+A [[Definition:Amicable Pair|amicable pair]] whose smallest [[Definition:Common Divisor of Integers|common]] [[Definition:Prime Factor|prime factor]] is greater than $3$ has the elements:
+:$m_1 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 101 \times 8693 \times 19 \, 479 \, 382 \, 229$
+and:
+:$m_2 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 365 \, 147 \times 47 \, 307 \, 071 \, 129$
+This was at one point the smallest known counterexample to the observation that:
+:most [[Definition:Amicable Pair|amicable pairs]] consist of [[Definition:Even Integer|even integers]]
+:most of the rest, whose elements are [[Definition:Odd Integer|odd]], have both elements [[Definition:Divisor of Integer|divisible]] by $3$.
+\end{theorem}
+
+\begin{proof}
+It is to be demonstrated that these numbers are [[Definition:Amicable Pair|amicable]].
+From [[Sigma Function of Integer]]:
+:$\displaystyle \map \sigma n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
+where:
+:$\displaystyle n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$
+is the [[Definition:Prime Decomposition|prime decomposition]] of $n$.
+When $k_i = 1$ the individual factor becomes $\dfrac {p_i^2 - 1} {p_i - 1} = \dfrac {\paren {p_i + 1} \paren {p_i - 1} } {p_i - 1} = p_i + 1$.
+We establish the contributions to the [[Definition:Sigma Function|$\sigma$ values]] of $m_1$ and $m_2$ by taking the [[Definition:Prime Factor|prime factors]] in turn, and extracting the [[Definition:Prime Factor|prime factors]] of each result.
+First, the elements common to both:
+{{begin-eqn}}
+{{eqn | l = \map \sigma 5
+ | r = 6
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 3
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {7^2}
+ | r = \dfrac {7^3 - 1} {7 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {343 - 1} 6
+ | c =
+}}
+{{eqn | r = 57
+ | c =
+}}
+{{eqn | r = 3 \times 19
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {11^2}
+ | r = \dfrac {11^3 - 1} {11 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {1331 - 1} {10}
+ | c =
+}}
+{{eqn | r = 133
+ | c =
+}}
+{{eqn | r = 7 \times 19
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {13}
+ | r = 14
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 7
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {17}
+ | r = 18
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 3^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {19^3}
+ | r = \dfrac {19^4 - 1} {19 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {130 \, 321 - 1} {18}
+ | c =
+}}
+{{eqn | r = 7240
+ | c =
+}}
+{{eqn | r = 2^3 \times 5 \times 181
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {23}
+ | r = 24
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^3 \times 3
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {37}
+ | r = 38
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 19
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {181}
+ | r = 182
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 7 \times 13
+ | c =
+}}
+{{end-eqn}}
+This gives a [[Definition:Common Divisor of Integers|common factor]] of both $\sigma$ values of:
+{{begin-eqn}}
+{{eqn | r = 6 \times 57 \times 133 \times 14 \times 18 \times 7240 \times 24 \times 38 \times 182
+ | o =
+}}
+{{eqn | r = 2^{11} \times 3^5 \times 5 \times 7^3 \times 13 \times 19^3 \times 181
+ | c =
+}}
+{{end-eqn}}
+The remaining [[Definition:Prime Factor|prime factors]] of $m_1$:
+{{begin-eqn}}
+{{eqn | l = \map \sigma {101 \times 8693 \times 19 \, 479 \, 382 \, 229}
+ | r = \paren {101 + 1} \paren {8693 + 1} \paren {19 \, 479 \, 382 \, 229 + 1}
+ | c = [[Sigma Function of Square-Free Integer]]
+}}
+{{eqn | r = \paren {2 \times 3 \times 17} \paren {2 \times 3^3 \times 7 \times 23} \paren {2 \times 3 \times 5 \times 7 \times 11^2 \times 37 \times 20 \, 719}
+ | c =
+}}
+{{eqn | r = 2^3 \times 3^5 \times 5 \times 7^2 \times 11^2 \times 17 \times 23 \times 37 \times 20 \, 719
+ | c =
+}}
+{{end-eqn}}
+The remaining [[Definition:Prime Factor|prime factors]] of $m_2$:
+{{begin-eqn}}
+{{eqn | l = \map \sigma {365 \, 147 \times 47 \, 307 \, 071 \, 129}
+ | r = \paren {365 \, 147 + 1} \paren {47 \, 307 \, 071 \, 129 + 1}
+ | c = [[Sigma Function of Square-Free Integer]]
+}}
+{{eqn | r = \paren {2^2 \times 3^4 \times 7^2 \times 23} \paren {2 \times 3 \times 5 \times 11^2 \times 17 \times 37 \times 20 \, 719}
+ | c =
+}}
+{{eqn | r = 2^3 \times 3^5 \times 5 \times 7^2 \times 11^2 \times 17 \times 23 \times 37 \times 20 \, 719
+ | c =
+}}
+{{end-eqn}}
+and so:
+{{begin-eqn}}
+{{eqn | l = \map \sigma {m_1} = \map \sigma {m_2}
+ | r = \paren {2^{11} \times 3^5 \times 5 \times 7^3 \times 13 \times 19^3 \times 181} \paren {2^3 \times 3^5 \times 5 \times 7^2 \times 11^2 \times 17 \times 23 \times 37 \times 20 \, 719}
+ | c =
+}}
+{{eqn | r = 2^{14} \times 3^{10} \times 5^2 \times 7^5 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 20 \, 719
+ | c =
+}}
+{{end-eqn}}
+It remains to be shown that:
+:$\map \sigma {m_1} = \map \sigma {m_2} = m_1 + m_2$
+So:
+{{begin-eqn}}
+{{eqn | l = m_1 + m_2
+ | r = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times \paren {101 \times 8693 \times 19 \, 479 \, 382 \, 229 + 365 \, 147 \times 47 \, 307 \, 071 \, 129}
+ | c =
+}}
+{{eqn | r = 6921698310331805 \times \paren {17102761241386397 + 17274035101540963}
+ | c =
+}}
+{{eqn | r = 6921698310331805 \times 34376796342927360
+ | c =
+}}
+{{eqn | r = 237945813161460881075326612684800
+ | c =
+}}
+{{eqn | r = 2^{14} \times 3^{10} \times 5^2 \times 7^5 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 20719
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Amicable Triplet}
+Tags: Amicable Triplets
+
+\begin{theorem}
+Let $m_1, m_2, m_3 \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
+{{TFAE|def = Amicable Triplet}}
+\end{theorem}
+
+\begin{proof}
+For $n \in \Z_{>0}$, let $s \left({n}\right)$ denote the [[Definition:Aliquot Sum|aliquot sum]] of [[Definition:Strictly Positive Integer|(strictly) positive integer]] $n$.
+The [[Definition:Integer Addition|sum]] of all the [[Definition:Divisor of Integer|divisors]] of a [[Definition:Strictly Positive Integer|(strictly) positive integer]] $n$ is $\sigma \left({n}\right)$, where $\sigma$ is the [[Definition:Sigma Function|$\sigma$ function]].
+The [[Definition:Aliquot Sum|aliquot sum]] of $n$ is the [[Definition:Integer Addition|sum]] of the [[Definition:Divisor of Integer|divisors]] of $n$ with $n$ excluded.
+Thus:
+:$s \left({n}\right) = \sigma \left({n}\right) - n$
+Suppose:
+{{begin-eqn}}
+{{eqn | l = s \left({m_1}\right)
+ | r = m_2 + m_3
+}}
+{{eqn | l = s \left({m_2}\right)
+ | r = m_1 + m_3
+}}
+{{eqn | l = s \left({m_3}\right)
+ | r = m_1 + m_2
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \sigma \left({m_1}\right) - m_1
+ | r = m_2 + m_3
+ | c = {{Defof|Proper Divisor of Integer}}
+}}
+{{eqn | ll= \leadsto
+ | l = \sigma \left({m_1}\right)
+ | r = m_1 + m_2 + m_3
+ | c =
+}}
+{{end-eqn}}
+Similarly:
+{{begin-eqn}}
+{{eqn | l = \sigma \left({m_2}\right) - m_2
+ | r = m_1 + m_3
+ | c = {{Defof|Proper Divisor of Integer}}
+}}
+{{eqn | ll= \leadsto
+ | l = \sigma \left({m_2}\right)
+ | r = m_1 + m_2 + m_3
+ | c =
+}}
+{{end-eqn}}
+and:
+{{begin-eqn}}
+{{eqn | l = \sigma \left({m_3}\right) - m_3
+ | r = m_1 + m_2
+ | c = {{Defof|Proper Divisor of Integer}}
+}}
+{{eqn | ll= \leadsto
+ | l = \sigma \left({m_3}\right)
+ | r = m_1 + m_2 + m_3
+ | c =
+}}
+{{end-eqn}}
+Thus:
+:$\sigma \left({m_1}\right) = \sigma \left({m_2}\right) = \sigma \left({m_3}\right) = m_1 + m_2 + m_3$
+The argument reverses.
+{{qed}}
+[[Category:Amicable Triplets]]
+qc6ebahwfbfp6nhfz4z3ayqnqab1fcm
+\end{proof}<|endoftext|>
+\section{Largest Number not Expressible as Sum of Less than 37 Positive Fifth Powers}
+Tags: Fifth Powers, Hilbert-Waring Theorem, 223
+
+\begin{theorem}
+The largest [[Definition:Positive Integer|positive integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of less than $37$ [[Definition:Positive Integer|positive]] [[Definition:Fifth Power|fifth powers]] is $223$:
+:$223 = 31 \times 1^5 + 6 \times 2^5$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|It needs to be shown that there are no larger numbers with this property.}}
+\end{proof}<|endoftext|>
+\section{Smallest Triple of Consecutive Sums of Squares}
+Tags: Sums of Squares
+
+\begin{theorem}
+The smallest [[Definition:Ordered Triple|triple]] of consecutive [[Definition:Positive Integer|positive integers]] each of which is the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]] is:
+:$\tuple {232, 233, 234}$
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = 232
+ | r = 14^2 + 6^2
+ | c =
+}}
+{{eqn | l = 233
+ | r = 13^2 + 8^2
+ | c =
+}}
+{{eqn | l = 234
+ | r = 15^2 + 3^2
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown this is the smallest such triple.}}
+\end{proof}<|endoftext|>
+\section{No Quadruple of Consecutive Sums of Squares Exists}
+Tags: Sums of Squares
+
+\begin{theorem}
+It is not possible for a [[Definition:Ordered Quadruple|quadruple]] of consecutive [[Definition:Positive Integer|positive integers]] each of which is the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]].
+\end{theorem}
+
+\begin{proof}
+$4$ consecutive [[Definition:Positive Integer|positive integers]] will be in the forms:
+{{begin-eqn}}
+{{eqn | l = n_0
+ | o = \equiv
+ | r = 0
+ | rr= \pmod 4
+ | c =
+}}
+{{eqn | l = n_1
+ | o = \equiv
+ | r = 1
+ | rr= \pmod 4
+ | c =
+}}
+{{eqn | l = n_2
+ | o = \equiv
+ | r = 2
+ | rr= \pmod 4
+ | c =
+}}
+{{eqn | l = n_3
+ | o = \equiv
+ | r = 3
+ | rr= \pmod 4
+ | c =
+}}
+{{end-eqn}}
+in some order.
+But from [[Sum of Two Squares not Congruent to 3 modulo 4]], $n_3$ cannot be the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]].
+The result follows.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Directed iff Filtered in Dual Ordered Set}
+Tags: Dual Ordered Sets
+
+\begin{theorem}
+Let $\struct {S, \preceq_1}$ be an [[Definition:Ordered Set|ordered set]].
+Let $\struct {S, \preceq_2}$ be a [[Definition:Dual Ordered Set|dual ordered set]] of $\struct {S, \preceq_1}$
+Let $X \subseteq S$.
+Then:
+:$X$ is [[Definition:Directed Subset|directed]] in $\struct {S, \preceq_1}$
+{{iff}}:
+:$X$ is [[Definition:Filtered Subset|filtered]] in $\struct {S, \preceq_2}$.
+\end{theorem}
+
+\begin{proof}
+=== Sufficient Condition ===
+Assume that:
+:$X$ is [[Definition:Directed Subset|directed]] in $\struct {S, \preceq_1}$
+Thus $X$ is [[Definition:Non-Empty Set|non-empty]].
+Let $x, y \in S$.
+By definition of [[Definition:Directed Subset|directed]]:
+:$\exists z \in S: x \preceq_1 z \land y \preceq_1 z$
+Thus by definition of [[Definition:Dual Ordered Set|dual ordered set]]:
+:$z \preceq_2 x \land z \preceq_2 y$
+{{qed|lemma}}
+=== Necessary Condition ===
+Assume that:
+:$X$ is [[Definition:Filtered Subset|filtered]] in $\struct {S, \preceq_2}$
+Thus $X$ is [[Definition:Non-Empty Set|non-empty]].
+Let $x, y \in S$.
+By definition of [[Definition:Filtered Subset|filtered]]:
+:$\exists z \in S: z \preceq_2 x \land z \preceq_2 y$
+Thus by definition of [[Definition:Dual Ordered Set|dual ordered set]]:
+:$x \preceq_1 z \land y \preceq_1 z$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum of Two Squares not Congruent to 3 modulo 4}
+Tags: Sums of Squares
+
+\begin{theorem}
+Let $n \in \Z$ such that $n = a^2 + b^2$ where $a, b \in \Z$.
+Then $n$ is not [[Definition:Congruence Modulo Integer|congruent modulo $4$]] to $3$.
+\end{theorem}
+
+\begin{proof}
+Let $n \equiv 3 \pmod 4$.
+{{AimForCont}} $n$ can be expressed as the sum of two [[Definition:Square Number|squares]]:
+:$n = a^2 + b^2$.
+From [[Square Modulo 4]], either $a^2 \equiv 0$ or $a^2 \equiv 1 \pmod 4$.
+Similarly for $b^2$.
+So $a^2 + b^2 \not \equiv 3 \pmod 4$ whatever $a$ and $b$ are.
+Thus $n$ cannot be the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]].
+The result follows by [[Proof by Contradiction]].
+{{qed}}
+[[Category:Sums of Squares]]
+jm7po2fj9tqvqo4v2sirg2sru4hrvcp
+\end{proof}<|endoftext|>
+\section{Numbers not Expressible as Sum of Less than 9 Positive Cubes}
+Tags: Hilbert-Waring Theorem, Cube Numbers, 23, 239
+
+\begin{theorem}
+The following are the only [[Definition:Positive Integer|positive integers]] cannot be expressed as the [[Definition:Integer Addition|sum]] of less than $9$ [[Definition:Positive Integer|positive]] [[Definition:Cube Number|cubes]]:
+{{begin-eqn}}
+{{eqn | l = 23
+ | r = 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
+ | c =
+}}
+{{eqn | l = 239
+ | r = 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|It needs to be demonstrated that these are the only ones.}}
+\end{proof}<|endoftext|>
+\section{Largest Prime Factor of n squared plus 1}
+Tags: Number Theory, 239
+
+\begin{theorem}
+Let $n \in \Z$ be greater than $239$.
+Then the largest [[Definition:Prime Factor|prime factor]] of $n^2 + 1$ is at least $17$.
+\end{theorem}
+
+\begin{proof}
+We note that for $n = 239$ we have:
+{{begin-eqn}}
+{{eqn | r = 239^2 + 1
+ | o =
+ | c =
+}}
+{{eqn | r = 57122
+ | c =
+}}
+{{eqn | r = 2 \times 13^4
+ | c =
+}}
+{{end-eqn}}
+Thus the largest [[Definition:Prime Factor|prime factor]] of $239^2 + 1$ is $13$.
+{{ProofWanted|Now the rest of the result is to be shown.}}
+\end{proof}<|endoftext|>
+\section{Solution of Ljunggren Equation}
+Tags: Diophantine Equations, 239, 13
+
+\begin{theorem}
+The only solutions of the [[Definition:Ljunggren Equation|Ljunggren equation]]:
+:$x^2 + 1 = 2 y^4$
+are:
+:$x = 1, y = 1$
+:$x = 239, y = 13$
+{{OEIS|A229384}}
+\end{theorem}
+
+\begin{proof}
+Setting $x = 1$:
+{{begin-eqn}}
+{{eqn | r = 1^2 + 1
+ | o =
+ | c =
+}}
+{{eqn | r = 2
+ | c =
+}}
+{{eqn | r = 2 \times 1^4
+ | c =
+}}
+{{end-eqn}}
+and so $y = 1$.
+Setting $x = 239$:
+{{begin-eqn}}
+{{eqn | r = 239^2 + 1
+ | o =
+ | c =
+}}
+{{eqn | r = 57122
+ | c =
+}}
+{{eqn | r = 2 \times 13^4
+ | c =
+}}
+{{end-eqn}}
+and so $y = 13$.
+{{ProofWanted|It remains to be shown this is the only solution. Perhaps trial and error for $y$ going from $0$ up to $13$ and then using [[Largest Prime Factor of n squared plus 1|Largest Prime Factor of $n^2 + 1$]]?}}
+\end{proof}<|endoftext|>
+\section{Solutions of Diophantine Equation x^4 + y^4 = z^2 + 1 for x = 239}
+Tags: Diophantine Equations, 239
+
+\begin{theorem}
+Consider the [[Definition:Indeterminate Equation|indeterminate]] [[Definition:Diophantine Equation|Diophantine equation]]:
+:$x^4 + y^4 = z^2 + 1$
+When $x = 239$ and $x > y$, there are $3$ solutions:
+{{begin-eqn}}
+{{eqn | l = 239^4 + 104^4
+ | r = 58 \, 136^2 + 1
+ | c =
+}}
+{{eqn | l = 239^4 + 143^4
+ | r = 60 \, 671^2 + 1
+ | c =
+}}
+{{eqn | l = 239^4 + 208^4
+ | r = 71 \, 656^2 + 1
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|It remains to be shown there are no other solutions. This can be achieved by brute force. For this page, it is then adequate to present a pseudocode program to illustrate it.}}
+\end{proof}<|endoftext|>
+\section{Number with over 240 Divisors is greater than 1,000,000}
+Tags: Tau Function, 240
+
+\begin{theorem}
+Let $n \in \Z$ be an [[Definition:Positive Integer|integer]] with over $240$ [[Definition:Divisor of Integer|divisors]].
+Then:
+:$n > 1 \, 000 \, 000$
+\end{theorem}<|endoftext|>
+\section{Smallest Numbers with 240 Divisors}
+Tags: Tau Function
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|integers]] with $240$ [[Definition:Divisor of Integer|divisors]] are:
+:$720 \, 720, 831 \, 600, 942 \, 480, 982 \, 800, 997 \, 920, \ldots$
+\end{theorem}
+
+\begin{proof}
+Let $\map \tau n$ denote the [[Definition:Tau Function|$\tau$ (tau) function]] of $n$:
+:$\map \tau n$ is the number of [[Definition:Divisor of Integer|divisors]] of $n$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \map \tau {720 \, 720}
+ | r = 240
+ | c = {{TauLink|720,720|720 \, 720}}
+}}
+{{eqn | l = \map \tau {831 \, 600}
+ | r = 240
+ | c = {{TauLink|831,600|831 \, 600}}
+}}
+{{eqn | l = \map \tau {942 \, 480}
+ | r = 240
+ | c = {{TauLink|942,480|942 \, 480}}
+}}
+{{eqn | l = \map \tau {982 \, 800}
+ | r = 240
+ | c = {{TauLink|982,800|982 \, 800}}
+}}
+{{eqn | l = \map \tau {997 \, 920}
+ | r = 240
+ | c = {{TauLink|997,920|997 \, 920}}
+}}
+{{end-eqn}}
+{{ProofWanted|Prove that these are indeed the smallest}}
+\end{proof}<|endoftext|>
+\section{Sequence of 4 Consecutive Integers with Equal Number of Divisors}
+Tags: Tau Function
+
+\begin{theorem}
+The following [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|integers]] are [[Definition:Set|sets]] of $4$ consecutive [[Definition:Positive Integer|integers]] which all have the same number of [[Definition:Divisor of Integer|divisors]]:
+:$\map \tau m = \map \tau {m + 1} = \map \tau {m + 2} = \map \tau {m + 3}$
+where $\map \tau n$ denotes the [[Definition:Tau Function|$\tau$ function]].
+:$242, 243, 244, 245, 3655, 3656, 3657, 3658, 4503, 4504, 4505, 4506, \ldots$
+{{OEIS|A039665}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \map \tau {242}
+ | r = 6
+ | c = {{TauLink|242}}
+}}
+{{eqn | l = \map \tau {243}
+ | r = 6
+ | c = {{TauLink|243}}
+}}
+{{eqn | l = \map \tau {244}
+ | r = 6
+ | c = {{TauLink|244}}
+}}
+{{eqn | l = \map \tau {245}
+ | r = 6
+ | c = {{TauLink|245}}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \tau {3655}
+ | r = 8
+ | c = {{TauLink|3655}}
+}}
+{{eqn | l = \map \tau {3656}
+ | r = 8
+ | c = {{TauLink|3656}}
+}}
+{{eqn | l = \map \tau {3657}
+ | r = 8
+ | c = {{TauLink|3657}}
+}}
+{{eqn | l = \map \tau {3658}
+ | r = 8
+ | c = {{TauLink|3658}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes/Sequence}
+Tags: Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] which are both the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Square Number|square numbers]] in two [[Definition:Distinct|distinct]] ways and also the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Cube Number|cube numbers]] begins:
+:$65, 250, \ldots$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 65
+ | r = 4^3 + 1^3
+ | c =
+}}
+{{eqn | r = 8^2 + 1^2
+ | c =
+}}
+{{eqn | r = 7^2 + 4^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 250
+ | r = 5^3 + 5^3
+ | c =
+}}
+{{eqn | r = 15^2 + 5^2
+ | c =
+}}
+{{eqn | r = 13^2 + 9^2
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum of 3 Cubes in 2 Distinct Ways}
+Tags: Sums of Cubes
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] which can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Cube Number|cubes numbers]] in two or more different ways begins:
+{{begin-eqn}}
+{{eqn | l = 251
+ | r = 5^3 + 5^3 + 1^3
+ | rr= = 6^3 + 3^3 + 2^3
+}}
+{{end-eqn}}
+{{OEIS|???}}
+\end{theorem}<|endoftext|>
+\section{78,557 is Sierpiński}
+Tags: Sierpiński Numbers of the Second Kind, 78,557
+
+\begin{theorem}
+$78 \, 557$ is a [[Definition:Sierpiński Number of the Second Kind|Sierpiński number of the second kind]].
+\end{theorem}
+
+\begin{proof}
+When considering $\bmod {36}$, every [[Definition:Positive Integer|positive integer]] $n$ can be written in one of the forms:
+:$2 k, 4 k + 1, 3 k + 1, 12 k + 11, 18 k + 15, 36 k + 27, 9 k + 3$
+$\begin{array}{|c|c|}
+\hline n \bmod {36} & \text { Form } \\
+\hline 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34 & 2 k \\
+\hline 1, 5, 9, 13, 17, 21, 25, 29, 33 & 4 k + 1 \\
+\hline 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34 & 3 k + 1 \\
+\hline 11, 23, 35 & 12 k + 11 \\
+\hline 15, 33 & 18 k + 15 \\
+\hline 27 & 36 k + 27 \\
+\hline 3, 12, 21, 30 & 9 k + 3\\
+\hline
+\end{array}$
+As seen by inspection, every number less than $36$ is included in the table.
+And we have:
+{{begin-eqn}}
+{{eqn | l = 78557 \times 2^{2 k} + 1
+ | o = \equiv
+ | r = 78557 \times 1^k + 1
+ | rr = \pmod 3
+ | c = [[Fermat's Little Theorem]] and [[Congruence of Powers]]
+}}
+{{eqn | o = \equiv
+ | r = 2 + 1
+ | rr = \pmod 3
+ | c = [[Congruence of Product]]
+}}
+{{eqn | o = \equiv
+ | r = 0
+ | rr = \pmod 3
+}}
+{{eqn | l = 78557 \times 2^{4 k + 1} + 1
+ | o = \equiv
+ | r = 78557 \times 1^k \times 2 + 1
+ | rr = \pmod 5
+ | c = [[Fermat's Little Theorem]] and [[Congruence of Powers]]
+}}
+{{eqn | o = \equiv
+ | r = 2 \times 2 + 1
+ | rr = \pmod 5
+ | c = [[Congruence of Product]]
+}}
+{{eqn | o = \equiv
+ | r = 0
+ | rr = \pmod 5
+}}
+{{eqn | l = 78557 \times 2^{3 k + 1} + 1
+ | r = 78557 \times 8^k \times 2 + 1
+}}
+{{eqn | o = \equiv
+ | r = 78557 \times 1^k \times 2 + 1
+ | rr = \pmod 7
+ | c = [[Congruence of Powers]]
+}}
+{{eqn | o = \equiv
+ | r = 3 \times 2 + 1
+ | rr = \pmod 7
+ | c = [[Congruence of Product]]
+}}
+{{eqn | o = \equiv
+ | r = 0
+ | rr = \pmod 7
+}}
+{{eqn | l = 78557 \times 2^{12 k + 11} + 1
+ | o = \equiv
+ | r = 78557 \times 1^k \times 2^{11} + 1
+ | rr = \pmod {13}
+ | c = [[Fermat's Little Theorem]] and [[Congruence of Powers]]
+}}
+{{eqn | o = \equiv
+ | r = 11 \times 2048 + 1
+ | rr = \pmod {13}
+ | c = [[Congruence of Product]]
+}}
+{{eqn | o = \equiv
+ | r = 0
+ | rr = \pmod {13}
+}}
+{{eqn | l = 78557 \times 2^{18 k + 15} + 1
+ | o = \equiv
+ | r = 78557 \times 1 \times 2^{15} + 1
+ | rr = \pmod {19}
+ | c = [[Fermat's Little Theorem]] and [[Congruence of Powers]]
+}}
+{{eqn | o = \equiv
+ | r = 11 \times 32768 + 1
+ | rr = \pmod {19}
+ | c = [[Congruence of Product]]
+}}
+{{eqn | o = \equiv
+ | r = 0
+ | rr = \pmod {19}
+}}
+{{eqn | l = 78557 \times 2^{36 k + 27} + 1
+ | o = \equiv
+ | r = 78557 \times 1^k \times 2^{27} + 1
+ | rr = \pmod {37}
+ | c = [[Fermat's Little Theorem]] and [[Congruence of Powers]]
+}}
+{{eqn | o = \equiv
+ | r = 6 \times 512^3 + 1
+ | rr = \pmod {37}
+ | c = [[Congruence of Product]]
+}}
+{{eqn | o = \equiv
+ | r = 6 \times \paren {-6}^3 + 1
+ | rr = \pmod {37}
+ | c = [[Congruence of Powers]]
+}}
+{{eqn | o = \equiv
+ | r = 0
+ | rr = \pmod {37}
+}}
+{{eqn | l = 78557 \times 2^{9 k + 3} + 1
+ | o = \equiv
+ | r = 78557 \times 512^k \times 2^3 + 1
+}}
+{{eqn | o = \equiv
+ | r = 78557 \times 1^k \times 8 + 1
+ | rr = \pmod {73}
+ | c = [[Congruence of Powers]]
+}}
+{{eqn | o = \equiv
+ | r = 9 \times 8 + 1
+ | rr = \pmod {73}
+ | c = [[Congruence of Product]]
+}}
+{{eqn | o = \equiv
+ | r = 0
+ | rr = \pmod {73}
+}}
+{{end-eqn}}
+Therefore $78557 \times 2^n + 1$ is always [[Definition:Divisor of Integer|divisible]] by one of $\set {3, 5, 7, 13, 19, 37, 73}$.
+Hence they are [[Definition:Composite Number|composite]] for every $n$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Existence of Infinite Number of Numbers that are Riesel, Carmichael and Sierpiński}
+Tags: Sierpiński Numbers of the Second Kind, Riesel Numbers, Carmichael Numbers
+
+\begin{theorem}
+There exists an [[Definition:Infinite Set|infinite number]] of [[Definition:Integer|integers]] which are simultaneously [[Definition:Sierpiński Number|Sierpiński]], [[Definition:Riesel Number|Riesel]] and [[Definition:Carmichael Number|Carmichael]].
+\end{theorem}<|endoftext|>
+\section{Equivalence of Definitions of Balanced Prime}
+Tags: Balanced Primes
+
+\begin{theorem}
+The following definitions of a [[Definition:Balanced Prime|balanced prime]] are [[Definition:Logical Equivalence|equivalent]]:
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = p_n
+ | r = \dfrac {p_{n - 1} + p_{n + 1} } 2
+ | c = [[Definition:Balanced Prime/Definition 1|Definition 1]]
+}}
+{{eqn | ll= \iff
+ | l = 2 p_n
+ | r = p_{n - 1} + p_{n + 1}
+ | c =
+}}
+{{eqn | ll= \iff
+ | l = p_n - p_{n - 1}
+ | r = p_{n + 1} - p_n
+ | c = [[Definition:Balanced Prime/Definition 3|Definition 3]]: equal [[Definition:Prime Gap|prime gaps]]
+}}
+{{eqn | r = d
+ | c = for some $d \in \Z$
+}}
+{{eqn | ll= \iff
+ | l = p_{n - 1} + d
+ | r = p_n
+ | c = [[Definition:Balanced Prime/Definition 2|Definition 2]]
+}}
+{{eqn | l = p_{n - 1} + 2 d
+ | r = p_{n + 1}
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Balanced Primes]]
+a8nmmveebhimoee5r36swim77fm2unp
+\end{proof}<|endoftext|>
+\section{Divisor of Fermat Number}
+Tags: Fermat Numbers, Divisor of Fermat Number
+
+\begin{theorem}
+Let $F_n$ be a [[Definition:Fermat Number|Fermat number]].
+Let $m$ be [[Definition:Divisor of Integer|divisor]] of $F_n$.
+\end{theorem}<|endoftext|>
+\section{Prime Decomposition of 6th Fermat Number}
+Tags: 18,446,744,073,709,551,617, 274,177, 67,280,421,310,721, Fermat Numbers
+
+\begin{theorem}
+The [[Definition:Prime Decomposition|prime decomposition]] of the $6$th [[Definition:Fermat Number|Fermat number]] is given by:
+{{begin-eqn}}
+{{eqn | l = 2^{\paren {2^6} } + 1
+ | r = 18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 617
+ | c = [[Definition:Fermat Number/Sequence|Sequence of Fermat Numbers]]
+}}
+{{eqn | r = 274 \, 177 \times 67 \, 280 \, 421 \, 310 \, 721
+ | c =
+}}
+{{eqn | r = \paren {3^2 \times 7 \times 17 \times 2^8 + 1} \times \paren {5 \times 47 \times 373 \times 2 \, 998 \, 279 \times 2^8 + 1}
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}<|endoftext|>
+\section{Prime Decomposition of 7th Fermat Number}
+Tags: 340,282,366,920,938,463,463,374,607,431,768,211,457, 59,649,589,127,497,217, 5,704,689,200,685,129,054,721, Fermat Numbers
+
+\begin{theorem}
+The [[Definition:Prime Decomposition|prime decomposition]] of the $7$th [[Definition:Fermat Number|Fermat number]] is given by:
+{{begin-eqn}}
+{{eqn | l = 2^{\paren {2^7} } + 1
+ | r = 340 \, 282 \, 366 \, 920 \, 938 \, 463 \, 463 \, 374 \, 607 \, 431 \, 768 \, 211 \, 457
+ | c = [[Definition:Fermat Number/Sequence|Sequence of Fermat Numbers]]
+}}
+{{eqn | r = 59 \, 649 \, 589 \, 127 \, 497 \, 217 \times 5 \, 704 \, 689 \, 200 \, 685 \, 129 \, 054 \, 721
+ | c =
+}}
+{{eqn | r = \paren {116 \, 503 \, 103 \, 764 \, 643 \times 2^9 + 1} \paren {3^5 \times 5 \times 12497 \times 733803 839347 \times 2^9 + 1}
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}<|endoftext|>
+\section{Filtered iff Directed in Dual Ordered Set}
+Tags: Dual Ordered Sets
+
+\begin{theorem}
+Let $\left({S, \preceq_1}\right)$ be an [[Definition:Ordered Set|ordered set]].
+Let $\left({S, \preceq_2}\right)$ be a [[Definition:Dual Ordered Set|dual ordered set]] of $\left({S, \preceq_1}\right)$
+Let $X \subseteq S$.
+Then $X$ is [[Definition:Filtered Subset|filtered]] in $\left({S, \preceq_1}\right)$ {{iff}}
+$X$ is [[Definition:Directed Subset|directed]] in $\left({S, \preceq_2}\right)$
+\end{theorem}
+
+\begin{proof}
+By [[Dual of Dual Ordering]]:
+:$\left({S, \preceq_1}\right)$ is [[Definition:Dual Ordered Set|dual]] of $\left({S, \preceq_2}\right)$
+Thus by [[Directed iff Filtered in Dual Ordered Set]]:
+:$X$ is [[Definition:Filtered Subset|filtered]] in $\left({S, \preceq_1}\right)$ {{iff}} $X$ is [[Definition:Directed Subset|directed]] in $\left({S, \preceq_2}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Filters equal Ideals in Dual Ordered Set}
+Tags: Dual Ordered Sets
+
+\begin{theorem}
+Let $L_1 = \left({S, \preceq_1}\right)$ be an [[Definition:Ordered Set|ordered set]].
+Let $L_2 = \left({S, \preceq_2}\right)$ be a [[Definition:Dual Ordered Set|dual ordered set]] of $L_1$.
+Then $\mathit{Filt}\left({L_1}\right) = \mathit{Ids}\left({L_2}\right)$
+where
+:$\mathit{Filt}\left({L_1}\right)$ denotes the [[Definition:Set of Sets|set]] of all [[Definition:Filter|filters]] of $L_1$,
+:$\mathit{Ids}\left({L_2}\right)$ denotes the [[Definition:Set of Sets|set]] of all [[Definition:Ideal (Order Theory)|ideals]] of $L_2$.
+\end{theorem}
+
+\begin{proof}
+Let $x$ be a [[Definition:Set|set]].
+By definition of $\mathit{Filt}$:
+:$x \in \mathit{Filt}\left({L_1}\right) \iff x$ is a [[Definition:Filter|filter]] on $L_1$.
+By [[Filter is Ideal in Dual Ordered Set]]:
+:$x \in \mathit{Filt}\left({L_1}\right) \iff x$ is an [[Definition:Ideal (Order Theory)|ideal]] in $L_2$.
+By definition of $\mathit{Ids}$:
+:$x \in \mathit{Filt}\left({L_1}\right) \iff x \in \mathit{Ids}\left({L_2}\right)$
+Hence by definition of [[Definition:Set Equality|set equality]]:
+:$\mathit{Filt}\left({L_1}\right) = \mathit{Ids}\left({L_2}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Pépin's Test}
+Tags: Fermat Numbers
+
+\begin{theorem}
+Let $F_n = 2^{2^n} + 1$ be a [[Definition:Fermat Number|Fermat number]].
+Then $F_n$ is [[Definition:Prime Number|prime]] {{iff}}:
+:$3^{\paren {F_n - 1} / 2} \equiv -1 \pmod {F_n}$
+\end{theorem}
+
+\begin{proof}
+=== Sufficient Condition ===
+Let this [[Definition:Congruence Modulo Integer|congruence]] hold:
+:$3^{\paren {F_n - 1} / 2} \equiv -1 \pmod {F_n}$
+Then:
+:$3^{F_n - 1} \equiv 1 \pmod {F_n}$
+Thus the [[Definition:Multiplicative Order of Integer|multiplicative order of $3$ modulo $F_n$]] is a [[Definition:Divisor of Integer|divisor]] of $F_n - 1 = 2^{2^n}$.
+This is a [[Definition:Integer Power|power of $2$]].
+On the other hand, the [[Definition:Multiplicative Order of Integer|multiplicative order of $3$ modulo $F_n$]] is not a [[Definition:Divisor of Integer|divisor]] of $\dfrac {F_n - 1} 2$.
+Therefore it must be equal to $F_n - 1$.
+In particular, there are at least $F_n - 1$ numbers below $F_n$ which are [[Definition:Coprime Integers|coprime]] to $F_n<$.
+This can happen only if $F_n$ is [[Definition:Prime Number|prime]].
+{{qed|lemma}}
+=== Necessary Condition ===
+Let $F_n$ be [[Definition:Prime Number|prime]].
+By [[Euler's Criterion]]:
+:$3^{\paren {F_n - 1} / 2} \equiv \paren {\dfrac 3 {F_n} } \pmod {F_n}$
+$\left({\dfrac 3 {F_n} }\right)$ denotes the [[Definition:Legendre Symbol|Legendre symbol]].
+By repeated [[Definition:Square (Algebra)|squaring]]:
+:$2^{2^n} \equiv 1 \pmod 3$
+Thus:
+:$F_n \equiv 2 \pmod 3$
+and so:
+:$\paren {\dfrac {F_n} 3} = -1$
+As $F_n \equiv 1 \pmod 4$, it follows from the [[Law of Quadratic Reciprocity]] that:
+:$\paren {\dfrac 3 {F_n} } = -1$
+{{qed}}
+{{Namedfor|Jean François Théophile Pépin|cat = Pépin}}
+\end{proof}<|endoftext|>
+\section{Triangular Fermat Number}
+Tags: Fermat Numbers, Triangular Numbers
+
+\begin{theorem}
+The only one [[Definition:Fermat Number|Fermat number]] which is [[Definition:Triangular Number|triangular]] is $3$.
+\end{theorem}
+
+\begin{proof}
+Let $F_n$ be a [[Definition:Fermat Number|Fermat number]] which is [[Definition:Triangular Number|triangular]].
+We use the weaker form of [[Divisor of Fermat Number]]:
+:Every [[Definition:Divisor of Integer|divisor]] of $F_n$ is of the form $k \times 2^{n + 1} + 1$
+as this formulation is valid for all $n \in \N$.
+From [[Closed Form for Triangular Numbers]]:
+:$F_n = \dfrac {m \paren {m + 1} } 2$
+Either $m$ or $m + 1$ is [[Definition:Even Integer|even]].
+Suppose $m$ is [[Definition:Even Integer|even]].
+Then:
+{{begin-eqn}}
+{{eqn | ll = \exists k \in \N:
+ | l = \frac m 2
+ | r = k \times 2^{n + 1} + 1
+ | c = [[Divisor of Fermat Number]]
+}}
+{{eqn | lll = \leadsto
+ | l = m + 1
+ | r = 2 \paren {k \times 2^{n + 1} + 1} + 1
+}}
+{{eqn | r = k \times 2^{n + 2} + 3
+}}
+{{eqn | ll = \exists j \in \N:
+ | l = m + 1
+ | r = j \times 2^{n + 1} + 1
+ | c = [[Divisor of Fermat Number]]
+}}
+{{eqn | lll = \leadsto
+ | l = k \times 2^{n + 2} + 2
+ | r = j \times 2^{n + 1}
+}}
+{{eqn | lll = \leadsto
+ | l = k \times 2^{n + 1} + 1
+ | r = j \times 2^n
+}}
+{{end-eqn}}
+As the {{LHS}} is always [[Definition:Odd Integer|odd]], we must have $2^n = 1$.
+This forces $n = 0$, with $F_n = 3$.
+We check that $3$ is indeed a [[Definition:Triangular Number|triangular number]].
+Suppose $m + 1$ is [[Definition:Even Integer|even]].
+Then:
+{{begin-eqn}}
+{{eqn | ll = \exists k \in \N:
+ | l = \frac {m + 1} 2
+ | r = k \times 2^{n + 1} + 1
+ | c = [[Divisor of Fermat Number]]
+}}
+{{eqn | lll = \leadsto
+ | l = m
+ | r = 2 \paren {k \times 2^{n + 1} + 1} - 1
+}}
+{{eqn | r = k \times 2^{n + 2} + 1
+}}
+{{eqn | l = F_n
+ | r = \frac {m \paren {m + 1} } 2
+}}
+{{eqn | lll = \leadsto
+ | l = 2^{2^n} + 1
+ | r = \paren {k \times 2^{n + 2} + 1} \paren {k \times 2^{n + 1} + 1}
+}}
+{{eqn | r = k^2 \times 2^{2 n + 3} + k \paren {2^{n + 2} + 2^{n + 1} } + 1
+}}
+{{eqn | lll = \leadsto
+ | l = 2^{2^n}
+ | r = k^2 \times 2^{2 n + 3} + k \paren {2^{n + 2} + 2^{n + 1} }
+}}
+{{eqn | r = k \times 2^{n + 1} \times \paren {k \times 2^{n + 2} + 3 }
+}}
+{{end-eqn}}
+As $k \times 2^{n + 2} + 3 > 1$ and is [[Definition:Odd Integer|odd]], $k \times 2^{n + 2} + 3$ cannot be a [[Definition:Integer Power|power]] of $2$.
+So $m + 1$ cannot be [[Definition:Even Integer|even]], and we have exhausted all possibilities.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Fermat Number is not Square}
+Tags: Fermat Numbers, Square Numbers, Fermat Number is not Square
+
+\begin{theorem}
+There exist no [[Definition:Fermat Number|Fermat numbers]] which are [[Definition:Square Number|square]].
+\end{theorem}
+
+\begin{proof}
+Recall the definition of [[Definition:Fermat Number|Fermat numbers]]:
+:$F_n = 2^{(2^n)}+1$, where $n = 0, 1, 2, \ldots$
+=== Marginal Case ===
+$F_0 = 3$ is not a [[Definition:Square Number|square]].
+=== General Case ===
+It will be shown that [[Definition:Fermat Number|Fermat numbers]] lie between $2$ consecutive [[Definition:Square Number|squares]], thus cannot be itself a [[Definition:Square Number|square]]:
+{{begin-eqn}}
+{{eqn | l = \left({2^{ \left({2^{n-1} }\right)} }\right)^2
+ | r = 2^{\left({2^n}\right)}
+ | c = [[Index Laws/Product of Indices]]
+}}
+{{eqn | o = <
+ | r = 2^{\left({2^n}\right)} + 1
+ | c = adding a positive term
+}}
+{{eqn | r = F_n
+ | c =
+}}
+{{eqn | o = <
+ | r = 2^{\left({2^n}\right)} + 2^{\left({2^{n-1} }\right) + 1} + 1
+ | c = adding another positive term
+}}
+{{eqn | r = \left({2^{\left({2^{n-1} }\right)} + 1}\right)^2
+ | c = [[Completing the Square]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Fermat Number is not Cube}
+Tags: Fermat Numbers, Cube Numbers, Fermat Number is not Cube
+
+\begin{theorem}
+There exist no [[Definition:Fermat Number|Fermat numbers]] which are [[Definition:Cube Number|cubes]].
+\end{theorem}
+
+\begin{proof}
+Direct consequence of [[Fermat Number is not Perfect Power]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Construction of Regular 257-Gon}
+Tags: Regular Polygons, 257
+
+\begin{theorem}
+It is possible to construct a [[Definition:Regular Polygon|regular polygon]] with $257$ [[Definition:Side of Polygon|sides]]) using a [[Definition:Compass and Straightedge Construction|compass and straightedge construction]].
+\end{theorem}
+
+\begin{proof}
+From [[Construction of Regular Prime p-Gon Exists iff p is Fermat Prime|Construction of Regular Prime $p$-Gon Exists iff $p$ is Fermat Prime]] it is known that this construction is possible.
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Ideals equal Filters in Dual Ordered Set}
+Tags: Dual Ordered Sets
+
+\begin{theorem}
+Let $L_1 = \left({S, \preceq_1}\right)$ be an [[Definition:Ordered Set|ordered set]].
+Let $L_2 = \left({S, \preceq_2}\right)$ be a [[Definition:Dual Ordered Set|dual ordered set]] of $L_1$.
+Then $\mathit{Ids}\left({L_1}\right) = \mathit{Filt}\left({L_2}\right)$
+where
+:$\mathit{Filt}\left({L_2}\right)$ denotes the [[Definition:Set of Sets|set]] of all [[Definition:Filter|filters]] of $L_2$,
+:$\mathit{Ids}\left({L_1}\right)$ denotes the [[Definition:Set of Sets|set]] of all [[Definition:Ideal (Order Theory)|ideals]] of $L_1$.
+\end{theorem}
+
+\begin{proof}
+Let $x$ be a [[Definition:Set|set]].
+By definition of $\mathit{Filt}$:
+:$x \in \mathit{Filt}\left({L_2}\right) \iff x$ is a [[Definition:Filter|filter]] on $L_2$.
+By [[Ideal is Filter in Dual Ordered Set]]:
+:$x \in \mathit{Filt}\left({L_2}\right) \iff x$ is an [[Definition:Ideal (Order Theory)|ideal]] in $L_1$.
+By definition of $\mathit{Ids}$:
+:$x \in \mathit{Filt}\left({L_2}\right) \iff x \in \mathit{Ids}\left({L_1}\right)$
+Hence by definition of [[Definition:Set Equality|set equality]]:
+:$\mathit{Ids}\left({L_1}\right) = \mathit{Filt}\left({L_2}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Power Set is Filter in Lattice of Power Set}
+Tags: Power Set
+
+\begin{theorem}
+Let $X$ be a [[Definition:Set|set]].
+Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be a [[Definition:Inclusion Ordered Set|inclusion]] [[Definition:Lattice|lattice]] of [[Definition:Power Set|power set]] of $X$.
+Then $\mathcal P\left({X}\right)$ is a [[Definition:Filter|filter]] on $L$.
+\end{theorem}
+
+\begin{proof}
+=== Filtered ===
+By [[Set is Element of its Power Set]]:
+:$X \in \mathcal P\left({X}\right)$
+Thus by definition:
+:$\mathcal P\left({X}\right)$ is a [[Definition:Non-Empty Set|non-empty set]].
+Let $x, y \in \mathcal P\left({X}\right)$.
+By [[Intersection is Subset]]:
+:$x \cap y \subseteq x$ and $x \cap y \subseteq y$
+By [[Subset Relation is Transitive]]:
+:$x \cap y \in \mathcal P\left({X}\right)$
+Thus
+:$\exists z \in \mathcal P\left({X}\right): z \subseteq x \land z \subseteq y$
+{{qed|lemma}}
+=== Upper ===
+Thus we have:
+:$\forall x, y \in \mathcal P\left({X}\right): x \subseteq y \implies y \in \mathcal P\left({X}\right)$
+{{qed|lemma}}
+Thus by definition of [[Definition:Filter in Ordered Set|filter in ordered set]]:
+:$\mathcal P\left({X}\right)$ is a [[Definition:Filter|filter]] on $L$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Triplet in Arithmetic Sequence with equal Sigma}
+Tags: Sigma Function, Arithmetic Sequences, 267, 295, 323
+
+\begin{theorem}
+The smallest [[Definition:Ordered Triple|triple]] of [[Definition:Positive Integer|integers]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] which have the same [[Definition:Sigma Function|$\sigma$ (sigma) value]] is:
+:$\map \sigma {267} = \map \sigma {295} = \map \sigma {323} = 360$
+\end{theorem}
+
+\begin{proof}
+We have that:
+{{begin-eqn}}
+{{eqn | l = 295 - 267
+ | r = 28
+ | c =
+}}
+{{eqn | l = 323 - 295
+ | r = 28
+ | c =
+}}
+{{end-eqn}}
+demonstrating that $267, 295, 323$ are in [[Definition:Arithmetic Sequence|arithmetic sequence]] with [[Definition:Common Difference|common difference]] $28$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \map \sigma {267}
+ | r = 360
+ | c = {{SigmaLink|267}}
+}}
+{{eqn | l = \map \sigma {295}
+ | r = 360
+ | c = {{SigmaLink|295}}
+}}
+{{eqn | l = \map \sigma {323}
+ | r = 360
+ | c = {{SigmaLink|323}}
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that these are the smallest.}}
+\end{proof}<|endoftext|>
+\section{Singleton of Set is Filter in Lattice of Power Set}
+Tags: Power Set
+
+\begin{theorem}
+Let $X$ be a [[Definition:Set|set]].
+Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be an [[Definition:Inclusion Ordered Set|inclusion]] [[Definition:Lattice|lattice]] of [[Definition:Power Set|power set]] of $X$.
+Then $\left\{ {X}\right\}$ is a [[Definition:Filter|filter]] on $L$.
+\end{theorem}
+
+\begin{proof}
+By [[Singleton is Directed and Filtered Subset]]:
+:$\left\{ {X}\right\}$ is [[Definition:Filtered Subset|filtered]].
+We will prove that
+:$\left\{ {X}\right\}$ is an [[Definition:Upper Set|upper set]].
+Let $x \in \left\{ {X}\right\}$, $y \in \mathcal P\left({X}\right)$ such that
+:$x \subseteq y$
+By definition of [[Definition:Singleton|singleton]]:
+:$x = X$
+By definition of [[Definition:Power Set|power set]]:
+:$y \subseteq X$
+By definition of [[Definition:Set Equality|set equality]]:
+:$y = X$
+Thus:
+:$y \in \left\{ {X}\right\}$
+{{qed|lemma}}
+Thus by definition of [[Definition:Filter in Ordered Set|filter in ordered set]]:
+:$\left\{ {X}\right\}$ is a [[Definition:Filter|filter]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Existence of Arbitrarily Long Aliquot Sequences}
+Tags: Aliquot Sequences
+
+\begin{theorem}
+It is possible to construct an [[Definition:Aliquot Sequence|aliquot sequence]] of arbitrary [[Definition:Length of Sequence|length]] which is [[Definition:Monotone Sequence|monotonically]] [[Definition:Increasing Sequence|increasing]].
+\end{theorem}
+
+\begin{proof}
+Let $p_i$ be the $i$th [[Definition:Prime Number|prime]].
+Construct the sequence $\sequence {t_i}_{t \mathop = 1}^\infty$ of [[Definition:Natural Numbers|natural numbers]] $t_i$ recursively as follows:
+:$t_1 = 2$
+:$t_{i + 1} = \map \phi {p_i^{t_i + 1} } - 1 = p_i^{t_i} \paren {p_i - 1} - 1$ for $i \ge 1$
+We claim that this sequence has the property:
+:$(1): \quad p_i^{t_i + 1} \divides \map \sigma {p_{i + 1}^{t_{i + 1} } }$ for $i \ge 1$
+For $i = 1$, $p_1 = 2$ and $t_1 = 2$.
+:$t_2 = 2^2 \paren {2 - 1} - 1 = 3$
+:$\map \sigma {p_2^{t_2} } = \map \sigma {3^3} = 40$
+which is a multiple of $2^3 = 8$.
+For $i > 1$:
+{{begin-eqn}}
+{{eqn | l = \paren {p_{i + 1} - 1} \map \sigma {p_{i + 1}^{t_{i + 1} } }
+ | r = p_{i + 1}^{t_{i + 1} + 1} - 1
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = p_{i + 1}^{\map \phi {p_i^{t_i + 1} } } - 1
+}}
+{{eqn | r = 0
+ | rr = \pmod {p_i^{t_i + 1} }
+ | c = [[Euler's Theorem]]
+}}
+{{end-eqn}}
+For $i > 1$, $p_{i + 1} - 1 > 2$.
+Hence $p_{i + 1} - 1$ is [[Definition:Even Integer|even]] and [[Definition:Composite Number|composite]].
+Since $p_i$ is [[Definition:Odd Integer|odd]] and $p_i > \dfrac {p_{i + 1} - 1} 2$ by [[Bertrand-Chebyshev Theorem]]:
+:$p_{i + 1} - 1$ and $p_i^{t_i + 1}$ are [[Definition:Coprime Integers|coprime]].
+By [[Euclid's Lemma]], $p_i^{t_i + 1} \divides \map \sigma {p_{i + 1}^{t_{i + 1} } }$.
+We define for $l \ge 1$: $A_l = \set {m \in \N: p_i^{t_i} \divides m, p_i^{t_i + 1} \nmid m, \forall i: 1 \le i \le l}$.
+Denote $\map s n$ the [[Definition:Aliquot Sum|aliquot sum]] of $n$.
+Then for $m \in A_l$ where $l \ge 2$, we claim:
+{{begin-eqn}}
+{{eqn | n = 2
+ | l = \map s m
+ | o = >
+ | r = m
+}}
+{{eqn | n = 3
+ | l = \map s m
+ | o = \in
+ | r = A_{l - 1}
+}}
+{{end-eqn}}
+Let $m \in A_l$ where $l \ge 2$.
+From definition we have:
+{{begin-eqn}}
+{{eqn | l = p_1^{t_1}
+ | r = 4 \divides m
+}}
+{{eqn | l = p_2^{t_2}
+ | r = 27 \divides m
+}}
+{{end-eqn}}
+Hence $m$ is a multiple of $27 \times 4 = 108$.
+Since $108$ is [[Definition:Abundant Number|abundant]], $m$ is also [[Definition:Abundant Number|abundant]] by [[Multiple of Abundant Number is Abundant]].
+$\map s m > m$ follows from definition of [[Definition:Abundant Number/Definition 3|abundant numbers]].
+Write $m = p_1^{t_1} \cdots p_l^{t_l} k$, where $k$ is not [[Definition:Divisor of Integer|divisible]] by $p_1, \dots, p_l$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \map s m
+ | r = \map \sigma m - m
+}}
+{{eqn | r = \map \sigma {p_1^{t_1} } \map \sigma {p_2^{t_2} } \cdots \map \sigma {p_l^{t_l} } \map \sigma k - m
+ | c = [[Sigma Function is Multiplicative]]
+}}
+{{eqn | r = K p_1^{t_1 + 1} \cdots p_{l - 1}^{t_{l - 1} + 1} - p_1^{t_1} \cdots p_l^{t_l} k
+ | c = for some $K \in \N$, by $(1)$
+}}
+{{eqn | r = p_1^{t_1} \cdots p_{l - 1}^{t_{l - 1} } \paren {K p_1 \cdots p_{l - 1} - p_l^{t_l} k}
+}}
+{{end-eqn}}
+Since $K p_1 \cdots p_{l - 1}$ is [[Definition:Divisor of Integer|divisible]] by $p_1, \dots, p_{l - 1}$ while $p_l^{t_l} k$ isn't:
+:$p_1, \dots, p_{l - 1} \nmid \paren {K p_1 \cdots p_{l - 1} - p_l^{t_l} k}$
+Hence for each $i$ where $1 \le i \le l - 1$:
+:$p_i^{t_i} \divides \map s m, p_i^{t_i + 1} \nmid \map s m$
+This shows that $\map s m \in A_{l - 1}$.
+Hence to create an [[Definition:Aliquot Sequence|aliquot sequence]] of [[Definition:Length of Sequence|length]] $l$, one picks an element $m$ from $A_l$.
+Applying $(2)$ and $(3)$ repeatedly, we see that:
+:$m, \map s m, \map s {\map s m}, \dots, \map {s^l} m$
+is a [[Definition:Monotone Sequence|monotonically]] [[Definition:Increasing Sequence|increasing]] [[Definition:Aliquot Sequence|aliquot sequence]] of [[Definition:Length of Sequence|length]] $l$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Consecutive Triple of Repeated Digit-Products}
+Tags: 281, 282, 283, Recreational Mathematics
+
+\begin{theorem}
+The [[Definition:Ordered Triple|triplet]] of [[Definition:Integer|integers]] $281, 282, 283$ have the property that if their [[Definition:Digit|digits]] are [[Definition:Integer Multiplication|multiplied]], and the process repeated on the result until only $1$ [[Definition:Digit|digit]] remains, that final [[Definition:Digit|digit]] is the same for all three, that is, $6$.
+There does not exist an [[Definition:Ordered Quadruple|set of four]] consecutive [[Definition:Integer|integers]] which also all end up at the same single [[Definition:Digit|digit]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | ll= 281:
+ | l = 2 \times 8 \times 1
+ | r = 16
+ | c =
+}}
+{{eqn | l = 1 \times 6
+ | r = 6
+ | c =
+}}
+{{eqn | ll= 282:
+ | l = 2 \times 8 \times 2
+ | r = 32
+ | c =
+}}
+{{eqn | l = 3 \times 2
+ | r = 6
+ | c =
+}}
+{{eqn | ll= 283:
+ | l = 2 \times 8 \times 3
+ | r = 48
+ | c =
+}}
+{{eqn | l = 4 \times 8
+ | r = 32
+ | c =
+}}
+{{eqn | l = 3 \times 2
+ | r = 6
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Consecutive Powerful Numbers}
+Tags: Powerful Numbers
+
+\begin{theorem}
+The following [[Definition:Ordered Pair|pairs]] are of consecutive [[Definition:Positive Integer|positive integers]] both of which are [[Definition:Powerful Number|powerful]]:
+:$\left({8, 9}\right), \left({288, 289}\right), \left({675, 676}\right), \left({9800, 9801}\right), \left({332 \, 928, 332 \, 929}\right), \ldots$
+{{OEIS|A060355}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 8
+ | r = 2^3
+}}
+{{eqn | l = 9
+ | r = 3^2
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 288
+ | r = 2^5 \times 3^2
+}}
+{{eqn | l = 289
+ | r = 17^2
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 675
+ | r = 3^3 \times 5^2
+}}
+{{eqn | l = 676
+ | r = 2^2 \times 13^2
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 9800
+ | r = 2^3 \times 5^2 \times 7^2
+}}
+{{eqn | l = 9801
+ | r = 3^4 \times 11^2
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 332 \, 928
+ | r = 2^7 \times 3^2 \times 17^2
+}}
+{{eqn | l = 332 \, 929
+ | r = 577^2
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Multiple of 9 with all Digits Even}
+Tags: Recreational Mathematics, 9, 288, Classic Problems, Smallest Multiple of 9 with all Digits Even
+
+\begin{theorem}
+$288$ is the smallest [[Definition:Integer Multiple|integer multiple of $9$]] all of whose [[Definition:Digit|digits]] are [[Definition:Even Integer|even]].
+\end{theorem}
+
+\begin{proof}
+By the brute force technique:
+{{begin-eqn}}
+{{eqn | l = 1 \times 9
+ | r = 9
+}}
+{{eqn | l = 2 \times 9
+ | r = 18
+}}
+{{eqn | l = 3 \times 9
+ | r = 27
+}}
+{{eqn | l = 4 \times 9
+ | r = 36
+}}
+{{eqn | l = 5 \times 9
+ | r = 45
+}}
+{{eqn | l = 6 \times 9
+ | r = 54
+}}
+{{eqn | l = 7 \times 9
+ | r = 63
+}}
+{{eqn | l = 8 \times 9
+ | r = 72
+}}
+{{eqn | l = 9 \times 9
+ | r = 81
+}}
+{{eqn | l = 10 \times 9
+ | r = 90
+}}
+{{eqn | l = 11 \times 9
+ | r = 99
+}}
+{{eqn | l = 12 \times 9
+ | r = 108
+}}
+{{end-eqn}}
+All [[Definition:Integer Multiple|integer multiples of $9$]] $k \times 9$ for $k = 12$ through to $k = 22$ begin with $1$, and so have at least one [[Definition:Odd Integer|odd]] [[Definition:Digit|digit]].
+Then:
+{{begin-eqn}}
+{{eqn | l = 22 \times 9
+ | r = 198
+}}
+{{eqn | l = 23 \times 9
+ | r = 207
+}}
+{{eqn | l = 24 \times 9
+ | r = 216
+}}
+{{eqn | l = 25 \times 9
+ | r = 225
+}}
+{{eqn | l = 26 \times 9
+ | r = 234
+}}
+{{eqn | l = 27 \times 9
+ | r = 243
+}}
+{{eqn | l = 28 \times 9
+ | r = 252
+}}
+{{eqn | l = 39 \times 9
+ | r = 261
+}}
+{{eqn | l = 30 \times 9
+ | r = 270
+}}
+{{eqn | l = 31 \times 9
+ | r = 279
+}}
+{{eqn | l = 32 \times 9
+ | r = 288
+}}
+{{end-eqn}}
+Hence the result.
+{{qed}}
+\end{proof}
+
+\begin{proof}
+Let $n$ be the smallest [[Definition:Integer Multiple|integer multiple of $9$]] all of whose [[Definition:Digit|digits]] are [[Definition:Even Integer|even]].
+From [[Divisibility by 9]], the [[Definition:Digit|digits]] of $n$ must add to an [[Definition:Integer Multiple|integer multiple of $9$]].
+But from [[Sum of Even Integers is Even]], the [[Definition:Digit|digits]] of $n$ must add to an [[Definition:Even Integer|even]] [[Definition:Integer Multiple|integer multiple of $9$]]: $18, 36, 54$, etc.
+There is only $1$ [[Definition:Integer Multiple|integer multiple of $9$]] with $2$ [[Definition:Digit|digits]] whose [[Definition:Digit|digits]] add to an [[Definition:Even Integer|even]] [[Definition:Integer Multiple|integer multiple of $9$]], and that is $99$.
+Thus we have to look at $3$-[[Definition:Digit|digit]] [[Definition:Integer|integers]].
+The following [[Definition:Set|sets]] of $3$ [[Definition:Even Integer|even]] [[Definition:Digit|digits]] add to $18$:
+:$\set {2, 8, 8}$
+:$\set {4, 6, 8}$
+:$\set {6, 6, 6}$
+and that seems to be about it.
+There are no [[Definition:Set|sets]] of $3$ [[Definition:Even Integer|even]] [[Definition:Digit|digits]] which add up to $36$ or higher.
+The result follows by inspection.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Multiple of 9 with all Digits Even/Historical Note}
+Tags: Historical Notes
+
+\begin{theorem}
+{{AuthorRef|David Wells}}, in his {{BookLink|Curious and Interesting Numbers|David Wells|ed = 2nd|edpage = Second Edition}} of $1997$, attributes this to an '''{{AuthorRef|A.J. Turner}}''', but it has not so far been possible to determine who this refers to.
+This question seems to be a classic that regularly crops up in compendia of puzzles.
+\end{theorem}<|endoftext|>
+\section{Kaprekar's Symmetry}
+Tags: Kaprekar Numbers
+
+\begin{theorem}
+Let $n$ be a [[Definition:Kaprekar Number|Kaprekar number]] with $D$ [[Definition:Digit|digits]].
+Then $10^D - n$ is also a [[Definition:Kaprekar Number|Kaprekar number]].
+\end{theorem}
+
+\begin{proof}
+Since $n$ is a [[Definition:Kaprekar Number|Kaprekar number]] of $D$ [[Definition:Digit|digits]]:
+:$\begin {cases} n^2 = a \times 10^D + b \\ n = a + b \end {cases}$
+for some [[Definition:Positive Integer|positive integers]] $a$ and $b$, $b < 10^D$.
+Hence:
+{{begin-eqn}}
+{{eqn | l = \paren {10^D - n}^2
+ | r = 10^{2 D} - 2 n \times 10^D + n^2
+}}
+{{eqn | r = 10^{2 D} - 2 \paren {a + b} 10^D + a \times 10^D + b
+}}
+{{eqn | r = 10^D \paren {10^D - a - 2 b} + b
+}}
+{{end-eqn}}
+and we have:
+:$\paren {10^D - a - 2 b} + b = 10^D - a - b = 10^D - n$
+Finally we check that $10^D - a - 2 b \ge 0$:
+{{AimForCont}} $10^D - a - 2 b \le -1$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \paren {10^D - n}^2
+ | r = 10^D \paren {10^D - a - 2 b} + b
+}}
+{{eqn | o = \le
+ | r = -10^D + b
+}}
+{{eqn | o = <
+ | r = 0
+}}
+{{end-eqn}}
+but [[Definition:Square Number|squares]] are [[Definition:Positive Integer|positive]], a [[Definition:Contradiction|contradiction]].
+Hence $10^D - n$ is also a [[Definition:Kaprekar Number|Kaprekar number]] of $D$ [[Definition:Digit|digits]].
+{{qed}}
+{{Namedfor|Dattathreya Ramchandra Kaprekar|cat = Kaprekar}}
+\end{proof}<|endoftext|>
+\section{Cyclic Permutation of Kaprekar Number}
+Tags: Kaprekar Numbers, Cyclic Permutation of Kaprekar Number
+
+\begin{theorem}
+Let $n$ be a [[Definition:Kaprekar Number|Kaprekar number]] of $k$ [[Definition:Digit|digits]].
+Let $m$ be an [[Definition:Integer|integer]] formed from a [[Definition:Cyclic Permutation|cyclic permutation]] of the [[Definition:Digit|digits]] of $n$.
+Let $m$ be [[Definition:Square (Algebra)|squared]] and the result split into $2$ parts, where the $2$nd part is of $k$ [[Definition:Digit|digits]].
+Let these two parts be [[Definition:Integer Addition|added]], in the way of operating on a [[Definition:Kaprekar Number|Kaprekar number]].
+If the result is more than $k$ [[Definition:Digit|digits]] long, split that into $2$ parts, where the $2$nd part is of $k$ [[Definition:Digit|digits]], and [[Definition:Integer Addition|add]] the parts.
+The result will be another [[Definition:Cyclic Permutation|cyclic permutation]] of the [[Definition:Digit|digits]] of $n$.
+\end{theorem}<|endoftext|>
+\section{Difference between Kaprekar Number and Square is Multiple of Repunit}
+Tags: Kaprekar Numbers
+
+\begin{theorem}
+Let $n$ be a [[Definition:Kaprekar Number|Kaprekar number]] of $m$ [[Definition:Digit|digits]].
+Then:
+: $n^2 - n = k R_m$
+where:
+:$R_m$ is the [[Definition:Repunit|$m$-digit repunit]]
+:$k$ is an [[Definition:Integer|integer]].
+\end{theorem}
+
+\begin{proof}
+Since $n$ is a [[Definition:Kaprekar Number|Kaprekar number]] of $m$ [[Definition:Digit|digits]]:
+:$\begin {cases} n^2 = a \times 10^m + b \\ n = a + b \end {cases}$
+for some [[Definition:Positive Integer|positive integers]] $a$ and $b$.
+Hence:
+{{begin-eqn}}
+{{eqn | l = n^2 - n
+ | r = a \times 10^m + b - a - b
+}}
+{{eqn | r = a \paren {10^m - 1}
+}}
+{{eqn | r = 9 a R_m
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Filters of Lattice of Power Set form Bounded Above Ordered Set}
+Tags: Power Set
+
+\begin{theorem}
+Let $X$ be a [[Definition:Set|set]].
+Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be an [[Definition:Inclusion Ordered Set|inclusion]] [[Definition:Lattice|lattice]] of [[Definition:Power Set|power set]] of $X$.
+Let $F = \left({\mathit{Filt}\left({L}\right), \subseteq}\right)$ be an [[Definition:Inclusion Ordered Set|inclusion ordered set]],
+where $\mathit{Filt}\left({L}\right)$ denotes the [[Definition:Set of Sets|set]] of all [[Definition:Filter|filters]] on $L$.
+Then $F$ is [[Definition:Bounded Above Set|bounded above]] and $\top_F = \mathcal P\left({X}\right)$
+where $\top_F$ denotes the [[Definition:Greatest Element|greatest element]] of $F$.
+\end{theorem}
+
+\begin{proof}
+By [[Power Set is Filter in Lattice of Power Set]]:
+:$\mathcal P\left({X}\right)$ is a [[Definition:Filter|filter]] on $L$.
+Let $A \in \mathit{Filt}\left({L}\right)$.
+Thus by definition of [[Definition:Filter|filter]]:
+:$A \subseteq \mathcal P\left({X}\right)$
+Thus by definitions:
+:$F$ is [[Definition:Bounded Above Set|bounded above]] and $\top_F = \mathcal P\left({X}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Filters of Lattice of Power Set form Bounded Below Ordered Set}
+Tags: Power Set
+
+\begin{theorem}
+Let $X$ be a [[Definition:Set|set]].
+Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be an [[Definition:Inclusion Ordered Set|inclusion]] [[Definition:Lattice|lattice]] of [[Definition:Power Set|power set]] of $X$.
+Let $F = \left({\mathit{Filt}\left({L}\right), \subseteq}\right)$ be an [[Definition:Inclusion Ordered Set|inclusion ordered set]],
+where $\mathit{Filt}\left({L}\right)$ denotes the [[Definition:Set of Sets|set]] of all [[Definition:Filter|filters]] on $L$.
+Then $F$ is [[Definition:Bounded Below Set|bounded below]] and $\bot_F = \left\{{X}\right\}$
+where $\bot_F$ denotes the [[Definition:Smallest Element|smallest element]] of $F$.
+\end{theorem}
+
+\begin{proof}
+By [[Singleton of Set is Filter in Lattice of Power Set]]:
+:$\left\{{X}\right\}$ is a [[Definition:Filter|filter]] on $L$.
+Let $A \in \mathit{Filt}\left({L}\right)$.
+By definition of [[Definition:Non-Empty Set|non-empty set]]:
+:$\exists x: x \in A$
+By definition of [[Definition:Power Set|power set]]:
+:$x \subseteq X$
+By definition of [[Definition:Upper Set|upper set]]:
+:$X \in A$
+Thus by definitions of [[Definition:Singleton|singleton]] and [[Definition:Subset|subset]]:
+:$\left\{{X}\right\} \subseteq A$
+Thus by definitions:
+:$F$ is [[Definition:Bounded Below Set|bounded below]] and $\bot_F = \left\{{X}\right\}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Palindromic Primes in Base 10 and Base 2}
+Tags: Palindromic Numbers, Prime Numbers, 2, 10
+
+\begin{theorem}
+The following $n \in \Z$ are [[Definition:Prime Number|prime numbers]] which are [[Definition:Palindromic Number|palindromic]] in both [[Definition:Decimal Notation|decimal]] and [[Definition:Binary Notation|binary]]:
+:$3, 5, 7, 313, 7 \, 284 \, 717 \, 174 \, 827, 390 \, 714 \, 505 \, 091 \, 666 \, 190 \, 505 \, 417 \, 093, \ldots$
+{{OEIS|A046472}}
+It is not known whether there are any more.
+\end{theorem}
+
+\begin{proof}
+:{| border="1"
+|-
+! align="center" style = "padding: 2px 10px" | $n_{10}$
+! align="center" style = "padding: 2px 10px" | $n_2$
+|-
+| align="right" style = "padding: 2px 10px" | $3$
+| align="right" style = "padding: 2px 10px" | $11$
+|-
+| align="right" style = "padding: 2px 10px" | $5$
+| align="right" style = "padding: 2px 10px" | $101$
+|-
+| align="right" style = "padding: 2px 10px" | $7$
+| align="right" style = "padding: 2px 10px" | $111$
+|-
+| align="right" style = "padding: 2px 10px" | $313$
+| align="right" style = "padding: 2px 10px" | $100 \, 111 \, 001$
+|-
+| align="right" style = "padding: 2px 10px" | $7 \, 284 \, 717 \, 174 \, 827$
+| align="right" style = "padding: 2px 10px" | $1 \, 101 \, 010 \, 000 \, 000 \, 011 \, 010 \, 111 \, 110 \, 101 \, 100 \, 000 \, 000 \, 101 \, 011$
+|-
+| align="right" style = "padding: 2px 10px" | $390 \, 714 \, 505 \, 091 \, 666 \, 190 \, 505 \, 417 \, 093$
+| align="right" style = "padding: 2px 10px" | $10 \, 100 \, 001 \, 100 \, 110 \, 001 \, 000 \, 000 \, 111 \, 100 \, 001 \, 100 \, 011 \, 000 \, 111 \, 011 \, 100 \, 011 \, 000 \, 110 \, 000 \, 111 \, 100 \, 000 \, 010 \, 001 \, 100 \, 110 \, 000 \, 101$
+|}
+The last two numbers have $43$ and $89$ [[Definition:Digit|digits]] (in [[Definition:Binary Notation|binary]]) respectively .
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Ideals form Complete Lattice}
+Tags: Order Theory
+
+\begin{theorem}
+Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Join Semilattice|join semilattice]].
+Let $\mathcal I = \struct {\mathit{Ids}\left({L}\right), \subseteq}$ be an [[Definition:Inclusion Ordered Set|inclusion ordered set]],
+where $\mathit{Ids}\left({L}\right)$ denotes the [[Definition:Set of Sets|set]] of all [[Definition:Ideal (Order Theory)|ideals]] in $L$.
+Then $\mathcal I$ is [[Definition:Complete Lattice|complete lattice]].
+\end{theorem}
+
+\begin{proof}
+Let $X \subseteq \mathit{Ids}\left({L}\right)$.
+By [[Intersection of Semilattice Ideals is Ideal/Set of Sets]]:
+:$\bigcap X$ is an [[Definition:Ideal (Order Theory)|ideal]].
+By [[Intersection is Largest Subset/General Result]]:
+:$\forall A \in \mathit{Ids}\left({L}\right):\left({\forall I \in X: A \subseteq I}\right) \iff A \subseteq \bigcap X$
+Thus by definition:
+:$X$ admits an [[Definition:Infimum of Set|infimum]].
+Thus by dual to [[Lattice is Complete iff it Admits All Suprema]]:
+:$\mathcal I$ is [[Definition:Complete Lattice|complete lattice]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Filters form Complete Lattice}
+Tags: Complete Lattices
+
+\begin{theorem}
+Let $L = \left({S, \wedge, \preceq}\right)$ be a [[Definition:Bounded Above Set|bounded above]] [[Definition:Meet Semilattice|meet semilattice]].
+Let $F = \left({\mathit{Filt}\left({L}\right), \subseteq}\right)$ be an [[Definition:Inclusion Ordered Set|inclusion ordered set]],
+where $\mathit{Filt}\left({L}\right)$ denotes [[Definition:Set of Sets|set]] of all [[Definition:Filter|filters]] on $L$.
+Then $F$ is a [[Definition:Complete Lattice|complete lattice]].
+\end{theorem}
+
+\begin{proof}
+Define $L^{-1} := \left({S, \succeq}\right)$ be a [[Definition:Dual Ordered Set|dual ordered set]] of $L$.
+By [[Dual Pairs (Order Theory)]]:
+:$L^{-1}$ is a [[Definition:Bounded Below Set|bounded below]] [[Definition:Meet Semilattice|meet semilattice]].
+By [[Filters equal Ideals in Dual Ordered Set]]:
+:$\mathit{Filt}\left({L}\right) = \mathit{Ids}\left({L^{-1} }\right)$
+where $\mathit{Ids}\left({L^{-1} }\right)$ denotes the [[Definition:Set of Sets|set]] of all [[Definition:Ideal (Order Theory)|ideals]] in $L^{-1}$.
+Thus by [[Ideals form Complete Lattice]]:
+:$F$ is a [[Definition:Complete Lattice|complete lattice]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{319 is not Expressible as Sum of Fewer than 19 Fourth Powers}
+Tags: Fourth Powers, Hilbert-Waring Theorem, 319
+
+\begin{theorem}
+:$319 = 15 \times 1^4 + 3 \times 2^4 + 4^4$
+or:
+:$319 = 12 \times 1^4 + 4 \times 2^4 + 3 \times 3^4$
+\end{theorem}
+
+\begin{proof}
+First note that $5^4 = 625 > 319$.
+Then note that $2 \times 4^4 = 512 > 319$.
+Hence any expression of $319$ as [[Definition:Fourth Power|fourth powers]] uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.
+For the remainder, using $2^4$ uses fewer [[Definition:Fourth Power|fourth powers]] than $16$ instances of $1^4$.
+Now we have:
+{{begin-eqn}}
+{{eqn | l = 319
+ | r = 4^4 + 3 \times 2^4 + 15 \times 1^4
+}}
+{{eqn | r = 3 \times 3^4 + 4 \times 2^4 + 12 \times 1^4
+}}
+{{eqn | r = 2 \times 3^4 + 9 \times 2^4 + 13 \times 1^4
+}}
+{{eqn | r = 1 \times 3^4 + 14 \times 2^4 + 14 \times 1^4
+}}
+{{eqn | r = 0 \times 3^4 + 19 \times 2^4 + 15 \times 1^4
+}}
+{{end-eqn}}
+and it can be seen that the first two use the least number of [[Definition:Fourth Power|fourth powers]], at $19$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Positive Integers not Expressible as Sum of Fewer than 19 Fourth Powers}
+Tags: Fourth Powers, Hilbert-Waring Theorem
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integers]] are the only ones which cannot be expressed as the [[Definition:Integer Addition|sum]] of fewer than $19$ [[Definition:Fourth Power|fourth powers]]:
+=== [[Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers|$79$ as Sum of $19$ Fourth Powers]] ===
+{{:Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers}}
+=== [[159 is not Expressible as Sum of Fewer than 19 Fourth Powers|$159$ as Sum of $19$ Fourth Powers]] ===
+{{:159 is not Expressible as Sum of Fewer than 19 Fourth Powers}}
+=== [[319 is not Expressible as Sum of Fewer than 19 Fourth Powers|$319$ as Sum of $19$ Fourth Powers]] ===
+{{:319 is not Expressible as Sum of Fewer than 19 Fourth Powers}}
+=== [[399 is not Expressible as Sum of Fewer than 19 Fourth Powers|$399$ as Sum of $19$ Fourth Powers]] ===
+{{:399 is not Expressible as Sum of Fewer than 19 Fourth Powers}}
+\end{theorem}<|endoftext|>
+\section{Sum of 2 Squares in 3 Distinct Ways}
+Tags: Square Numbers, Sums of Squares, Sum of 2 Squares in 3 Distinct Ways
+
+\begin{theorem}
+== [[Sum of 2 Squares in 3 Distinct Ways/Examples|Examples]] ==
+{{:Sum of 2 Squares in 3 Distinct Ways/Examples}}
+\end{theorem}<|endoftext|>
+\section{Existence of n such that M - 2^n or M + 2^n has no Prime factors less than 331}
+Tags: Not Every Number is the Sum or Difference of Two Prime Powers, 331
+
+\begin{theorem}
+Let $M \in \Z$ be an [[Definition:Integer|integer]].
+Then there exists a [[Definition:Positive Integer|positive integer]] $n \in \Z_{\ge 0}$ such that either $M - 2^n$ or $M + 2^n$ has no [[Definition:Prime Factor|prime factors]] less than $331$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Arises during the course of [[Not Every Number is the Sum or Difference of Two Prime Powers]] and may be either a lemma or corollary -- hence may require a rename. Work is needed to analyse that result.}}
+\end{proof}<|endoftext|>
+\section{Completely Irreducible implies Infimum differs from Element}
+Tags: Meet Irreducible
+
+\begin{theorem}
+Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
+Let $p \in S$ such that
+:$p$ is [[Definition:Completely Irreducible|completely irreducible]].
+Then $\map \inf {p^\succeq \setminus \set p} \ne p$
+where $p^\succeq$ donotes the [[Definition:Upper Closure of Element|upper closure of]] $p$.
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Completely Irreducible|completely irreducible]]:
+:$p^\succeq \setminus \set p$ admits a [[Definition:Minimum Element|minimum]].
+Then:
+:$p^\succeq \setminus \set p$ admits a [[Definition:Infimum of Set|infimum]] and $\map \inf {p^\succeq \setminus \set p} \in p^\succeq \setminus \set p$
+By definition of [[Definition:Set Difference|difference]]:
+:$\map \inf {p^\succeq \setminus \set p} \notin \set p$
+Thus by definition of [[Definition:Singleton|singleton]]:
+:$\map \inf {p^\succeq \setminus \set p} \ne p$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Not Every Number is the Sum or Difference of Two Prime Powers}
+Tags: Not Every Number is the Sum or Difference of Two Prime Powers
+
+\begin{theorem}
+Not every [[Definition:Positive Integer|positive integer]] can be expressed in the form $p^m \pm q^n$ where $p, q$ are [[Definition:Prime Number|prime]] and $m, n$ are [[Definition:Positive Integer|positive integers]].
+\end{theorem}
+
+\begin{proof}
+What is to be demonstrated is that there exist [[Definition:Odd Integer|odd integers]] which cannot be expressed as $2^m \pm q^n$.
+{{ProofWanted|I have the paper downloaded and I am studying it.}}
+\end{proof}<|endoftext|>
+\section{Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element}
+Tags: Meet Irreducible
+
+\begin{theorem}
+Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
+Let $p \in S$.
+Then $p$ is [[Definition:Completely Irreducible|completely irreducible]] {{iff}}
+:$\exists q \in S: p \prec q \land \paren {\forall s \in S: p \prec s \implies q \preceq s} \land p^\succeq = \set p \cup q^\succeq$
+where $p^\succeq$ denotes the [[Definition:Upper Closure of Element|upper closure of]] $p$.
+\end{theorem}
+
+\begin{proof}
+=== Sufficient Condition ===
+Assume that:
+:$p$ is [[Definition:Completely Irreducible|completely irreducible]].
+By definition of [[Definition:Completely Irreducible|completely irreducible]]:
+:$p^\succeq \setminus \set p$ admits a [[Definition:Minimum Element|minimum]].
+By definitions of [[Definition:Minimum Element|minimum]] and [[Definition:Infimum of Set|infimum]]:
+:$p^\succeq \setminus \set p$ admits an [[Definition:Infimum of Set|infimum]].
+Define $q = \map \inf {p^\succeq \setminus \set p}$
+By definition of [[Definition:Minimum Element|minimum]]:
+:$q \in p^\succeq \setminus \set p$
+By definition of [[Definition:Set Difference|difference]]:
+:$q \in p^\succeq$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$p \preceq q$
+By [[Completely Irreducible implies Infimum differs from Element]]:
+:$q \ne p$
+Thus by definition of [[Definition:Strictly Precede|strictly precede]]:
+:$p \prec q$
+We will prove that
+:$\set p \cup q^\succeq \subseteq p^\succeq$
+Let $x \in \set p \cup q^\succeq$
+By definition of [[Definition:Set Union|union]]:
+:$x \in \set p$ or $x \in q^\succeq$
+In the first case:
+:$x \in \set p$
+By definition of [[Definition:Singleton|singleton]]:
+:$x = p$
+By definition of [[Definition:Reflexivity|reflexivity]]:
+:$p \preceq x$
+Thus by definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$x \in p^\succeq$
+{{qed|lemma}}
+In the second case:
+:$x \in q^\succeq$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$q \preceq x$
+By definition of [[Definition:Transitivity|transitivity]]:
+:$p \preceq x$
+Thus by definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$x \in p^\succeq$
+{{qed|lemma}}
+By definition of [[Definition:Infimum of Set|infimum]]:
+:$q$ is [[Definition:Lower Bound of Set|lower bound]] for $p^\succeq \setminus \set p$
+We will prove that:
+:$\forall s \in S: p \prec s \implies q \preceq s$
+Let $s \in S$ such that:
+:$p \prec s$
+By definition of [[Definition:Strictly Precede|strictly precede]]:
+:$p \preceq s$ and $p \ne s$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$s \in p^\succeq$
+By definition of [[Definition:Singleton|singleton]]:
+:$s \notin \set p$
+By definition of [[Definition:Set Difference|difference]]:
+:$s \in p^\succeq \setminus \set p$
+Thus by definition of [[Definition:Lower Bound of Set|lower bound]]:
+:$q \preceq s$
+{{qed|lemma}}
+We will prove that:
+:$p^\succeq \subseteq \set p \cup q^\succeq$
+Let $x \in p^\succeq$.
+By definition of [[Definition:Singleton|singleton]]:
+:$x = p$ or $x \in p^\succeq$ and $x \notin \set p$
+By definition of [[Definition:Set Difference|difference]]:
+:$x = p$ or $x \in p^\succeq \setminus \set p$
+By definitions of [[Definition:Infimum of Set|infimum]] and [[Definition:Lower Bound of Set|lower bound]]:
+:$x = p$ or $q \preceq x$
+By definitions of [[Definition:Singleton|singleton]] and [[Definition:Upper Closure of Element|upper closure of element]]:
+:$x \in \set p$ or $x \in q^\succeq$
+Thus by definition of [[Definition:Set Union|union]]:
+:$x \in \set p \cup q^\succeq$
+{{qed|lemma}}
+Thus by definition of [[Definition:Set Equality|set equality]]:
+:$p^\succeq = \set p \cup q^\succeq$
+{{qed|lemma}}
+=== Necessary Condition ===
+Assume that:
+:$\exists q \in S: p \prec q \land \paren {\forall s \in S: p \prec s \implies q \preceq s} \land p^\succeq = \set p \cup q^\succeq$
+We will prove that:
+:$q$ is [[Definition:Lower Bound of Set|lower bound]] for $p^\succeq \setminus \set p$
+Let $a \in p^\succeq \setminus \set p$.
+By definition of [[Definition:Set Difference|difference]]:
+:$a \in p^\succeq$ and $a \notin \set p$
+By definitions of [[Definition:Upper Closure of Element|upper closure of element]] and [[Definition:Singleton|singleton]]:
+:$p \preceq a$ and $a \ne p$
+By definition of [[Definition:Strictly Precede|strictly precede]]:
+:$p \prec a$
+Thus by assumption:
+:$q \preceq a$
+{{qed|lemma}}
+By definition of [[Definition:Reflexivity|reflexivity]]:
+:$q \preceq q$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$q \in q^\succeq$
+By definition of [[Definition:Set Union|union]]:
+:$q \in \set p \cup q^\succeq$
+By definition of [[Definition:Strictly Precede|strictly precede]]:
+:$p \ne q$
+By definition of [[Definition:Singleton|singleton]]:
+:$q \notin \set p$
+By definition of [[Definition:Set Difference|difference]]:
+:$q \in p^\succeq \setminus \set p$
+We will prove that:
+:$\forall b \in S: b$ is [[Definition:Lower Bound of Set|lower bound]] for $p^\succeq \setminus \set p \implies b \preceq q$
+Let $b \in S$ such that:
+:$b$ is [[Definition:Lower Bound of Set|lower bound]] for $p^\succeq \setminus \set p$
+Thus by definition of [[Definition:Lower Bound of Set|lower bound]]:
+:$b \preceq q$
+{{qed|lemma}}
+By definition of [[Definition:Infimum of Set|infimum]]:
+:$q = \map \inf {p^\succeq \setminus \set p}$
+By definition:
+:$p^\succeq \setminus \set p$ admits a [[Definition:Minimum Element|minimum]].
+Thus by definition:
+:$p$ is [[Definition:Completely Irreducible|completely irreducible]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Poulet Number/Examples/341}
+Tags: Poulet Numbers, 341
+
+\begin{theorem}
+The smallest [[Definition:Poulet Number|Poulet number]] is $341$:
+:$2^{341} \equiv 2 \pmod {341}$
+despite the fact that $341$ is not [[Definition:Prime Number|prime]]:
+:$341 = 11 \times 31$
+\end{theorem}
+
+\begin{proof}
+We have that:
+{{begin-eqn}}
+{{eqn | l = 2^{341}
+ | r = 4 \, 479 \, 489 \, 484 \, 355 \, 608 \, 421 \, 114 \, 884 \, 561 \, 136 \, 888 \, 556 \, 243 \, 290 \, 994 \, 469 \, 299 \, 069 \, 799 \, 978 \, 201 \, 927 \, 583 \, 742 \, 360 \, 321 \, 890 \, 761 \, 754 \, 986 \, 543 \, 214 \, 231 \, 552
+ | c =
+}}
+{{eqn | r = 4 \, 479 \, 489 \, 484 \, 355 \, 608 \, 421 \, 114 \, 884 \, 561 \, 136 \, 888 \, 556 \, 243 \, 290 \, 994 \, 469 \, 299 \, 069 \, 799 \, 978 \, 201 \, 927 \, 583 \, 742 \, 360 \, 321 \, 890 \, 761 \, 754 \, 986 \, 543 \, 214 \, 231 \, 550 + 2
+ | c =
+}}
+{{eqn | r = 341 \times 13 \, 136 \, 332 \, 798 \, 696 \, 798 \, 888 \, 899 \, 954 \, 724 \, 741 \, 608 \, 669 \, 335 \, 164 \, 206 \, 654 \, 835 \, 981 \, 818 \, 117 \, 894 \, 215 \, 788 \, 100 \, 763 \, 407 \, 304 \, 286 \, 671 \, 514 \, 789 \, 484 \, 550 + 2
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 2^{341}
+ | o = \equiv
+ | r = 1 \pmod {341}
+ | c =
+}}
+{{end-eqn}}
+{{finish|It remains to be shown that it is the smallest.}}
+\end{proof}<|endoftext|>
+\section{Fourth Powers which are Sum of 4 Fourth Powers}
+Tags: Fourth Powers
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integers]] are such that their [[Definition:Fourth Power|fourth powers]] can be expressed as the [[Definition:Integer Addition|sum]] of the [[Definition:Fourth Power|fourth powers]] of $4$ other [[Definition:Positive Integer|positive integers]] with no [[Definition:Common Divisor of Integers|common factors]]:
+:$353, 651, 2487, 2501, 2829, \ldots$
+{{OEIS|A039664}}
+\end{theorem}<|endoftext|>
+\section{Sums of both 2 and 3 Consecutive Squares}
+Tags: Sums of Squares
+
+\begin{theorem}
+The following are the smallest [[Definition:Positive Integer|positive integers]] that are the [[Definition:Integer Addition|sum]] of both $2$ and $3$ consecutive non-zero [[Definition:Square Number|square numbers]]:
+:$365, 35 \, 645, 3 \, 492 \, 725, 342 \, 251 \, 285, 33 \, 537 \, 133 \, 085, 3 \, 286 \, 296 \, 790 \, 925, \ldots$
+{{OEIS|A007667}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 365
+ | r = 10^2 + 11^2 + 12^2
+ | c =
+}}
+{{eqn | r = 13^2 + 14^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 35 \, 645
+ | r = 108^2 + 109^2 + 110^2
+ | c =
+}}
+{{eqn | r = 133^2 + 134^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 3 \, 492 \, 725
+ | r = 1078^2 + 1079^2 + 1080^2
+ | c =
+}}
+{{eqn | r = 1321^2 + 1322^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 342 \, 251 \, 285
+ | r = 10 \, 680^2 + 10 \, 681^2 + 10 \, 682^2
+ | c =
+}}
+{{eqn | r = 13 \, 081^2 + 13 \, 082^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 33 \, 537 \, 133 \, 085
+ | r = 105 \, 730^2 + 105 \, 731^2 + 105 \, 732^2
+ | c =
+}}
+{{eqn | r = 129 \, 493^2 + 129 \, 494^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 3 \, 286 \, 296 \, 790 \, 925
+ | r = 1 \, 046 \, 628^2 + 1 \, 046 \, 629^2 + 1 \, 046 \, 630^2
+ | c =
+}}
+{{eqn | r = 1 \, 281 \, 853^2 + 1 \, 281 \, 854^2
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|Needs to be proved that there are no more in between these}}
+\end{proof}<|endoftext|>
+\section{Sturm-Liouville Problem/Unit Weight Function}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $P, Q: \R \to \R$ be [[Definition:Real Function|real mappings]] such that $P$ is [[Definition:Smooth Real Function|smooth]] and [[Definition:Positive Real Function|positive]], while $Q$ is [[Definition:Continuous Real Function|continuous]]:
+:$\map P x \in C^\infty$
+:$\map P x > 0$
+:$\map Q x \in C^0$
+Let the [[Definition:Sturm-Liouville Equation|Sturm-Liouville equation]], with $\map w x = 1$, be of the form:
+:$-\paren {P y'}' + Q y = \lambda y$
+where $\lambda \in \R$.
+Let it satisfy the following [[Definition:Boundary Condition|boundary conditions]]:
+:$\map y a = \map y b = 0$
+Then all [[Definition:Solution of Differential Equation|solutions]] of the [[Definition:Sturm-Liouville Equation|Sturm-Liouville equation]], together with their [[Definition:Eigenvalue|eigenvalues]], form [[Definition:Infinite Sequence|infinite sequences]] $\sequence {y^{\paren n} }$ and $\sequence {\lambda^{\paren n} }$.
+Furthermore, each $\lambda^{\paren n}$ corresponds to an [[Definition:Eigenfunction|eigenfunction]] $y^{\paren n}$, unique up to a [[Definition:Constant|constant]] [[Definition:Factor (Algebra)|factor]].
+\end{theorem}
+
+\begin{proof}
+=== Outline ===
+Firstly, an equivalence between the [[Definition:Sturm-Liouville Equation|Sturm-Liouville equation]] and [[Definition:Minimum Value of Functional|minimisation]] of [[Definition:Real Functional|functional]] $\displaystyle J \sqbrk y = \int_a^b \paren {P y'^2 + Q y^2} \rd x$ problems is established.
+Then, the [[Definition:Lower Bound|lower bound]] of the [[Definition:Real Functional|functional]] $J$ is found, thus allowing $J$ to have a finite [[Definition:Minimum Value of Functional|minimisation]].
+Afterwards, a trial [[Definition:Minimizing Sequence of Functional|minimizing sequence]] is chosen, and $J$ becomes a [[Definition:Function|function]] of expansion [[Definition:Coefficient|coefficients]]. The more [[Definition:Coefficient|coefficients]], (possibly) the lower value.
+[[Definition:Sequence|Sequences]] of trial [[Definition:Mapping|mappings]] $\sequence {y_n^{\paren 1} }$ and values $\sequence {\lambda_n^{\paren 1} }$ of $J$ are introduced.
+[[Definition:Convergence|Convergence]] of $\sequence {\lambda_n^{\paren 1} }$ is shown at once.
+As for $\sequence {y_n^{\paren 1} }$, convergence to $y^{\paren 1}$ is proved for its [[Definition:Subsequence|subsequence]].
+The rest of arguments rest upon this weaker result.
+Furthermore, $\lambda^{\paren 1}$ and $y^{\paren 1}$ are observed to satisfy [[Definition:Sturm-Liouville Equation|Sturm-Liouville equation]].
+[[Definition:Convergence|Convergence]] of the original [[Definition:Sequence|sequence]] $\sequence {y_n^{\paren 1} }$ is secured.
+Finally, construction of the rest of [[Definition:Eigenfunction|eigenfunctions]] and [[Definition:Eigenvalue|eigenvalues]] is described.
+{{finish|Maybe add a diagrammatical presentation of the whole proof with Lemmas and somehow enumerated intermediate passages}}
+=== Lemma 1 ===
+The given [[Definition:Sturm-Liouville Equation|Sturm-Liouville equation]] is an [[Definition:Euler's Equation for Vanishing Variation|Euler equation]] of the following [[Definition:Real Functional|functional]]:
+:$\ds J \sqbrk y = \int_a^b \paren {P y'^2 + Qy^2} \rd x$
+constrained by a [[Simplest Variational Problem with Subsidiary Conditions|subsidiary condition]]:
+:$\ds \int_a^b y^2 \rd x = 1$
+=== Proof ===
+According to [[Simplest Variational Problem with Subsidiary Conditions]], the following equation must hold:
+:$F_y - \dfrac \rd {\rd x} F_{y'} + \lambda \paren {G_y - \dfrac \rd {\rd x} G_{y'} } = 0$
+where:
+:$F = P y'^2 + Q y^2$
+:$G = y^2$
+Then the [[Definition:Euler's Equation for Vanishing Variation|Euler equation]] reads:
+:$2 Q y - 2 \paren {P y'}' + 2 \lambda y = 0$
+Division by $2$ and rearrangement of terms yields the desired result.
+{{qed|lemma}}
+By [[Necessary Condition for Integral Functional to have Extremum for given function]], if $y$ is an [[Definition:Extremum of Functional|extremum]] of $J$, it is also a [[Definition:Solution of Differential Equation|solution]] of the [[Definition:Sturm-Liouville Equation|Sturm-Liouville equation]].
+=== Lemma 2 ===
+$J$ is bounded from below.
+=== Proof ===
+Since $Q$ is continuous on an interval, it is bounded.
+Since $P > 0$, it holds that:
+:$\displaystyle \int_a^b \paren {P y'^2 + Q y^2} \rd x > \int_a^b Q y^2 \rd x \ge M \int_a^b y^2 \rd x = M$
+where:
+:$\displaystyle M = \min_{a \mathop \le x \mathop \le b} \map Q x$
+Therefore, $J$ is bounded from below.
+{{qed|lemma}}
+Introduce a new [[Definition:Variable|variable]]:
+:$t := \pi \dfrac {x - a} {b - a}$
+Then the [[Definition:Closed Real Interval|interval]] of consideration $\closedint a b$ is mapped onto $\closedint 0 \pi$.
+Choose [[Definition:Ritz Sequence|Ritz sequence]] $\sequence {\map {\phi_n} t} = \sequence {\sin n t}$, where $n \in \N$.
+=== Lemma 3 ===
+The [[Definition:Element|elements]] of the [[Definition:Sequence|sequence]] $\sequence {\sin n t}$ are [[Definition:Orthogonal|orthogonal]] on the interval $\closedint 0 \pi$:
+:$\displaystyle \int_0^\pi \map \sin {k t} \map \sin {l t} \rd x = \frac \pi 2 \delta_{k l}$
+=== Proof ===
+The product involves two [[Definition:Element|elements]] of the [[Definition:Sequence|sequence]] $\sequence {\map \sin {n t} }$.
+Their indices either match each other or not.
+Suppose $k = l$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \int_0^\pi \map {\sin^2} {k t} \rd t
+ | r = \int_0^\pi \frac {1 - \map \cos {2 k t} } 2 \rd t
+ | c = [[Square of Sine]]
+}}
+{{eqn | r = \intlimits {\frac \pi 2 - \frac {\map \sin {2 k t} } {4 k} } 0 \pi
+}}
+{{eqn | r = \frac \pi 2
+ | c = [[Sine of Multiple of Pi]]
+}}
+{{end-eqn}}
+Suppose $k \ne l$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \int_0^\pi \map \sin {l t} \map \sin {k t} \rd t
+ | r = \int_0^\pi \frac {\map \cos {\paren {l - k} t} - \map \cos {\paren {l + k} t} } 2 \rd t
+ | c = [[Simpson's Formula for Sine by Sine]]
+}}
+{{eqn | r = \intlimits {\frac 1 2 \paren {\frac {\map \sin {\paren {l - k} t} } {l - k} - \frac {\map \sin {\paren {l + k} t} } {l + k} } } 0 \pi
+}}
+{{eqn | r = 0
+ | c = [[Sine of Multiple of Pi]]
+}}
+{{end-eqn}}
+By [[Proof by Cases]], the statement is proved.
+{{qed|lemma}}
+Let the trial [[Definition:Solution of Differential Equation|solution]] be of the following form:
+:$\displaystyle \map y x = \sum_{k \mathop = 1}^n \alpha_k \map \sin {k \map t x}$
+Trial [[Definition:Solution of Differential Equation|solution]] has to satisfy [[Definition:Boundary Condition|boundary]] and [[Simplest Variational Problem with Subsidiary Conditions|subsidiary conditions]].
+[[Definition:Boundary Condition|Boundary conditions]] are satisfied without further requirements.
+[[Simplest Variational Problem with Subsidiary Conditions|Subsidiary condition]] results into an additional constraint on [[Definition:Coefficient|coefficients]] $\alpha_k$:
+{{begin-eqn}}
+{{eqn | l = \int_a^b \paren {\sum_{k \mathop = 1}^n \alpha_k \map \sin {k \map t x} }^2 \rd x
+ | r = \frac \pi {b - a} \int_0^\pi \paren {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} }^2 \rd t
+}}
+{{eqn | r = \frac \pi {b - a} \int_0^\pi \paren {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} } \paren {\sum_{l \mathop = 1}^n \alpha_l \sin l x} \rd t
+ | c = Second summation label $k$ relabelled as $l$
+}}
+{{eqn | r = \frac \pi {b - a} \sum_{k, l \mathop = 1}^n \alpha_k \alpha_l \int_0^\pi \map \sin {k t} \map \sin {l t} \rd t
+}}
+{{eqn | r = \frac \pi {b - a} \sum_{k, l \mathop = 1}^n \alpha_k \alpha_l \frac \pi 2 \delta_{k l}
+ | c = as $\displaystyle \int_0^\pi \map \sin {k t} \map \sin {l t} \rd x = \frac \pi 2 \delta_{k l}$
+}}
+{{eqn | r = \frac \pi {b - a} \frac \pi 2 \sum_{k \mathop = 1}^n \alpha_k^2
+}}
+{{eqn | r = 1
+}}
+{{end-eqn}}
+All the points $\boldsymbol \alpha$ constitute a set $\sigma_n$ which is a surface of an $n$-dimensional sphere, defined by the [[Simplest Variational Problem with Subsidiary Conditions|subsidiary condition]].
+For the assumed trial [[Definition:Mapping|mapping]] the [[Definition:Real Functional|functional]] $\map {J_n} {\boldsymbol \alpha}$ reads as:
+:$\ds \map {J_n} {\boldsymbol \alpha} = \frac \pi {b - a} \int_0^\pi \sqbrk {\map P {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} }'^2 + \map Q {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} }^2} \rd t$
+The [[Definition:Integrand|integrand]] is a [[Definition:Degree of Polynomial|second order]] [[Definition:Polynomial (Analysis)|polynomial]] {{WRT}} the [[Definition:Vector (Linear Algebra)|components]] of $\boldsymbol \alpha$.
+Hence, $J$ is [[Definition:Continuous Real Function|continuous]] {{WRT}} the [[Definition:Vector (Linear Algebra)|components]] of $\boldsymbol \alpha$.
+The [[Definition:Vector (Linear Algebra)|components]] of $\boldsymbol \alpha$ constitute a [[Definition:Closed Set (Topology)|closed]] and bounded [[Definition:Set|set]].
+By definition, $\sigma_n$ is a [[Definition:Compact Topological Subspace|compact set]].
+Thus, $\map {J_n} {\boldsymbol \alpha}$ is [[Definition:Continuous Real Function on Interval|continuous]] on $\sigma_n$.
+By [[Continuous Function on Compact Subspace of Euclidean Space is Bounded]], $\map {J_n} {\boldsymbol \alpha}$ has a [[Definition:Minimum Value|minimum]] on $\sigma_n$.
+Let $\map {y_n^{\paren 1} } x$ be defined as:
+:$\displaystyle \map {y_n^{\paren 1} } x = \sum_{k \mathop = 1}^n \alpha_k^{\paren 1} \sin {k \map t x}$
+for which $\map {J_n} {\boldsymbol \alpha}$ achieves the [[Definition:Minimum Value|minimum]] $\lambda_n^{\paren 1}$, unrelated to $\lambda$.
+Then the $n$-th [[Definition:Element|element]] of the [[Definition:Sequence|sequence]] $\sequence {y_n^{\paren 1} }$ corresponds to the $n$-th [[Definition:Element|element]] of the [[Definition:Sequence|sequence]] of [[Definition:Minimum Value of Functional|minima]] $\sequence {\lambda_n^{\paren 1} }$ of $\map {J_n} {\boldsymbol \alpha}$.
+Since $\sigma_n \subset \sigma_{n + 1}$, where $\sigma_n$ has $\alpha_{n + 1} = 0$, it holds that:
+:$\map {J_n} {\alpha_1, \ldots, \alpha_n} = \map {J_{n + 1} } {\alpha_1, \ldots, \alpha_n, 0}$
+By [[Ritz Method implies Not Worse Approximation with Increased Number of Functions]]:
+:$\lambda_{n + 1}^{\paren 1} \le \lambda_n^{\paren 1}$
+Therefore, by increasing the domain of definition of $y_n^{\paren 1}$ through additional summands, the [[Definition:Minimum Value of Functional|minima]] of $\map {J_n} {\boldsymbol \alpha}$ cannot increase.
+{{explain|Define "domain of definition"}}
+From the last inequality and $J$ being bounded from below it follows, that the following limit exists:
+:$\displaystyle \lambda^{\paren 1} = \lim_{n \mathop \to \infty} \lambda_n^{\paren 1}$
+=== Lemma 4 ===
+The sequence $\sequence {y_n^{\paren 1} }$ contains a [[Definition:Uniform Convergence|uniformly convergent]] [[Definition:Subsequence|subsequence]].
+=== Proof ===
+The [[Definition:Sequence|sequence]]
+:$\displaystyle \lambda_n^{\paren 1} = \frac \pi {b - a} \int_0^\pi \paren {P {y_n^{\paren 1} }'^2 + Q {y_n^{\paren 1} }^2} \rd t$
+is [[Definition:Convergence|convergent]] with its limit being $\lambda^{\paren 1}$.
+Hence, it is [[Definition:Bounded Sequence|bounded]]:
+:$\displaystyle \frac \pi {b - a} \int_0^\pi \paren {P {y_n^{\paren 1} }'^2 + Q {y_n^{\paren 1} }^2} \rd t \le M $
+Furthermore:
+{{begin-eqn}}
+{{eqn | l = \frac \pi {b - a} \int_0^\pi P {y_n^{\paren 1} }'^2 \rd t
+ | o = \le
+ | r = M - \frac \pi {b - a} \int_0^\pi Q {y_n^{\paren 1} }^2 \rd t
+}}
+{{eqn | o = \le
+ | r = M + \frac \pi {b - a} \size {\int_0^\pi Q {y_n^{\paren 1} }^2 \rd t}
+}}
+{{eqn | o = \le
+ | r = M + \max_{a \mathop \le x \mathop \le b} \size {\map Q x} \size {\int_a^b {y_n^{\paren 1} }^2 \rd x}
+}}
+{{eqn | r = M + \max_{a \mathop \le x \mathop \le b} \size {\map Q x}
+ | c = as $\displaystyle \int_a^b y^2 \rd x = 1$, where $y = y _n^{\paren 1}$ minimizes $\map {J_n} {\boldsymbol \alpha}$
+}}
+{{eqn | r = M_1
+}}
+{{end-eqn}}
+Consequently:
+:$\displaystyle \frac \pi {b - a} \min_{a \mathop \le x \mathop \le b} \map P x \int_0^\pi \map { {y_n^{\paren 1} }'^2} t \rd t \le \frac \pi {b - a} \int_0^\pi P \map { {y_n^{\paren 1} }'^2} t \rd t \le M_1$
+For [[Definition:Positive Real Number|positive]] $P$, division by $P$ does not affect the direction of inequality.
+It follows that:
+:$\displaystyle \int_0^\pi \map { {y_n^{\paren 1} }'^2} t \rd t \le \frac {b - a} \pi \frac {M_1} {\min_{a \mathop \le x \mathop \le b} } \map P x = M_2$
+Consider squared absolute value of $y_n^{\paren 1}$.
+Then, for $0 \le t \le \pi$:
+{{begin-eqn}}
+{{eqn | l = \size {\map {y_n^{\paren 1} } t}^2
+ | r = \size {\map {y_n^{\paren 1} } t - 0}^2
+ | c = as $\map {y_n^{\paren 1} } {t = 0} = 0$
+}}
+{{eqn | r = \size {\map {y_n^{\paren 1} } t - \map {y_n^{\paren 1} } 0}^2
+}}
+{{eqn | r = \size {\int_0^t \map { {y_n^{\paren 1} }'} \zeta \rd \zeta}^2
+}}
+{{eqn | o = \le
+ | r = \int_0^t \map { {y_n^{\paren 1} }'^2} \zeta \rd \zeta \int_0^t \rd \zeta
+ | c = [[Cauchy-Bunyakovsky-Schwarz Inequality for Definite Integrals]]:
+}}
+{{eqn | o = \le
+ | r = \int_0^\pi \map { {y_n^{\paren 1} }'^2} \zeta \rd \zeta \int_0^\pi \rd \zeta
+ | c = as all [[Definition:Integrand|integrands]] are non-negative
+}}
+{{eqn | o = \le
+ | r = M_2 \pi
+}}
+{{end-eqn}}
+In other words:
+:$\forall t \in \closedint 0 \pi, n \in \N : \size {\map {y_n^{\paren 1} } t - \map {y_n^{\paren 1} } 0} \le \sqrt {M_2 \pi}$
+Thus $\sequence {y_n^{\paren 1} }$ is [[Definition:Uniformly Bounded|uniformly bounded]].
+In addition to this, for $0 \le t_1, t_2 \le \pi$:
+{{begin-eqn}}
+{{eqn | l = \size {\map {y_n^{\paren 1} } {t_2} - \map {y_n^{\paren 1} } {t_1} }^2
+ | r = \size {\int_{t_1}^{t_2} \map { {y_n^{\paren 1} }'} t \rd t}^2
+ | c = [[Fundamental Theorem of Calculus]]
+}}
+{{eqn | o = \le
+ | r = \int_{t_1}^{t_2} {y_n^{\paren 1} }'^2 \rd x \size {\int_{t_1}^{t_2} \rd t}
+ | c = [[Cauchy-Bunyakovsky-Schwarz Inequality for Definite Integrals]]
+}}
+{{eqn | o = \le
+ | r = \int_0^\pi {y_n^{\paren 1} }'^2 \rd x \size {\int_{t_1}^{t_2} \rd t}
+}}
+{{eqn | o = \le
+ | r = M_2 \size {t_2 - t_1}
+}}
+{{end-eqn}}
+Let $\epsilon$ be any [[Definition:Strictly Positive Real Number|strictly positive real number]] such that $\epsilon = \sqrt {M_2 \delta}$, where $\delta$ is a [[Definition:Strictly Positive Real Number|strictly positive real number]].
+Suppose $\delta$ is such that $\size {t_2 - t_1} < \delta$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \size {\map {y_n^{\paren 1} } {t_2} - \map {y_n^{\paren 1} } {t_1} }
+ | o = \le
+ | r = \sqrt {M_2 \size {t_2 - t_1} }
+}}
+{{eqn | o = <
+ | r = \sqrt {M_2 \delta}
+}}
+{{eqn | r = \epsilon
+}}
+{{end-eqn}}
+In other words:
+:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall n \in \N: \forall t_1, t_2 \in \closedint 0 \pi: \size {t_2 - t_1} < \delta \implies \size {\map {y_n^{\paren 1} } {t_2} - \map {y_n^{\paren 1} } {t_1} } < \epsilon$
+where [[Definition:Metric Induced by Norm|metric is induced by norm]].
+Thus $\sequence {y_n^{\paren 1} }$ is [[Definition:Uniformly Equicontinuous|uniformly equicontinuous]].
+By [[Definition:Arzela's Theorem|Arzela's Theorem]], there exists a [[Definition:Uniform Convergence|uniformly convergent]] [[Definition:Subsequence|subsequence]] $\sequence {y_{n_m}^{\paren 1} }$ from $\sequence {y_n^{\paren 1} }$.
+{{qed|lemma}}
+Denote:
+:$\displaystyle \map {y^{\paren 1} } x = \lim_{m \mathop \to \infty} \map {y_{n_m}^{\paren 1} } x$
+Now a proposition is established, needed for the upcoming Lemma 6.
+=== Lemma 5 ===
+Let $\map y t$ be [[Definition:Continuity|continuous]] in $\closedint 0 \pi$.
+Suppose:
+:$\displaystyle \forall h \in C^2 \openint 0 \pi : \map h 0 = \map h \pi = \map {h'} 0 = \map {h'} \pi = 0 : \int_0^\pi \sqbrk {-\paren {P h'}' + Q_1 h} y \rd t = 0$
+Then $\map y t \in C^2 \openint 0 \pi$ and:
+:$-\paren {P y'}' + Q_1 y = 0$
+=== Proof ===
+By [[Integration by Parts|Integration by parts]], [[Product Rule for Derivatives]], boundary conditions for $h$, and noticing that:
+{{begin-eqn}}
+{{eqn | l = \map \d {\int_0^t \map {f'} \xi \rd \xi}
+ | r = \map \d {\map f t - \map f 0}
+ | c = [[Fundamental Theorem of Calculus]]
+}}
+{{eqn | r = \map {f'} t \rd t
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \d {\int_0^t \paren {\int_0^\xi \map g \zeta \rd \zeta} \rd \xi}
+ | r = \paren {\int_0^t \map g \zeta \rd \zeta} \rd t
+}}
+{{eqn | r = \paren {\int_0^t \map g \xi \rd \xi} \rd t
+}}
+{{end-eqn}}
+the previous [[Definition:Definite Integral|integral]] can be rewritten as:
+{{begin-eqn}}
+{{eqn | l = \int_0^\pi \paren {-\paren {P h'}' + Q_1 h} y \rd t
+ | r = -\int_0^\pi P h'' y \rd t - \int_0^\pi P' h' y \rd t + \int_0^\pi Q_1 h y \rd t
+}}
+{{eqn | r = -\int_0^\pi P h'' y \rd t - \int_0^\pi h' \rd \paren {\int_0^t P' y \rd \xi} + \int_0^\pi h \rd \paren {\int_0^t Q_1 y \rd \xi}
+}}
+{{eqn | r = -\int_0^\pi P h'' y \rd t - \paren {\intlimits {h' \int_0^t P' y \rd \xi} {t = 0} {t = \pi} - \int_0^\pi h'' \paren {\int_0^t P' y \rd \xi} \rd t} + \paren {\intlimits {h \int_0^t Q_1 y \rd \xi} {t \mathop = 0} {t \mathop = \pi} - \int_0^\pi h' \paren {\int_0^t Q_1 y \rd \xi} \rd t}
+}}
+{{eqn | r = -\int_0^\pi P h'' y \rd t + \int_0^\pi h'' \paren {\int_0^t P' y \rd \xi} \rd t - \int_0^\pi h' \paren {\int_0^t Q_1 y \rd \xi } \rd t
+}}
+{{eqn | r = -\int_0^\pi P h'' y \rd t + \int_0^\pi h'' \paren {\int_0^t P' y \rd \xi} \rd t - \paren {\int_0^\pi h' \rd \paren {\int_0^t \paren {\int_0^\xi Q_1 y \rd \zeta} \rd \xi} }
+}}
+{{eqn | r = -\int_0^\pi P h'' y \rd t + \int_0^\pi h'' \int_0^t P' y \rd \xi - \paren {\intlimits {h' \int_0^t \paren {\int_0^\xi Q_1 y \rd \zeta} \rd \xi} {t \mathop = 0} {t \mathop = \pi} - \int_0^\pi h'' \paren {\int_0^t \paren {\int_0^\xi Q_1 y \rd \zeta} \rd \xi} \rd t}
+}}
+{{eqn | r = -\int_0^\pi P h'' y \rd t + \int_0^\pi h'' \int_0^t P' y \rd \xi + \int_0^\pi h'' \paren {\int_0^t \paren {\int_0^\xi Q_1 y \rd \zeta} \rd \xi} \rd t
+}}
+{{eqn | r = \int_0^\pi \paren {- P y + \int_0^t P' y \rd \xi + \int_0^t \paren {\int_0^\xi Q_1 y \rd \zeta} \rd \xi} h'' \rd t
+}}
+{{eqn | r = 0
+}}
+{{end-eqn}}
+From the [[Sturm-Liouville Problem/Unit Weight Function/Lemma|lemma]]:
+:$\displaystyle - P y + \int_0^t P' y \rd \zeta + \int_0^x \paren {\int_0^t Q_1 y \rd t} \rd \zeta = c_0 + c_1 t$
+The {{RHS}} as well as the second and third terms on the {{LHS}} are differentiable {{WRT}} $t$.
+Thus $\paren {P y}'$ exists.
+[[Definition:Differentiation|Differentiation]] {{WRT|Differentiation}} $t$ leads to:
+:$\displaystyle -\paren {P y}' + P' y + \int_0^t Q_1 y \rd \zeta = c_1$
+or:
+:$\displaystyle - P y' + \int_0^t Q_1 y \rd \zeta = c_1$
+The {{RHS}}, the second term on the {{LHS}} and $P$ are [[Definition:Continuity|continuous]] and [[Definition:Differentiable Mapping|differentiable]] {{WRT|Differentiation}} $t$, while $P$ is also [[Definition:Positive Real Number|positive]].
+Therefore, $y'$ exists and is [[Definition:Continuity|continuous]].
+Hence, $\paren {P y'}'$ exists and:
+:$-\paren {P y'}' + Q_1 y = 0$
+Furthermore, $P$ is [[Definition:Continuity|continuous]] and [[Definition:Differentiable Mapping|differentiable]], while $Q_1$ is continuous.
+Then $y''$ exists and is [[Definition:Continuity|continuous]].
+{{qed|lemma}}
+=== Lemma 6 ===
+$y^{\paren 1}$ together with $\lambda^{\paren 1}$ satisfy the [[Definition:Sturm-Liouville Equation|Sturm-Liouville equation]], where $\map w x = 1$:
+:$-\paren {P {y^{\paren 1} }' }' + Q y^{\paren 1} = \lambda^{\paren 1} y^{\paren 1}$
+=== Proof ===
+Let $J_n$, together with its subsidiary condition $\displaystyle \int_a^b y^2 \rd x = 1$, achieve a minimum for $\boldsymbol \alpha = \boldsymbol \alpha^{\paren 1}$.
+Then the necessary condition for its minimum is:
+{{begin-eqn}}
+{{eqn | l = \map {\dfrac \partial {\partial \alpha_r} } {\map {J_n} {\boldsymbol \alpha} - \lambda_n^{\paren 1} \int_a^b y^2 \rd x}
+ | r = \map {\dfrac \partial {\partial \alpha_r} } {\map {J_n} {\boldsymbol \alpha} - \lambda_n^{\paren 1} \frac \pi {b - a} \int_0^\pi \paren {\sum_{k \mathop = 1}^n \alpha_k \sin {k x} }^2 \rd x}
+}}
+{{eqn | r = 0
+}}
+{{end-eqn}}
+Notice, that:
+{{begin-eqn}}
+{{eqn | l = \dfrac \partial {\partial \alpha_r} \paren {\sum_{k \mathop = 1}^n \alpha_k \sin {k x} }^2
+ | r = \dfrac {\partial \paren {\sum_{k \mathop = 1}^n \alpha_k \sin {k x} }^2} {\partial \sum_{k = 1}^n \alpha_k \sin {k x} } \dfrac {\partial \sum_{j \mathop = 1}^n \alpha_j \sin {j x} } {\partial \alpha_r}
+ | c = [[Derivative of Composite Function]]
+}}
+{{eqn | r = 2 \sqbrk {\sum_{k \mathop = 1}^n \alpha_k \sin {k x} } \sum_{j = 1}^n \delta_j^r \sin {j x}
+ | c = as $\dfrac {\partial \alpha_j} {\partial \alpha_r} = \delta_j^r$
+}}
+{{eqn | r = 2 \sqbrk {\sum_{k \mathop = 1}^n \alpha_k \sin {k x} } \sin {r x}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \dfrac {\partial J_n} {\partial \alpha_r}
+ | r = \frac {\partial} {\partial \alpha_r} \frac \pi {b - a} \int_0^\pi \sqbrk {\map P {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} }'^2 + Q \paren {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} }^2} \rd t
+}}
+{{eqn | r = \frac \pi {b - a} \int_0^\pi \sqbrk {P \dfrac {\partial \paren {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} }'^2} {\partial \paren {\sum_{j \mathop = 1}^n \alpha_j \sin {j t} }'} \dfrac {\partial \paren {\sum_{j \mathop = 1}^n \alpha_j \sin {j t} }'} {\partial \alpha_r} + 2 Q \sqbrk {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} } \sin {r t} } \rd t
+ | c = [[Derivative of Composite Function]]
+}}
+{{eqn | r = \frac {2 \pi} {b - a} \int_0^\pi \sqbrk {\map P {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} }' \dfrac {\partial \paren {\sum_{j \mathop = 1}^n \alpha_j j \cos {j t} } }{\partial \alpha_r} + Q \sqbrk {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} } \sin {r t} } \rd t
+}}
+{{eqn | r = \frac {2 \pi} {b - a} \int_0^\pi \sqbrk {\map P {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} }' \paren {\sum_{j \mathop = 1}^n \delta_j^r j \cos {j t} } + Q \sqbrk {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} } \sin {r t} } \rd t
+}}
+{{eqn | r = \frac {2 \pi} {b - a} \int_0^\pi \sqbrk {\map P {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} }' \paren {r \cos {r t} } + Q \sqbrk {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} } \sin {r t} }\rd t
+}}
+{{eqn | r = \frac {2 \pi} {b - a} \int_0^\pi \sqbrk {\map P {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} }' \paren {\sin {r t} }' + Q \sqbrk {\sum_{k \mathop = 1}^n \alpha_k \sin {k t} } \sin {r t} } \rd t
+}}
+{{end-eqn}}
+This leads to a system of equations:
+:$\displaystyle \int_0^\pi \paren {\map P t \sqbrk {\sum_{k \mathop = 1}^n \alpha_k^{\paren 1} \paren {\sin {k t} }'} \paren {\sin {r x} }' + \sqbrk {Q - \lambda_n^{\paren 1} } \sqbrk {\sum_{k \mathop = 1}^n \alpha_k^{\paren 1} \sin {k t} } \sin {r t} } \rd t = 0$
+Multiplying each equation by an arbitrary constant $C_r^{\paren n}$ and summing over $r$ results in:
+:$\displaystyle \int_0^\pi \sqbrk {P y_n' h_n' + \paren {Q - \lambda_n^{\paren 1} y_n h_n} } \rd t = 0$
+where:
+:$\displaystyle \map {h_n} t = \sum_{r \mathop = 1}^n C_r^{\paren n} \sin {r t}$
+:$\displaystyle y_n = \sum_{k \mathop = 1}^n \alpha_k \sin {k t}$
+By [[Integration by Parts|Integration by parts]]:
+{{begin-eqn}}
+{{eqn | l = \int_0^\pi \sqbrk {P y_n' h_n' + \paren {Q - \lambda_n^{\paren 1} y_n h_n} } \rd t
+ | r = \int_0^\pi P h_n' \map {\rd y_n} t + \int_0^\pi \sqbrk {\paren {Q - \lambda_n^{\paren 1} y_n h_n} } \rd t
+}}
+{{eqn | r = P h_n' y_n \big \vert_0^\pi - \int_0^\pi y_n \rd P \map {h_n'} t + \int_0^\pi \sqbrk {\paren {Q - \lambda_n^{\paren 1} y_n h_n} } \rd t
+}}
+{{eqn | r = \int_0^\pi \sqbrk { - \paren {P h_n'}' + \paren {Q - \lambda_n^{\paren 1} y_n h_n} } y_n \rd t
+}}
+{{end-eqn}}
+Consider all real mappings $h$ such that:
+:$\map h x \in C^2 \openint 0 \pi$
+and satisfying the [[Definition:Boundary Condition|boundary conditions]].
+Then $C_r^{\paren n}$ can be chosen, such that:
+:$\displaystyle \lim_{n \mathop \to \infty} \int_0^\pi \size {\map {h_n} x - \map h x}^2 \rd x = 0$
+:$\displaystyle \lim_{n \mathop \to \infty} \int_0^\pi \size {\map {h_n'} x - \map {h'} x}^2 \rd x = 0$
+:$\displaystyle \lim_{n \mathop \to \infty} \int_0^\pi \size {\map {h_n''} x - \map {h''} x}^2 \rd x = 0$
+{{finish|proof of the $C_r^{\paren n}$ segment}}
+Due to the existence of uniformly convergent subsequence, $y_n^{\paren 1}$ converges to $y^{\paren 1}$ uniformly on $\closedint 0 \pi$:
+:$\displaystyle \lim_{m \mathop \to \infty} \int_0^\pi \paren {-\paren {P h_{n_m}'}' + \paren {Q - \lambda_{n_m}^{\paren 1} } h_{n_m} } y_{n_m}^{\paren 1} \rd x = \int_0^\pi \paren { -\paren {P h'}' + \paren {Q - \lambda^{\paren 1} } h} y^{\paren 1} \rd x = 0$
+{{finish|prove}}
+By Lemma 5, where $Q_1 = Q - \lambda^{\paren 1}$, $y^{\paren 1} \in C^2 \closedint 0 \pi$ and satisfies [[Definition:Sturm-Liouville Equation|Sturm-Liouville equation]] with $w = 1$.
+{{qed|lemma}}
+=== Lemma 7 ===
+$\sequence {\map {y_n^{\paren 1} } x}$ [[Definition:Pointwise Convergence|pointwise converges]] to $\map {y^{\paren 1} } x$.
+=== Proof ===
+By [[Existence and Uniqueness of Solution for Linear Second Order ODE with two Initial Conditions]], where $\map R x = 0$, the [[Definition:Sturm-Liouville Equation|Sturm-Liouville equation]]
+:$-\paren {P y'}' + Q y = \lambda y$
+satisfying the [[Definition:Boundary Condition|boundary conditions]]:
+:$\map y 0 = \map y \pi = 0$
+and the [[Simplest Variational Problem with Subsidiary Conditions|subsidiary condition]]:
+:$\displaystyle \int_0^\pi \map {y^2} t = 1$
+is unique up to the [[Definition:Sign|sign]] of $y$.
+Let $\map {y^{\paren 1} } t$ be a [[Definition:Solution of Differential Equation|solution]] corresponding to $\lambda = \lambda^{\paren 1}$
+Due to the [[Simplest Variational Problem with Subsidiary Conditions|subsidiary condition]], the condition $\map {y^{\paren 1} } t = 0$ cannot hold in the entire [[Definition:Closed Real Interval|closed interval]] $\closedint 0 \pi$.
+Hence, the [[Definition:Set|set]] of [[Definition:Root of Polynomial|roots]] to this condition is [[Definition:Countable Set|countable]].
+Then:
+:$\exists t_0 \in \closedint 0 \pi: \map {y^{\paren 1} } {t_0} \ne 0$
+Choose the [[Definition:Sign of Number|sign]] so that $\map {y^{\paren 1} } {t_0} > 0$
+Similarly, let $\map {y_n^{\paren 1} } t$ be a [[Definition:Solution of Differential Equation|solution]] corresponding to $\lambda = \lambda_n^{\paren 1}$
+Choose the [[Definition:Sign of Number|signs]] so that:
+:$\forall n \in \N : \map {y_n^{\paren 1} } {t_0} \ge 0$
+Suppose $\map {y_n^{\paren 1} } t$ does not [[Definition:Pointwise Convergence|pointwise converge]] to $\map {y^{\paren 1} } t$.
+By [[Arzela's Theorem]] there exists another [[Definition:Subsequence|subsequence]] from $\sequence {\map {y_n^{\paren 1} } t}$, converging to another [[Definition:Solution of Differential Equation|solution]] $\overline y^{\paren 1}$, where $\lambda = \lambda^{\paren 1}$.
+Because of the uniqueness of [[Definition:Solution of Differential Equation|solutions]], except for the [[Definition:Sign of Number|sign]], both [[Definition:Solution of Differential Equation|solutions]] may differ only in their [[Definition:Sign of Number|signs]]:
+:$\map {\overline y^{\paren 1} } x = - \map {y^{\paren 1} } t$
+Therefore:
+:$\map {\overline y^{\paren 1} } {t_0} < 0$
+This is impossible, since:
+:$\forall n \in N : \map {y_n^{\paren 1} } {t_0} \ge 0$
+Therefore $\map {y_n^{\paren 1} } t$ [[Definition:Pointwise Convergence|pointwise converges]] to $\map {y^{\paren 1} } t$, provided $\map {y_n^{\paren 1} } t$ is chosen with the correct [[Definition:Sign of Number|sign]].
+{{finish|In fact, convergence is uniform; establish this result by a proof}}
+{{qed|lemma}}
+{{finish|Check how properly insert [[Limit of Subsequence equals Limit of Sequence]] as an alternative}}
+=== Lemma 8 ===
+[[Definition:Sequence|Sequences]] $\sequence {y^{\paren n} }$ and $\sequence {\lambda^{\paren n} }$ are [[Definition:Infinite Sequence|infinite]].
+=== Proof ===
+Suppose, $y^{\paren r}$ and $\lambda^{\paren r}$ are known.
+The next [[Definition:Eigenfunction|eigenfunction]] $y^{\paren {r + 1} }$ and the corresponding [[Definition:Eigenvalue|eigenvalue]] $\lambda^{\paren {r + 1} }$ can be found by [[Definition:Minimum Value of Functional|minimising]]
+:$\displaystyle J \sqbrk y = \int_0^\pi \paren {P y'^2 + Q y^2} \rd x $
+where [[Definition:Boundary Condition|boundary]] and [[Simplest Variational Problem with Subsidiary Conditions|subsidiary conditions]] are supplied with [[Definition:Orthogonal|orthogonality]] conditions:
+:$\forall m \in \N : {1 \le m \le r} : \displaystyle \int_0^\pi \map {y^{\paren m} } t \map {y^{\paren {r + 1} } } t \rd t = 0$
+The new [[Definition:Solution of Differential Equation|solution]] of the form:
+:$\displaystyle \map {y_n^{\paren {r + 1} } } t = \sum_{k \mathop = 1}^n \alpha_k^{\paren {r + 1} } \sin {k t}$
+is now also [[Definition:Orthogonal|orthogonal]] to [[Definition:Mapping|mappings]]:
+:$\displaystyle \map {y_n^{\paren m} } t = \sum_{k \mathop = 1}^n \alpha_k^{\paren m} \sin {k t}$
+This results into:
+:$\displaystyle \sum_{k \mathop = 1}^n \alpha_k^{\paren {r + 1} } \int_0^\pi \sin {k t} \paren {\sum_{l \mathop = 1}^n \alpha_l^{\paren m} \sin {l t} } \rd t = \frac \pi 2 \sum_{k \mathop = 1}^n \alpha_k^{\paren {r + 1} } \alpha_k^{\paren m} = 0$
+These [[Definition:Equation|equations]] describe $r$ distinct [[Definition:Dimension|$\paren {n - 1}$-dimensional]] [[Definition:Hyperplane|hyperplanes]], passing through the [[Definition:Origin|origin of coordinates]] in $n$ [[Definition:Dimension|dimensions]].
+These [[Definition:Hyperplane|hyperplanes]] intersect the [[Definition:Sphere|sphere]] $\sigma_n$, resulting in an [[Definition:Dimension|$\paren {n - r}$-dimensional]] [[Definition:Sphere|sphere]] $\hat \sigma_{n - r}$.
+By definition, it is a [[Definition:Compact Topological Subspace|compact set]].
+By [[Continuous Function on Compact Subspace of Euclidean Space is Bounded]], $\map {J_n} {\boldsymbol \alpha}$ has a [[Definition:Minimum Value|minimum]] on $\hat {\sigma}_{n - r}$.
+Denote it as $\lambda_n^{\paren {r + 1} }$.
+By [[Ritz Method implies Not Worse Approximation with Increased Number of Functions]]:
+:$\displaystyle \lambda_{n + 1}^{\paren {r + 1} } \le \lambda_n^{\paren {r + 1} }$
+This, together with $J$ being [[Definition:Bounded Below Mapping|bounded from below]], implies:
+:$\displaystyle \lambda^{\paren {r + 1} } = \lim_{n \mathop \to \infty} \lambda_n^{\paren {r + 1} }$
+Additional constraints may or may not affect the new [[Definition:Minimum Value|minimum]]:
+:$\lambda^{\paren r} \le \lambda^{\paren {r + 1} }$
+Let:
+:$\displaystyle \map {y_n^{\paren {r + 1} } } t = \sum_{k \mathop = 1}^n \alpha_k^{\paren {r + 1} } \sin {k t}$
+$y^{\paren {r + 1} }$ satisfies [[Definition:Sturm-Liouville Equation|Sturm-Liouville equation]] together with [[Definition:Boundary Condition|boundary]], [[Simplest Variational Problem with Subsidiary Conditions|subsidiary]] and [[Definition:Orthogonal|orthogonality]] conditions.
+By Lemma 7, which is not affected by additional constraints, $\sequence {y_n^{\paren {r + 1} } }$ [[Definition:Uniform Convergence|uniformly converges]] to $y^{\paren {r + 1} }$.
+Thus, $y^{\paren {r + 1} }$ is an [[Definition:Eigenfunction|eigenfunction]] of [[Definition:Sturm-Liouville Equation|Sturm-Liouville equation]] with an [[Definition:Eigenvalue|eigenvalue]] $\lambda^{\paren {r + 1} }$.
+[[Definition:Orthogonal|Orthogonal]] [[Definition:Mapping|mappings]] are [[Definition:Linearly Independent|linearly independent]].
+Each [[Definition:Eigenvalue|eigenvalue]] corresponds only to one [[Definition:Eigenfunction|eigenfunction]], unique up to a [[Definition:Constant|constant]] factor.
+Thus:
+:$\lambda^{\paren r} < \lambda^{\paren {r + 1} }$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Completely Irreducible implies Meet Irreducible}
+Tags: Meet Irreducible
+
+\begin{theorem}
+Let $L = \left({S, \wedge, \preceq}\right)$ be a [[Definition:Meet Semilattice|meet semilattice]].
+Let $p \in S$.
+Then if $p$ is [[Definition:Completely Irreducible|completely irreducible]], then $p$ is [[Definition:Meet Irreducible|meet irreducible]].
+\end{theorem}
+
+\begin{proof}
+Assume that
+:$p$ is [[Definition:Completely Irreducible|completely irreducible]].
+By [[Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element]]
+:$\exists q \in S: p \prec q \land \left({\forall s \in S: p \prec s \implies q \preceq s}\right) \land p^\succeq = \left\{ {p}\right\} \cup q^\succeq$
+where $p^\succeq$ denotes the [[Definition:Upper Closure of Element|upper closure of]] $p$.
+Let $a, b \in S$ such that
+:$p = a \wedge b$
+Aiming for a [[Definition:Contradiction|contradiction]] suppose that
+:$a \ne p$ and $b \ne p$
+By [[Meet Precedes Operands]]:
+:$p \preceq b$
+By definition of [[Definition:Strictly Precede|strictly precede]]:
+:$p \prec b$
+Then
+:$q \preceq b$
+By definition of [[Definition:Strictly Precede|strictly precede]]:
+:$p \preceq q$
+By [[Meet Precedes Operands]]:
+:$p \preceq a$
+By definition of [[Definition:Strictly Precede|strictly precede]]:
+:$p \prec a$
+Then
+:$q \preceq a$
+By [[Meet Semilattice is Ordered Structure]]:
+:$q \preceq p$
+By definition of [[Definition:Antisymmetric Relation|antisymmetry]]:
+:$p = q$
+Thus this contradicts $p \prec q$ by definition of [[Definition:Strictly Precede|strictly precede]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Discrepancy between Julian Year and Tropical Year}
+Tags: Calendars
+
+\begin{theorem}
+The [[Definition:Julian Year|Julian year]] and the [[Definition:Tropical Year|tropical year]] differ such that the [[Definition:Julian Calendar|Julian calendar]] becomes $1$ [[Definition:Day|day]] further out approximately every $128$ [[Definition:Year|years]].
+\end{theorem}
+
+\begin{proof}
+By definition, the [[Definition:Length of Time|length]] $Y_T$ of the [[Definition:Tropical Year|tropical year]] is defined as $\approx 365 \cdotp 24219 \, 878$ [[Definition:Day|days]]
+By definition of the [[Definition:Julian Year|Julian year]]:
+:$Y_J = \begin{cases} 366 \, \text {days} & : 4 \divides y \\
+365 \, \text {days} & : 4 \nmid y \end{cases}$
+where:
+:$Y_J$ denotes the [[Definition:Length of Time|length]] of the [[Definition:Julian Year|Julian year]] in [[Definition:Day|days]]
+:$y$ denotes the number of the [[Definition:Year|year]]
+:$4 \divides y$ denotes that $y$ is [[Definition:Divisor of Integer|divisible]] by $4$
+:$4 \nmid y$ denotes that $y$ is not [[Definition:Divisor of Integer|divisible]] by $4$.
+Thus:
+{{begin-eqn}}
+{{eqn | l = 4 Y_J
+ | r = 3 \times 365 + 366
+ | c = considering $4$ consecutive [[Definition:Julian Year|Julian years]]
+}}
+{{eqn | ll= \leadsto
+ | l = Y_J
+ | r = \frac {1461} 4
+ | c =
+}}
+{{eqn | r = 365 \cdotp 25
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = Y_J - Y_T
+ | o = \approx
+ | r = 365 \cdotp 25 - 365 \cdotp 24219 \, 878
+ | c =
+}}
+{{eqn | o = \approx
+ | r = 0 \cdotp 0078013
+ | c =
+}}
+{{eqn | o = \approx
+ | r = \frac 1 {128 \cdotp 18}
+ | c =
+}}
+{{end-eqn}}
+Thus the [[Definition:Arithmetic Mean|mean]] [[Definition:Julian Year|Julian year]] is $\dfrac 1 {128 \cdotp 18}$ [[Definition:Day|days]] longer than the [[Definition:Tropical Year|tropical year]].
+This means that after approximately $128$ [[Definition:Julian Year|Julian years]], the [[Definition:Julian Calendar|Julian calendar]] starts one [[Definition:Day|day]] later relative to the [[Definition:Tropical Year|tropical year]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Discrepancy between Gregorian Year and Tropical Year}
+Tags: Calendars
+
+\begin{theorem}
+The [[Definition:Gregorian Year|Gregorian year]] and the [[Definition:Tropical Year|tropical year]] differ such that the [[Definition:Gregorian Calendar|Gregorian calendar]] becomes $1$ [[Definition:Day|day]] further out approximately every $3318$ [[Definition:Year|years]].
+\end{theorem}
+
+\begin{proof}
+By definition, the [[Definition:Length of Time|length]] $Y_T$ of the [[Definition:Tropical Year|tropical year]] is defined as $\approx 365 \cdotp 24219 \, 878$ [[Definition:Day|days]]
+By definition of the [[Definition:Gregorian Year|Gregorian year]]:
+:$Y_G = \begin{cases} 366 \, \text {days} & : 400 \divides y \\
+365 \, \text {days} & : 400 \nmid y \text { and } 100 \divides y\\
+366 \, \text {days} & : 100 \nmid y \text { and } 4 \divides y\\
+365 \, \text {days} & : 4 \nmid y \end{cases}$
+where:
+:$Y_G$ denotes the [[Definition:Length of Time|length]] of the [[Definition:Gregorian Year|Gregorian year]] in [[Definition:Day|days]]
+:$y$ denotes the number of the [[Definition:Year|year]]
+:$4 \divides y$ denotes that $y$ is [[Definition:Divisor of Integer|divisible]] by $4$
+:$4 \nmid y$ denotes that $y$ is not [[Definition:Divisor of Integer|divisible]] by $4$.
+Thus in every $400$ [[Definition:Year|years]], there are:
+:$300$ [[Definition:Year|years]] which are not [[Definition:Divisor of Integer|divisible]] by $4$, which have $365$ days
+:$1$ [[Definition:Year|year]] which is [[Definition:Divisor of Integer|divisible]] by $400$, which has $366$ days
+:$3$ [[Definition:Year|years]] which are [[Definition:Divisor of Integer|divisible]] by $100$ but not $400$, which have $365$ days
+:The remaining $96$ [[Definition:Year|years]] which are [[Definition:Divisor of Integer|divisible]] by $4$ but not $100$, which have $366$ days.
+Thus:
+{{begin-eqn}}
+{{eqn | l = 400 Y_G
+ | r = 303 \times 365 + 97 \times 366
+ | c = considering $400$ consecutive [[Definition:Gregorian Year|Gregorian years]]
+}}
+{{eqn | ll= \leadsto
+ | l = Y_G
+ | r = \frac {110 \, 595 + 35 \, 502} {400}
+ | c =
+}}
+{{eqn | r = \frac {146 \, 097} {400}
+ | c =
+}}
+{{eqn | r = 365 \cdotp 2425
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = Y_G - Y_T
+ | o = \approx
+ | r = 365 \cdotp 2425 - 365 \cdotp 24219 \, 878
+ | c =
+}}
+{{eqn | o = \approx
+ | r = 0 \cdotp 0003013
+ | c =
+}}
+{{eqn | o = \approx
+ | r = \frac 1 {3318 \cdotp 95}
+ | c =
+}}
+{{end-eqn}}
+Thus the [[Definition:Arithmetic Mean|mean]] [[Definition:Gregorian Year|Gregorian year]] is $\dfrac 1 {3318 \cdotp 95}$ [[Definition:Day|days]] longer than the [[Definition:Tropical Year|tropical year]].
+This means that after approximately $3318$ [[Definition:Gregorian Year|Gregorian years]], the [[Definition:Gregorian Calendar|Gregorian calendar]] starts one [[Definition:Day|day]] later relative to the [[Definition:Tropical Year|tropical year]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Completely Irreducible and Subset Admits Infimum Equals Element implies Element Belongs to Subset}
+Tags: Meet Irreducible
+
+\begin{theorem}
+Let $L = \left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
+Let $x \in S$ such that
+:$x$ is [[Definition:Completely Irreducible|completely irreducible]].
+Let $X \subseteq S$ such that
+:$X$ admits an [[Definition:Infimum of Set|infimum]] and $x = \inf X$
+Then $x \in X$
+\end{theorem}
+
+\begin{proof}
+Aiming for a [[Definition:Contradiction|contradiction]] suppose that
+:$x \notin X$
+By [[Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element]]
+:$\exists q \in S: x \prec q \land \left({\forall s \in S: x \prec s \implies q \preceq s}\right) \land x^\succeq = \left\{ {x}\right\} \cup q^\succeq$
+where $x^\succeq$ denotes the [[Definition:Upper Closure of Element|upper closure of]] $x$.
+We will prove that
+:$X \subseteq q^\succeq$
+Let $y \in X$.
+By definitions of [[Definition:Infimum of Set|infimum]] and [[Definition:Lower Bound of Set|lower bound]]:
+:$x \preceq y$
+By definition of [[Definition:Strictly Precede|strictly precede]]:
+:$x \prec y$
+By existence of $q$:
+:$q \preceq y$
+Thus by definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$y \in q^\succeq$
+{{qed|lemma}}
+By [[Infimum of Upper Closure of Element]]:
+:$\inf\left({q^\succeq}\right) = q$
+By [[Infimum of Subset]]:
+:$q \preceq x$
+Thus this contrsdicts $x \prec q$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Order Generating Includes Completely Irreducible Elements}
+Tags: Meet Irreducible
+
+\begin{theorem}
+Let $\left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
+Let $X \subseteq S$ be an [[Definition:Order Generating|order generating]] [[Definition:Subset|subset]] of $S$.
+Let $X \in S$ be a [[Definition:Completely Irreducible|completely irreducible]] [[Definition:Element|element]] of $S$.
+Then $x \in X$.
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Order Generating|order generating]]:
+:$x^\succeq \cap X$ admits an [[Definition:Infimum of Set|infimum]] and $x = \inf\left({x^\succeq \cap X}\right)$
+By [[Completely Irreducible and Subset Admits Infimum Equals Element implies Element Belongs to Subset]]:
+:$x \in x^\succeq \cap X$
+Thus by definition of [[Definition:Set Intersection|intersection]]:
+:$x \in X$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{399 is not Expressible as Sum of Fewer than 19 Fourth Powers}
+Tags: Fourth Powers, Hilbert-Waring Theorem, 399
+
+\begin{theorem}
+:$399 = 14 \times 1^4 + 3 \times 2^4 + 3^4 + 4^4$
+or:
+:$399 = 11 \times 1^4 + 4 \times 2^4 + 4 \times 3^4$
+\end{theorem}
+
+\begin{proof}
+First note that $5^4 = 625 > 399$.
+Then note that $2 \times 4^4 = 512 > 399$.
+Hence any expression of $399$ as [[Definition:Fourth Power|fourth powers]] uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.
+For the remainder, using $2^4$ uses fewer [[Definition:Fourth Power|fourth powers]] than $16$ instances of $1^4$ does.
+Now we have:
+{{begin-eqn}}
+{{eqn | l = 399
+ | r = 4^4 + 3^4 + 3 \times 2^4 + 14 \times 1^4
+}}
+{{eqn | r = 4^4 + 8 \times 2^4 + 15 \times 1^4
+}}
+{{eqn | r = 4 \times 3^4 + 4 \times 2^4 + 11 \times 1^4
+}}
+{{eqn | r = 3 \times 3^4 + 9 \times 2^4 + 12 \times 1^4
+}}
+{{eqn | r = 2 \times 3^4 + 14 \times 2^4 + 13 \times 1^4
+}}
+{{eqn | r = 1 \times 3^4 + 19 \times 2^4 + 14 \times 1^4
+}}
+{{eqn | r = 0 \times 3^4 + 24 \times 2^4 + 15 \times 1^4
+}}
+{{end-eqn}}
+and it can be seen that the first and the third uses the least number of [[Definition:Fourth Power|fourth powers]], at $19$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Maximal implies Difference equals Intersection}
+Tags: Upper Closures
+
+\begin{theorem}
+Let $\left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
+Let $x, y \in S$ such that
+:$x$ is [[Definition:Maximal|maximal]] in $S \setminus y^\succeq$
+Then $x^\succeq \setminus \left\{ {x}\right\} = x^\succeq \cap y^\succeq$
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Maximal|maximal]]:
+:$x \in S \setminus y^\succeq$
+By definition of [[Definition:Set Difference|difference]]:
+:$x \notin y^\succeq$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$y \npreceq x$
+We will prove that
+:$x^\succeq \setminus \left\{ {x}\right\} \subseteq x^\succeq \cap y^\succeq$
+Let $a \in x^\succeq \setminus \left\{ {x}\right\}$
+By definition of [[Definition:Set Difference|difference]]:
+:$a \in x^\succeq$ and $a \notin \left\{ {x}\right\}$
+By definitions of [[Definition:Upper Closure of Element|upper closure of element]] and [[Definition:Singleton|singleton]]:
+:$x \preceq a$ and $a \ne x$
+By definition of [[Definition:Strictly Precede|strictly precede]]:
+:$x \prec a$
+By definition of [[Definition:Maximal|maximal]]:
+:$a \notin S \setminus y^\succeq$
+By definition of [[Definition:Set Difference|difference]]:
+:$a \in y^\succeq$
+Thus by definition of [[Definition:Set Intersection|intersection]]:
+:$a \in x^\succeq \cap y^\succeq$
+{{qed|lemma}}
+We will prove that
+:$x^\succeq \cap y^\succeq \subseteq x^\succeq \setminus \left\{ {x}\right\}$
+Let $a \in x^\succeq \cap y^\succeq$
+By definition of [[Definition:Set Intersection|intersection]]:
+:$a \in x^\succeq$ and $a \in y^\succeq$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$y \preceq a$
+Then
+:$a \ne x$
+By definition of [[Definition:Singleton|singleton]]:
+:$a \notin \left\{ {x}\right\}$
+Thus by definition of [[Definition:Set Difference|difference]]:
+:$a \in x^\succeq \setminus \left\{ {x}\right\}$
+{{qed|lemma}}
+Thus by definition of [[Definition:Set Equality|set equality]]:
+:$x^\succeq \setminus \left\{ {x}\right\} = x^\succeq \cap y^\succeq$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Square of n such that 2n-1 is Composite is not Sum of Square and Prime}
+Tags: Numbers not Sum of Square and Prime
+
+\begin{theorem}
+Let $n^2$ be a [[Definition:Square Number|square]] such that $2 n - 1$ is [[Definition:Composite Number|composite]].
+Then $n^2$ cannot be expressed as the [[Definition:Integer Addition|sum]] of a [[Definition:Square Number|square]] and a [[Definition:Prime Number|prime]].
+\end{theorem}
+
+\begin{proof}
+The case where $n = 1$ is trivial, as there are no [[Definition:Prime Number|prime numbers]] less than $1$.
+Let $n, m \in \Z$ be [[Definition:Integer|integers]] such that $n > 1$.
+Let $n^2 = m^2 + p$ where $p$ is [[Definition:Prime Number|prime]].
+Then:
+{{begin-eqn}}
+{{eqn | l = p
+ | r = n^2 - m^2
+ | c =
+}}
+{{eqn | r = \left({n + m}\right) \left({n - m}\right)
+ | c = [[Difference of Two Squares]]
+}}
+{{eqn | ll= \leadsto
+ | l = n - m
+ | r = 1
+ | c = {{Defof|Prime Number}}
+}}
+{{eqn | ll= \leadsto
+ | l = m
+ | r = n - 1
+ | c = {{Defof|Prime Number}}
+}}
+{{eqn | ll= \leadsto
+ | l = n + m
+ | r = 2 n - 1
+ | c =
+}}
+{{eqn | r = p
+ | c =
+}}
+{{end-eqn}}
+So if $2 n - 1$ is [[Definition:Composite Number|composite]], there exists no [[Definition:Prime Number|prime]] $p$ such that $n^2 = m^2 + p$.
+Further, if such a $p$ does exist, then $m = n - 1$, and so:
+:$n^2 = \left({n - 1}\right)^2 + p$
+{{qed}}
+[[Category:Numbers not Sum of Square and Prime]]
+qphnse71dg6iklevcnpp6qqp071fatz
+\end{proof}<|endoftext|>
+\section{40 times Heptagonal Numbers plus 9 gives Squares of Numbers ending in 7}
+Tags: Heptagonal Numbers, Square Numbers
+
+\begin{theorem}
+Consider the [[Definition:Heptagonal Number|heptagonal numbers]]:
+:$\displaystyle H_n = \sum_{k \mathop = 1}^n \left({5 k - 4}\right)$
+Let $S_n$ be the [[Definition:Integer Sequence|sequence]] defined as:
+:$\forall n \in \Z_{>1}: S_n = 40 \times H_n + 9$
+Then $S_n$ consists of the [[Definition:Square (Algebra)|squares]] of all the [[Definition:Positive Integer|positive integers]] which end in a $7$:
+:$49, 289, 729, 1369, 2209, 3249, 4489, 5929, 7569, \ldots$
+that is:
+:$7^2, 17^2, 27^2, 37^2, 47^2, 57^2, 67^2, 77^2, 87^2, 97^2, \ldots$
+{{OEIS|A017354}}
+\end{theorem}
+
+\begin{proof}
+For $n \in \Z_{>1}$, we have:
+{{begin-eqn}}
+{{eqn | l = 10 \times H_n + 9
+ | r = 40 \dfrac {n \left({5 n - 3}\right)} 2 + 9
+ | c = [[Closed Form for Heptagonal Numbers]]
+}}
+{{eqn | r = 20 n \left({5 n - 3}\right) + 9
+ | c =
+}}
+{{eqn | r = 100 n^2 - 60 n + 9
+ | c =
+}}
+{{eqn | r = \left({10 n - 3}\right)^2
+ | c =
+}}
+{{eqn | r = \left({10 \left({n - 1}\right) + 7}\right)^2
+ | c =
+}}
+{{end-eqn}}
+The [[Definition:Positive Integer|positive integers]] of the form $10 \left({n - 1}\right) + 7$ for $n \in \Z_{>1}$ are precisely the [[Definition:Positive Integer|positive integers]] which end in a $7$:
+:$7, 17, 27, 37, 47, \ldots$
+{{qed}}
+[[Category:Heptagonal Numbers]]
+[[Category:Square Numbers]]
+hsge2ptz9w33aiomxuqcxf5gltagzc9
+\end{proof}<|endoftext|>
+\section{Maximal implies Completely Irreducible}
+Tags: Meet Irreducible
+
+\begin{theorem}
+Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
+Let $p \in S$ such that
+:$\exists k \in S: p$ is [[Definition:Maximal Element|maximal]] in $S \setminus k^\succeq$
+Then $p$ is [[Definition:Completely Irreducible|completely irreducible]].
+\end{theorem}
+
+\begin{proof}
+By [[Join Succeeds Operands]]:
+:$k \preceq p \vee k$ and $p \preceq p \vee k$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$p \vee k \in k^\succeq$ and $p \vee k \in p^\succeq$
+By definition of [[Definition:Set Intersection|intersection]]:
+:$p \vee k \in p^\succeq \cap k^\succeq$
+By [[Maximal implies Difference equals Intersection]]:
+:$p^\succeq \setminus \set p = p^\succeq \cap k^\succeq$
+By [[Infimum of Intersection of Upper Closures equals Join Operands]]
+:$\map \inf {p^\succeq \cap k^\succeq} = p \vee k$
+By definition:
+:$p^\succeq \setminus \set p$ admits a [[Definition:Minimum Element|minimum]].
+Thus by definition:
+:$p$ is [[Definition:Completely Irreducible|completely irreducible]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Infimum of Intersection of Upper Closures equals Join Operands}
+Tags: Upper Closures
+
+\begin{theorem}
+Let $L = \left({S, \vee, \preceq}\right)$ be a [[Definition:Join Semilattice|join semilattice]].
+Let $x, y \in S$.
+Then $\inf\left({x^\succeq \cap y^\succeq}\right) = x \vee y$
+\end{theorem}
+
+\begin{proof}
+By [[Intersection of Upper Closures is Upper Closure of Join Operands]]:
+:$x^\succeq \cap y^\succeq = \left({x \vee y}\right)^\succeq$
+Thus by [[Infimum of Upper Closure of Element]]:
+:$\inf\left({x^\succeq \cap y^\succeq}\right) = x \vee y$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sturm-Liouville Problem/Unit Weight Function/Lemma}
+Tags: Calculus of Variations
+
+\begin{theorem}
+Let $\map \alpha x: \R \to \R$ such that $\map \alpha x \in C^2 \closedint a b$.
+Suppose:
+:$\displaystyle \forall \map h x \in C^2 \closedint a b: \map h a = \map h b = \map {h'} a = \map {h'} b = 0: \int_a^b \map \alpha x \, \map {h''} x \rd x = 0$
+Then:
+:$\forall x \in \closedint a b: \map \alpha x = c_0 + c_1 x$
+where $ c_0, c_1 \in \R $ are constants.
+\end{theorem}
+
+\begin{proof}
+Let $ c_0, c_1 $ be defined by the conditions:
+:$\displaystyle \int_a^b \paren {\map \alpha x - c_0 - c_1 x} \rd x = 0$
+:$\displaystyle \int_a^b \rd x \int_a^x \paren {\map \alpha \xi - c_0 - c_1 \xi} \rd \xi = 0$
+Suppose:
+:$\displaystyle \map h x = \int_a^x \xi \int_a^\xi \paren {\map \alpha t - c_0 - c_1 t} \rd t$
+This form satisfies conditions on $h$ in the theorem.
+Then:
+{{begin-eqn}}
+{{eqn | l = \int_a^b \paren {\map \alpha x - c_0 - c_1 x} \, \map {h''} x \rd x
+ | r = \int_a^b \map \alpha x \, \map {h''} x \rd x - c_0 \paren {\map {h'} b - \map {h'} a} - c_1 \int_a^b x \, \map {h''} x \rd x
+}}
+{{eqn | r = -c_1 \paren {b \, \map {h'} b - a \, \map {h'} a} - c_1 \paren {\map h b - \map h a}
+}}
+{{eqn | r = 0
+}}
+{{end-eqn}}
+On the other hand:
+{{begin-eqn}}
+{{eqn | l = \int_a^b \paren {\map \alpha x - c_0 - c_1 x} \, \map {h''} x \rd x
+ | r = \int_a^b \paren {\map \alpha x - c_0 - c_1 x}^2 \rd x
+}}
+{{eqn | r = 0
+}}
+{{end-eqn}}
+Therefore:
+:$\map \alpha x - c_0 - c_1 x = 0$
+or:
+:$\map \alpha x = c_0 + c_1 x$
+{{qed}}
+[[Category:Calculus of Variations]]
+47tzettuyi1mejel2x1if581rk915sv
+\end{proof}<|endoftext|>
+\section{Intersection of Upper Closures is Upper Closure of Join Operands}
+Tags: Upper Closures
+
+\begin{theorem}
+Let $L = \left({S, \vee, \preceq}\right)$ be a [[Definition:Join Semilattice|join semilattice]].
+Let $x, y \in S$.
+Then $\left({x \vee y}\right)^\succeq = x^\succeq \cap y^\succeq$
+\end{theorem}
+
+\begin{proof}
+We will prove that
+$\left({x \vee y}\right)^\succeq \subseteq x^\succeq \cap y^\succeq$
+Let $a \in \left({x \vee y}\right)^\succeq$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$x \vee y \preceq a$
+By [[Join Succeeds Operands]]:
+:$x \preceq x \vee y$ and $y \preceq x \vee y$
+By definition of [[Definition:Transitivity|transitivity]]:
+:$x \preceq a$ and $y \preceq a$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$a \in x^\succeq$ and $a \in y^\succeq$
+Thus by definition of [[Definition:Set Intersection|intersection]]:
+:$a \in x^\succeq \cap y^\succeq$
+{{qed|lemma}}
+We will prove that
+:$x^\succeq \cap y^\succeq \subseteq \left({x \vee y}\right)^\succeq$
+Let $a \in x^\succeq \cap y^\succeq$
+Thus bt definition of [[Definition:Set Intersection|intersection]]:
+:$a \in x^\succeq$ and $a \in y^\succeq$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$x \preceq a$ and $y \preceq a$
+By definitions of [[Definition:Supremum of Set|supremum]] and [[Definition:Upper Bound of Set|Upper bound]]:
+:$x \vee y \preceq a$
+Thus by definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$a \in \left({x \vee y}\right)^\succeq$
+{{qed|lemma}}
+Hence by definition of [[Definition:Set Equality|set equality]]:
+:$\left({x \vee y}\right)^\succeq = x^\succeq \cap y^\succeq$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Strictly Precede and Step Condition and not Precede implies Joins are equal}
+Tags: Join and Meet Semilattices
+
+\begin{theorem}
+Let $\struct {S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
+Let $p, q, u \in S$ be such that:
+:$p \prec q$ and $\paren {\forall s \in S: p \prec s \implies q \preceq s}$ and $u \npreceq p$
+Then:
+:$p \vee u = q \vee u$
+\end{theorem}
+
+\begin{proof}
+From the definition of [[Definition:Join (Order Theory)|join]], it is required to prove that:
+:$\forall s \in S: p \preceq s \land u \preceq s \implies q \vee u \preceq s$
+Let $s \in S$ be such that:
+:$p \preceq s$ and $u \preceq s$
+We have:
+:$p \ne s$
+By definition of [[Definition:Strictly Precede|strictly precede]]:
+:$p \prec s$
+By assumption:
+:$q \preceq s$
+Thus by definition of the [[Definition:Join (Order Theory)|join]] $q \vee u$:
+:$q \vee u \preceq s$
+and hence, as desired:
+:$q \vee u = p \vee u$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice}
+Tags: Continuous Lattices, Meet Irreducible
+
+\begin{theorem}
+Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Algebraic Ordered Set|algebraic]] [[Definition:Lattice|lattice]].
+Let $x, y \in S$ such that
+:$y \npreceq x$
+Then
+:$\exists p \in S: p$ is [[Definition:Completely Irreducible|completely irreducible]] $\land ~x \preceq p \land y \npreceq p$
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Algebraic Ordered Set|algebraic]]:
+:$\forall z \in S: z^\ll$ is [[Definition:Directed Subset|directed]]
+and
+:$L$ satisfies [[Definition:Axiom of Approximation|axiom of approximation]].
+By [[Axiom of Approximation in Up-Complete Semilattice]]:
+:$\exists k \in S: k \ll y \land k \npreceq x$
+By [[Algebraic iff Continuous and For Every Way Below Exists Compact Between]]:
+:$\exists z \in K\left({L}\right): k \preceq z \preceq y$
+By definition of [[Definition:Transitivity|transitivity]]:
+:$z \npreceq x$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$x \notin z^\succeq$
+By definition of [[Definition:Set Difference|difference]]:
+:$x \in S \setminus z^\succeq$
+By definition of [[Definition:Relative Complement|relative complement]]:
+:$x \in \complement_S\left({z^\succeq}\right)$
+By definition of [[Definition:Compact Subset of Lattice|compact subset]]:
+:$z$ is a [[Definition:Compact Element|compact element]].
+By [[Upper Closure of Element is Way Below Open Filter iff Element is Compact]]:
+:$z^\succeq$ is a [[Definition:Way Below Open|way below open]] [[Definition:Filter|filter]] on $L$.
+By [[Upper Way Below Open Subset Complement is Non Empty implies There Exists Maximal Element of Complement]]:
+:$\exists p \in S: x \preceq p \land p = \max\complement_S\left({z^\succeq}\right)$
+Thus by [[Maximal implies Completely Irreducible]]:
+:$p$ is [[Definition:Completely Irreducible|completely irreducible]].
+Thus $x \preceq p$
+Aiming for a [[Definition:Contradiction|contradiction]] suppose that
+:$y \preceq p$
+By definition of [[Definition:Transitivity|transitivity]]:
+:$z \preceq p$
+By definition of [[Definition:Maximum Element|maximum element]]:
+:$p \in \complement_S\left({z^\succeq}\right)$
+By definition of [[Definition:Relative Complement|relative complement]]:
+:$p \notin z^\succeq$
+Thus this contradicts $z \preceq p$ by definition of [[Definition:Upper Closure of Element|upper closure of element]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Set of All Completely Irreducible Elements is Smallest Order Generating}
+Tags: Meet Irreducible
+
+\begin{theorem}
+Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Algebraic Ordered Set|algebraic]] [[Definition:Lattice|lattice]].
+Then $\mathit{Irr}\left({L}\right)$ is [[Definition:Order Generating|order generating]] and
+:$\forall X \subseteq S: X$ is [[Definition:Order Generating|order generating]] $\implies \mathit{Irr}\left({L}\right) \subseteq X$
+where $\mathit{Irr}\left({L}\right)$ denotes the [[Definition:Set|set]] of all [[Definition:Completely Irreducible|completely irreducible elements]] of $L$.
+\end{theorem}
+
+\begin{proof}
+By [[Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice]]:
+:$\forall x, y \in S: y \npreceq x \implies \exists p \in \mathit{Irr}\left({L}\right): p \preceq x \land p \npreceq y$
+Thus by [[Order Generating iff Not Preceding implies There Exists Element Preceding and Not Preceding]]:
+:$\mathit{Irr}\left({L}\right)$ is [[Definition:Order Generating|order generating]].
+Let $X \subseteq S$ such that
+:$X$ is [[Definition:Order Generating|order generating]].
+Let $x \in \mathit{Irr}\left({L}\right)$.
+By definition of $\mathit{Irr}$:
+:$x$ is [[Definition:Completely Irreducible|completely irreducible]].
+Thus by [[Order Generating Includes Completely Irreducible Elements]]:
+:$x \in X$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Completely Irreducible Element equals Infimum of Subset implies Element Belongs to Subset}
+Tags: Meet Irreducible
+
+\begin{theorem}
+Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a [[Definition:Complete Lattice|complete lattice]].
+Let $X \subseteq S$, $p \in S$ such that
+:$p$ is [[Definition:Completely Irreducible|completely irreducible]] and $p = \inf X$
+Then $p \in X$
+\end{theorem}
+
+\begin{proof}
+Aiming for a [[Definition:Contradiction|contradiction]] suppose that
+:$p \notin X$
+By [[Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element]]:
+:$\exists q \in S: p \prec q \land \left({\forall s \in S: p \prec s \implies q \preceq s}\right) \land p^\succeq = \left\{ {p}\right\} \cup q^\succeq$
+where $p^\succeq$ denotes the [[Definition:Upper Closure of Element|upper closure of]] $p$.
+We will prove that
+:$q$ is [[Definition:Lower Bound of Set|lower bound]] for $X$
+Let $x \in X$.
+By definitions of [[Definition:Infimum of Set|infimum]] and [[Definition:Lower Bound of Set|lower bound]]
+:$p \preceq x$
+By assumption:
+:$x \ne p$
+By definition of [[Definition:Strictly Precede|strictly precede]]:
+:$p \prec x$
+Thus by step condition:
+:$q \preceq x$
+{{qed|lemma}}
+By definition of [[Definition:Strictly Precede|strictly precede]]:
+:$p \preceq q$
+We will prove that
+:$\forall b \in S: b$ is [[Definition:Lower Bound of Set|lower bound]] for $X \implies b \preceq q$
+Let $b \in S$ such that
+:$b$ is [[Definition:Lower Bound of Set|lower bound]] for $X$.
+By definition of [[Definition:Infimum of Set|infimum]]:
+:$b \preceq p$
+Thus by definition of [[Definition:Transitivity|transitivity]]:
+:$b \preceq q$
+{{qed|lemma}}
+By definition of [[Definition:Infimum of Set|infimum]]:
+:$q = \inf X = p$
+Thus this contradicts $p \ne q$ by definition of [[Definition:Strictly Precede|strictly precede]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Preimage of Lower Set under Increasing Mapping is Lower}
+Tags: Lower Sets
+
+\begin{theorem}
+Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be [[Definition:Preordered Set|preordered sets]].
+Let $f: S \to T$ be an [[Definition:Increasing Mapping|increasing mapping]].
+Let $X \subseteq T$ be a [[Definition:Lower Set|lower]] [[Definition:Subset|subset]] of $T$.
+Then $f^{-1} \sqbrk X$ is [[Definition:Lower Set|lower]]
+where $f^{-1} \sqbrk X$ denotes the [[Definition:Preimage of Subset under Mapping|preimage]] of $X$ under $f$.
+\end{theorem}
+
+\begin{proof}
+Let $x \in f^{-1} \sqbrk X$, $y \in S$ such that
+:$y \preceq x$
+By definition of [[Definition:Increasing Mapping|increasing mapping]]:
+:$\map f y \precsim \map f x$
+By definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
+:$\map f x \in X$
+By definition of [[Definition:Lower Set|lower set]]:
+:$\map f y \in X$
+Thus by definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
+:$y \in f^{-1} \sqbrk X$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Preimage of Upper Set under Increasing Mapping is Upper}
+Tags: Upper Sets
+
+\begin{theorem}
+Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be [[Definition:Preordered Set|preordered sets]].
+Let $f: S \to T$ be an [[Definition:Increasing Mapping|increasing mapping]].
+Let $X \subseteq T$ be a [[Definition:Upper Set|upper]] [[Definition:Subset|subset]] of $T$.
+Then $f^{-1} \sqbrk X$ is [[Definition:Upper Set|upper]]
+where $f^{-1} \sqbrk X$ denotes the [[Definition:Preimage of Subset under Mapping|preimage]] of $X$ under $f$.
+\end{theorem}
+
+\begin{proof}
+Let $x \in f^{-1} \sqbrk X$, $y \in S$ such that
+:$x \preceq y$
+By definition of [[Definition:Increasing Mapping|increasing mapping]]:
+:$\map f x \precsim \map f y$
+By definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
+:$\map f x \in X$
+By definition of [[Definition:Upper Set|upper set]]:
+:$\map f y \in X$
+Thus by definition of [[Definition:Preimage of Subset under Mapping|preimage of subset]]:
+:$y \in f^{-1} \sqbrk X$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Continuous implies Increasing in Scott Topological Lattices}
+Tags: Topological Order Theory
+
+\begin{theorem}
+Let $T_1 = \struct {S_1, \preceq_1, \tau_1}$ and $T_2 = \struct {S_2, \preceq_2, \tau_2}$ be [[Definition:Up-Complete|up-complete]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Scott Topology|Scott topologies]].
+Let $f: S_1 \to S_2$ be a [[Definition:Continuous (Topology)|continuous]] [[Definition:Mapping|mapping]].
+Then $f$ is an [[Definition:Increasing Mapping|increasing mapping]].
+\end{theorem}
+
+\begin{proof}
+Let $x, y \in S_1$ such that
+:$x \preceq_1 y$
+{{AimForCont}} that
+:$\map f x \npreceq_2 \map f y$
+By definition of [[Definition:Lower Closure of Element|lower closure of element]]:
+:$\map f x \notin \paren {\map f y}^\preceq$
+By definition of [[Definition:Relative Complement|relative complement]]:
+:$\map f x \in \relcomp {S_2} {\paren {\map f y}^\preceq}$
+By definition of [[Definition:Reflexivity|reflexivity]]:
+:$\map f y \preceq_2 \map f y$
+By definition of [[Definition:Lower Closure of Element|lower closure of element]]:
+:$\map f y \in \paren {\map f y}^{\preceq_2}$
+By [[Closure of Singleton is Lower Closure of Element in Scott Topological Lattice]]:
+:$\set {\map f y}^- = \paren {\map f y}^{\preceq_2}$
+By definition of [[Definition:Closure (Topology)|closure]]:
+:$\paren {\map f y}^{\preceq_2}$ is a [[Definition:Closed Set (Topology)|closed set]].
+By definition of [[Definition:Closed Set (Topology)|closed set]]:
+:$\relcomp {S_2} {\paren {\map f y}^{\preceq_2} }$ is a [[Definition:Open Set (Topology)|open set]].
+By definition of [[Definition:Continuous (Topology)|continuous]]:
+:$f^{-1} \sqbrk {\relcomp {S_2} {\paren {\map f y}^{\preceq_2} } }$ is an [[Definition:Open Set (Topology)|open set]].
+By definition of [[Definition:Scott Topology]]:
+:$f^{-1} \sqbrk {\relcomp {S_2} {\paren {\map f y}^{\preceq_2} } }$ is an [[Definition:Upper Set|upper set]].
+By definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
+:$x \in f^{-1} \sqbrk {\relcomp {S_2} {\paren {\map f y}^{\preceq_2} } }$
+By definition of [[Definition:Upper Set|upper set]]:
+:$y \in f^{-1} \sqbrk {\relcomp {S_2} {\paren {\map f y}^{\preceq_2} } }$
+By definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
+:$\map f y \in \relcomp {S_2} {\paren {\map f y}^{\preceq_2} }$
+Thus this contradicts $\map f y \in \paren {\map f y}^{\preceq_2}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Directed Suprema Preserving Mapping at Element is Supremum}
+Tags: Continuous Lattices
+
+\begin{theorem}
+Let $\left({S, \vee, \wedge, \preceq}\right)$ and $\left({T, \vee_2, \wedge_2, \precsim}\right)$ be [[Definition:Bounded Below Set|bounded below]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Lattice|lattices]].
+Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that
+:$f$ [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
+Let $x \in S$.
+Then $f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Continuous Ordered Set|continuous]]:
+:$x^\ll$ is [[Definition:Directed Subset|directed]]
+and
+:$\left({S, \vee, \wedge, \preceq}\right)$ is [[Definition:Up-Complete|up-complete]]
+and
+:$\left({S, \vee, \wedge, \preceq}\right)$ satisfies [[Definition:Axiom of Approximation|axiom of approximation]].
+By definition of [[Definition:Mapping Preserves Supremum/Directed|mapping preserves directed suprema]]:
+:$f$ [[Definition:Mapping Preserves Supremum/Subset|preserves the supremum]] of $x^\ll$.
+By definition of [[Definition:Up-Complete|up-complete]]:
+:$x^\ll$ admits a [[Definition:Supremum of Set|supremum]].
+Thus
+{{begin-eqn}}
+{{eqn | l = f\left({x}\right)
+ | r = f\left({\sup \left({x^\ll}\right)}\right)
+ | c = [[Definition:Axiom of Approximation|Axiom of Approximation]]
+}}
+{{eqn | r = \sup \left({f \left[{x^\ll}\right]}\right)
+ | c = {{Defof|Mapping Preserves Supremum/Subset|Mapping Preserves Supremum: Subset}}
+}}
+{{eqn | r = \sup \left({f \left[{\left\{ {w \in S: w \ll x}\right\} }\right]}\right)
+ | c = {{Defof|Way Below Closure}}
+}}
+{{eqn | r = \sup \left\{ {f \left({w}\right): w \in S \land w \ll x}\right\}
+ | c = {{Defof|Image of Subset under Mapping}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Preceding implies Way Below Closure is Subset of Way Below Closure}
+Tags: Way Below Relation
+
+\begin{theorem}
+Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
+Let $x, y \in S$ such that
+:$x \preceq y$
+Then $x^\ll \subseteq y^\ll$
+where $x^\ll$ denotes the [[Definition:Way Below Closure|way below closure]] of $x$.
+\end{theorem}
+
+\begin{proof}
+Let $z \in x^\ll$.
+By definition of [[Definition:Way Below Closure|way below closure]]:
+:$z \ll x$
+By [[Preceding and Way Below implies Way Below]] and definition of [[Definition:Reflexivity|reflexivity]]:
+:$z \ll y$
+Thus by definition of [[Definition:Way Below Closure|way below closure]]:
+:$z \in y^\ll$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Mapping at Element is Supremum implies Mapping is Increasing}
+Tags: Increasing Mappings
+
+\begin{theorem}
+Let $\left({S, \vee, \wedge, \preceq}\right)$ be a [[Definition:Lattice|lattice]].
+Let $\left({T, \vee_2, \wedge_2, \precsim}\right)$ be a [[Definition:Complete Lattice|complete lattice]].
+Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that
+:$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
+Then $f$ is an [[Definition:Increasing Mapping|increasing mapping]].
+\end{theorem}
+
+\begin{proof}
+Let $x, y \in S$ such that
+:$x \preceq y$
+By [[Preceding implies Way Below Closure is Subset of Way Below Closure]]:
+:$x^\ll \subseteq y^\ll$
+By definitions of [[Definition:Image of Subset under Mapping|image of set]] and [[Definition:Way Below Closure|way below closure]]:
+:$f\left[{x^\ll}\right] = \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
+and
+:$f\left[{y^\ll}\right] = \left\{ {f\left({w}\right): w \in S \land w \ll y}\right\}$
+where $f\left[{x^\ll}\right]$ denotes the [[Definition:Image of Subset under Mapping|image]] of $x^\ll$ under $f$.
+By [[Image of Subset under Relation is Subset of Image/Corollary 2]]:
+:$f\left[{x^\ll}\right] \subseteq f\left[{y^\ll}\right]$
+Thus
+{{begin-eqn}}
+{{eqn | l = f\left({x}\right)
+| r = \sup \left({f\left[{x^\ll}\right]}\right)
+| c = assumption
+}}
+{{eqn | o = \precsim
+| r = \sup \left({f\left[{y^\ll}\right]}\right)
+| c = [[Supremum of Subset]]
+}}
+{{eqn | r = f\left({y}\right)
+| c = assumption
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Semiperfect Number is not Deficient}
+Tags: Semiperfect Numbers, Deficient Numbers
+
+\begin{theorem}
+Let $n \in \Z_{>0}$ be a [[Definition:Semiperfect Number|semiperfect number]].
+Then $n$ is not [[Definition:Deficient Number|deficient]].
+\end{theorem}
+
+\begin{proof}
+Let $n$ be [[Definition:Semiperfect Number|semiperfect]].
+Then by definition, the [[Definition:Integer Addition|sum]] of the [[Definition:Aliquot Part|aliquot parts]] of $n$ is not less than $n$.
+The result follows by definition of [[Definition:Deficient Number|deficient]].
+{{qed}}
+[[Category:Semiperfect Numbers]]
+[[Category:Deficient Numbers]]
+hc85iiuau9aa2yjhlxau95ge4lnn53b
+\end{proof}<|endoftext|>
+\section{Perfect Number is Primitive Semiperfect}
+Tags: Perfect Numbers, Primitive Semiperfect Numbers
+
+\begin{theorem}
+Let $n \in \Z_{>0}$ be a [[Definition:Perfect Number|perfect number]].
+Then $n$ is also a [[Definition:Primitive Semiperfect Number|primitive semiperfect number]].
+\end{theorem}
+
+\begin{proof}
+Let $n$ be [[Definition:Perfect Number|perfect]].
+From [[Divisor of Perfect Number is Deficient]], all [[Definition:Divisor of Integer|divisors]] of $n$ are [[Definition:Deficient Number|deficient]].
+But from [[Semiperfect Number is not Deficient]], it follows that the [[Definition:Divisor of Integer|divisors]] of $n$ cannot be [[Definition:Semiperfect Number|semiperfect]].
+Hence the result, by definition of [[Definition:Primitive Semiperfect Number|primitive semiperfect number]].
+{{qed}}
+[[Category:Perfect Numbers]]
+[[Category:Primitive Semiperfect Numbers]]
+m2rwpvgnzeynsz9mte4tobkvu5pwm2i
+\end{proof}<|endoftext|>
+\section{Mapping at Element is Supremum implies Way Below iff There Exists Element that Way Below and Way Below}
+Tags: Way Below Relation, Continuous Lattices
+
+\begin{theorem}
+Let $\left({S, \vee, \wedge, \preceq}\right)$ be a [[Definition:Complete Lattice|complete lattice]].
+Let $\left({T, \vee_2, \wedge_2, \precsim}\right)$ be a [[Definition:Continuous Ordered Set|continuous]] [[Definition:Complete Lattice|complete lattice]].
+Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that
+:$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
+Let $x \in S, y \in T$.
+Then
+:$y \ll f\left({x}\right) \iff \exists w \in S: w \ll x \land y \ll f\left({w}\right)$
+\end{theorem}
+
+\begin{proof}
+By [[Mapping at Element is Supremum implies Mapping is Increasing]]:
+:$f$ is an [[Definition:Increasing Mapping|increasing mapping]].
+=== Sufficient Condition ===
+Let $y \ll f\left({x}\right)$
+By [[Way Below Closure is Directed in Bounded Below Join Semilattice]]:
+:$x^\ll$ is [[Definition:Directed Subset|directed]]
+where $x^\ll$ denotes the [[Definition:Way Below Closure|way below closure]] of $x$.
+By [[Image of Directed Subset under Increasing Mapping is Directed]]:
+:$f\left[{x^\ll}\right]$ is [[Definition:Directed Subset|directed]]
+where $f\left[{x^\ll}\right]$ denotes the [[Definition:Image of Subset under Mapping|image]] of $x^\ll$ under $f$.
+By assumption:
+:$f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
+By definitions of [[Definition:Image of Subset under Mapping|image of set]] and [[Definition:Way Below Closure|way below closure]]:
+:$f\left[{x^\ll}\right] = \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
+By [[Way Below iff Second Operand Preceding Supremum of Directed Set There Exists Element of Directed Set First Operand Way Below Element]]:
+:$\exists z \in f\left[{x^\ll}\right]: y \ll z$
+By definition of [[Definition:Image of Subset under Mapping|image of set]]:
+:$\exists w \in x^\ll: f\left({w}\right) = z$
+Thus by definition of [[Definition:Way Below Closure|way below closure]]:
+:$w \ll x$
+Thus $y \ll f\left({w}\right)$
+{{qed|lemma}}
+=== Necessary Condition ===
+Let
+:$\exists w \in S: w \ll x \land y \ll f\left({w}\right)$
+By [[Way Below implies Preceding]]:
+:$w \preceq x$
+By definition of [[Definition:Increasing Mapping|increasing mapping]]:
+:$f\left({w}\right) \precsim f\left({x}\right)$
+Thus by [[Preceding and Way Below implies Way Below]]:
+:$y \ll f\left({x}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Subset and Image Admit Suprema and Mapping is Increasing implies Supremum of Image Precedes Mapping at Supremum}
+Tags: Order Theory
+
+\begin{theorem}
+Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be [[Definition:Ordered Set|ordered sets]].
+Let $f: S \to T$ be a [[Definition:Increasing Mapping|increasing mapping]].
+Let $D \subseteq S$ such that
+:$D$ admits a [[Definition:Supremum of Set|supremum]] in $S$ and $f\left[{D}\right]$ admits a [[Definition:Supremum of Set|supremum]] in $T$.
+Then $\sup \left({f\left[{D}\right]}\right) \precsim f\left({\sup D}\right)$
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Supremum of Set|supremum]]:
+:$\sup D$ is [[Definition:Upper Bound of Set|upper bound]] for $D$.
+By [[Increasing Mapping Preserves Upper Bounds]]:
+:$f\left({\sup D}\right)$ is [[Definition:Upper Bound of Set|upper bound]] for $f\left[{D}\right]$.
+Thus by definition of [[Definition:Supremum of Set|supremum]]:
+:$\sup \left({f\left[{D}\right]}\right) \precsim f\left({\sup D}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Characteristic Subgroup of Normal Subgroup is Normal}
+Tags: Characteristic Subgroups, Normal Subgroups
+
+\begin{theorem}
+Let $G$ be a [[Definition:Group|group]].
+Let $N\leq G$ be [[Definition:Normal Subgroup|normal]].
+Let $H\leq N$ be [[Definition:Characteristic Subgroup|characteristic]].
+Then $H$ is [[Definition:Normal Subgroup|normal]] in $G$.
+\end{theorem}
+
+\begin{proof}
+Let $g \in G$.
+Because $N$ is [[Definition:Normal Subgroup|normal]], [[Definition:Conjugate of Group Subset|conjugation]] by $g$ is an [[Definition:Group Automorphism|automorphism]] of $N$.
+Because $H$ is [[Definition:Characteristic Subgroup|characteristic]] in $N$, $g H g^{-1} = H$.
+Thus $H$ is [[Definition:Normal Subgroup|normal]] in $G$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Center is Characteristic Subgroup}
+Tags: Characteristic Subgroups, Centers of Groups
+
+\begin{theorem}
+Let $G$ be a [[Definition:Group|group]].
+Then its [[Definition:Center of Group|center]] $\map Z G$ is [[Definition:Characteristic Subgroup|characteristic]] in $G$.
+\end{theorem}
+
+\begin{proof}
+Let $\phi$ be an [[Definition:Automorphism (Abstract Algebra)|automorphism]] of $G$.
+Let $x \in \map Z G, y \in G$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \map \phi x y
+ | r = \map \phi x \map \phi {\map {\phi^{-1} } y}
+ | c = [[Definition:Automorphism (Abstract Algebra)|automorphisms]] are [[Definition:Bijection|bijections]]
+}}
+{{eqn | r = \map \phi {x \map {\phi^{-1} } y}
+ | c = {{Defof|Group Homomorphism}}
+}}
+{{eqn | r = \map \phi {\map {\phi^{-1} } y x}
+ | c = {{Defof|Center of Group}}
+}}
+{{eqn | r = \map \phi {\map {\phi^{-1} } y} \map \phi x
+ | c = {{Defof|Group Homomorphism}}
+}}
+{{eqn | r = y \map \phi x
+}}
+{{end-eqn}}
+Hence $\map \phi x \in \map Z G$.
+So we have $\phi \sqbrk {\map Z G} \subseteq \map Z G$.
+Since $\phi^{-1}$ is also an [[Definition:Automorphism (Abstract Algebra)|automorphism]]:
+:$\phi^{-1} \sqbrk {\map Z G} \subseteq \map Z G$
+Since $\phi$ is a [[Definition:Bijection|bijection]]:
+:$\map Z G = \phi \sqbrk {\phi^{-1} \sqbrk {\map Z G}} \subseteq \phi \sqbrk {\map Z G}$
+Therefore we conclude that $\phi \sqbrk {\map Z G} = \map Z G$.
+Hence $\map Z G$ is [[Definition:Characteristic Subgroup|characteristic]] in $G$.
+{{qed}}
+[[Category:Characteristic Subgroups]]
+[[Category:Centers of Groups]]
+7j9sccn0viksp9srk03nwwlmiikg2gc
+\end{proof}<|endoftext|>
+\section{Semidirect Product of Groups is Group}
+Tags: Semidirect Products
+
+\begin{theorem}
+Let $H$ and $N$ be [[Definition:Group|groups]].
+Let $\operatorname{Aut}(N)$ denote the [[Definition:Automorphism Group of Group|automorphism group]] of $N$.
+Let $\phi : H\to \operatorname{Aut}(N)$ be a [[Definition:Group Homomorphism|group homomorphism]], that is, let $H$ [[Definition:Group Action|act]] on $N$.
+Let $N\rtimes_\phi H$ be the [[Definition:Semidirect Product|semidirect product]] of $N$ and $H$ with respect to $\phi$, that is:
+:$N\rtimes_\phi H = (N\times H, \circ)$ where
+:$(n_1, h_1) \circ (n_2, h_2) = (n_1\cdot \phi(h_1)(n_2), h_1\cdot h_2)$
+Then $N\rtimes_\phi H$ is a [[Definition:Group|group]].
+\end{theorem}
+
+\begin{proof}
+=== Associativity ===
+Let $(n_1,h_1),(n_2,h_2),(n_3,h_3)\in N\times H$. Then
+{{begin-eqn}}
+{{eqn | l = ((n_1, h_1) \circ (n_2, h_2)) \circ (n_3, h_3)
+ | r = ((n_1\cdot \phi_{h_1}(n_2), h_1\cdot h_2)) \circ (n_3,h_3)
+}}
+{{eqn | l =
+ | r = (n_1\cdot \phi_{h_1}(n_2) \cdot \phi_{h_1h_2}(n_3), h_1\cdot h_2\cdot h_3)
+}}
+{{eqn | l =
+ | r = (n_1\cdot \phi_{h_1}(n_2) \cdot \phi_{h_1}(\phi_{h_2}(n_3)), h_1\cdot h_2\cdot h_3)
+ | c = {{Defof|Group Action}}
+}}
+{{eqn | l =
+ | r = (n_1\cdot \phi_{h_1}(n_2\phi_{h_2}(n_3)), h_1\cdot h_2\cdot h_3)
+ | c = $H$ acts by [[Definition:Group Automorphism|automorphisms]]
+}}
+{{eqn | l =
+ | r = (n_1, h_1) \circ ((n_2, h_2) \circ (n_3, h_3))
+ | c =
+}}
+{{end-eqn}}
+{{qed|lemma}}
+=== Identity Element ===
+Let $e = (e_N,e_H)$ and $(n,h)\in N\times H$.
+Then
+{{begin-eqn}}
+{{eqn | l = e \circ (n,h)
+ | r = (e_N\cdot \phi_{e_H}(n), e_H\cdot h)
+}}
+{{eqn | l =
+ | r = (n, h)
+ | c = {{Defof|Group Action}}
+}}
+{{end-eqn}}
+and
+{{begin-eqn}}
+{{eqn | l = (n,h) \circ e
+ | r = (n\cdot \phi_{h}(e_N), h\cdot e_H)
+}}
+{{eqn | l =
+ | r = (n, h)
+ | c = $\phi_h(e_N)=e_N$ because $H$ acts by [[Definition:Group Automorphism|automorphisms]]
+}}
+{{end-eqn}}
+Thus $e$ is an [[Definition:Identity Element|identity element]] for $\circ$.
+{{qed|lemma}}
+=== Existence of inverse element ===
+Let $(n,h)\in N\times H$.
+We are looking for $(m,g)$ with $(n,h)\circ(m,g) = (e_N,e_H) = (m,g)\circ(n,h)$, that is:
+:$\begin{eqnarray}
+ n \phi_h(m) &= e_N \\
+ hg &= e_H \\
+ m \phi_g(n) &= e_N \\
+ gh &= e_H
+\end{eqnarray}$
+Thus we take $g=h^{-1}$.
+From the first equation we have $m=\phi_h^{-1}(n^{-1})$; from the third $m=\phi_{h^{-1}}(n)^{-1}$.
+These are equal because $H$ acts by [[Definition:Group Automorphism|automorphisms]].
+The inverse of $(n,h)$ is thus $(\phi_{h^{-1}}(n^{-1}),h^{-1})$.
+{{qed|lemma}}
+{{qed}}
+[[Category:Semidirect Products]]
+1u8lm7p5m9mp1f6iwtyhm10i6d3ytrl
+\end{proof}<|endoftext|>
+\section{Inverse of Element in Semidirect Product}
+Tags: Group Theory, Semidirect Products
+
+\begin{theorem}
+Let $N$ and $H$ be [[Definition:Group|groups]].
+Let $H$ [[Definition:Group Action by Automorphisms|act by automorphisms]] on $N$ via $\phi$.
+Let $N \rtimes_\phi H$ be the corresponding [[Definition:Semidirect Product|(outer) semidirect product]].
+Let $\tuple {n, h} \in N \rtimes_\phi H$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \tuple {n, h}^{-1}
+ | r = \tuple {\map {\phi_{h^{-1} } } {n^{-1} }, h^{-1} }
+ | c =
+}}
+{{eqn | r = \tuple {\paren {\map {\phi_{h^{-1} } } n}^{-1}, h^{-1} }
+ | c =
+}}
+{{eqn | r = \tuple {\map { {\phi_h}^{-1} } {n^{-1} }, h^{-1} }
+ | c =
+}}
+{{eqn | r = \tuple {\paren {\map { {\phi_h}^{-1} } n}^{-1}, h^{-1} }
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+Follows from [[Semidirect Product of Groups is Group]].
+The alternatives follow from the fact that $H$ [[Definition:Group Action by Automorphisms|acts by automorphisms]].
+{{qed}}
+{{finish|Expand the proof by demonstrating how it works.}}
+[[Category:Group Theory]]
+[[Category:Semidirect Products]]
+9mlhlpyesu2zbhormoglwgj83gye5uv
+\end{proof}<|endoftext|>
+\section{Semidirect Product with Trivial Action is Direct Product}
+Tags: Semidirect Products
+
+\begin{theorem}
+Let $H$ and $N$ be [[Definition:Group|groups]].
+Let $\Aut N$ denote the [[Definition:Automorphism Group of Group|automorphism group]] of $N$.
+Let $\phi: H \to \Aut N$ be defined as:
+:$\forall h \in H: \map \phi h = I_N$ for all $h \in H$
+where $I_N$ denotes the [[Definition:Identity Mapping|identity mapping]] on $N$.
+Let $N \rtimes_\phi H$ be the corresponding [[Definition:Semidirect Product|semidirect product]].
+Then $N \rtimes_\phi H$ is the [[Definition:External Direct Product|direct product]] of $N$ and $H$.
+\end{theorem}
+
+\begin{proof}
+Pick arbitrary $\tuple {n_1, h_1}, \tuple {n_2, h_2} \in N \rtimes_\phi H$.
+{{begin-eqn}}
+{{eqn | l = \tuple {n_1, h_1} \tuple {n_2, h_2}
+ | r = \tuple {n_1 \cdot \map \phi {h_1} \paren {n_2}, h_1 h_2}
+ | c = {{Defof|Semidirect Product}}
+}}
+{{eqn | r = \tuple {n_1 \cdot \map {I_N} {n_2}, h_1 h_2}
+ | c = Definition of $\phi$
+}}
+{{eqn | r = \tuple {n_1 n_2, h_1 h_2}
+ | c =
+}}
+{{end-eqn}}
+which meets the definition of [[Definition:External Direct Product|direct product]].
+{{qed}}
+[[Category:Semidirect Products]]
+rukq7wpt4k5wgd159gbyr0un0kx53rm
+\end{proof}<|endoftext|>
+\section{Semidirect Product is Abelian iff Components are Abelian and Action is Trivial}
+Tags: Semidirect Products
+
+\begin{theorem}
+Let $N$ and $H$ be [[Definition:Group|groups]].
+Let $H$ [[Definition:Group Action by Automorphisms|act by automorphisms]] on $N$ via $\phi$.
+Let $N\rtimes_\phi H$ be the corresponding [[Definition:Semidirect Product|semidirect product]].
+Then the following are equivalent:
+:$(1): \quad$ $N\rtimes_\phi H$ is [[Definition:Abelian Group|abelian]]
+:$(2): \quad$ $N$ and $H$ are [[Definition:Abelian Group|abelian]] and $H$ [[Definition:Trivial Group Action|acts trivially]]
+\end{theorem}
+
+\begin{proof}
+=== 1 implies 2 ===
+Let $n\in N$, $h\in H$.
+From $(n,e)(e,h) = (e,h)(n,e)$ we have $n\phi_e(e) = e\phi_h(n)$.
+Thus $H$ [[Definition:Trivial Group Action|acts trivially]].
+By [[Semidirect Product with Trivial Action is Direct Product]], $N\rtimes_\phi H = N\times H$.
+By [[External Direct Product of Abelian Groups is Abelian Group]], $N$ and $H$ are [[Definition:Abelian Group|abelian]].
+{{qed|lemma}}
+=== 2 implies 1 ===
+By [[Semidirect Product with Trivial Action is Direct Product]], $N\rtimes_\phi H = N\times H$.
+By [[External Direct Product of Abelian Groups is Abelian Group]], $N\rtimes_\phi H$ is [[Definition:Abelian Group|abelian]].
+{{qed|lemma}}
+{{qed}}
+[[Category:Semidirect Products]]
+0cxwo51lkiba91x6nn93qaiug5et4rg
+\end{proof}<|endoftext|>
+\section{Integers whose Squares end in 444}
+Tags: Square Numbers, Recreational Mathematics
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] whose [[Definition:Square (Algebra)|square]] ends in $444$ begins:
+:$38, 462, 538, 962, 1038, 1462, 1538, 1962, 2038, 2462, 2538, 2962, 3038, 3462, \ldots$
+{{OEIS|A039685}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 38^2
+ | r = 1444
+}}
+{{eqn | l = 462^2
+ | r = 213 \, 444
+}}
+{{eqn | l = 538^2
+ | r = 289 \, 444
+}}
+{{eqn | l = 962^2
+ | r = 925 \, 444
+}}
+{{eqn | l = 1038^2
+ | r = 1 \, 077 \, 444
+}}
+{{eqn | l = 1462^2
+ | r = 2 \, 137 \, 444
+}}
+{{eqn | l = 1538^2
+ | r = 2 \, 365 \, 444
+}}
+{{eqn | l = 1962^2
+ | r = 3 \, 849 \, 444
+}}
+{{eqn | l = 2038^2
+ | r = 4 \, 153 \, 444
+}}
+{{eqn | l = 2462^2
+ | r = 6 \, 061 \, 444
+}}
+{{eqn | l = 2538^2
+ | r = 6 \, 441 \, 444
+}}
+{{eqn | l = 2962^2
+ | r = 8 \, 773 \, 444
+}}
+{{eqn | l = 3038^2
+ | r = 9 \, 229 \, 444
+}}
+{{eqn | l = 3462^2
+ | r = 11 \, 985 \, 444
+}}
+{{end-eqn}}
+All such $n$ are of the form $500 m + 38$ or $500 m - 38$:
+{{begin-eqn}}
+{{eqn | l = \paren {500 m + 38}^2
+ | r = 250 \, 000 m^2 + 38 \, 000 m + 1444
+ | c =
+}}
+{{eqn | r = \paren {250 m^2 + 38 m + 1} + 444
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \paren {500 m - 38}^2
+ | r = 250 \, 000 m^2 - 38 \, 000 m + 1444
+ | c =
+}}
+{{eqn | r = \paren {250 m^2 - 38 m + 1} + 444
+ | c =
+}}
+{{end-eqn}}
+and it is seen that all such numbers end in $444$.
+Now we show that all such numbers are so expressed.
+In [[Squares Ending in Repeated Digits]], we have shown the only numbers with squares ending in $44$ ends in:
+:$12, 38, 62, 88$
+hence any number with square ending in $444$ must also end in those numbers.
+Suppose $\sqbrk {axy}^2 \equiv 444 \pmod {1000}$, where $a < 10$ and $\sqbrk {xy}$ is in the above list.
+For $\sqbrk {xy} = 12$:
+{{begin-eqn}}
+{{eqn | l = 444
+ | r = \paren {100 a + 12}^2
+ | rr = \pmod {200}
+}}
+{{eqn | r = 10000 a^2 + 2400 a + 144
+ | rr = \pmod {200}
+}}
+{{eqn | r = 144
+ | rr = \pmod {200}
+}}
+{{end-eqn}}
+This is a [[Definition:Contradiction|contradiction]].
+Similarly for $\sqbrk {xy} = 88$:
+{{begin-eqn}}
+{{eqn | l = 444
+ | r = \paren {100 a + 88}^2
+ | rr = \pmod {200}
+}}
+{{eqn | r = 10000 a^2 + 17600 a + 7744
+ | rr = \pmod {200}
+}}
+{{eqn | r = 144
+ | rr = \pmod {200}
+}}
+{{end-eqn}}
+Again, a [[Definition:Contradiction|contradiction]].
+For $\sqbrk {xy} = 38$:
+{{begin-eqn}}
+{{eqn | l = 444
+ | r = \paren {100 a + 38}^2
+ | rr = \pmod {1000}
+}}
+{{eqn | r = 10000 a^2 + 7600 a + 1444
+ | rr = \pmod {1000}
+}}
+{{eqn | r = 600 a + 444
+ | rr = \pmod {1000}
+}}
+{{end-eqn}}
+The solutions to $600 a \equiv 0 \pmod {1000}$ are $a = 0$ or $5$.
+Hence:
+:$\paren {500 n + 38}^2 \equiv 444 \pmod {1000}$
+Similarly, for $\sqbrk {xy} = 62$:
+{{begin-eqn}}
+{{eqn | l = 444
+ | r = \paren {100 a + 62}^2
+ | rr = \pmod {1000}
+}}
+{{eqn | r = 10000 a^2 + 12400 a + 3844
+ | rr = \pmod {1000}
+}}
+{{eqn | r = 400 a + 844
+ | rr = \pmod {1000}
+}}
+{{end-eqn}}
+The solutions to $400 a + 400 \equiv 0 \pmod {1000}$ are $a = 4$ or $9$.
+Hence:
+:$\paren {500 n + 462}^2 \equiv \paren {500 n - 38}^2 \equiv 444 \pmod {1000}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Subset and Image Admit Infima and Mapping is Increasing implies Infimum of Image Succeeds Mapping at Infimum}
+Tags: Order Theory
+
+\begin{theorem}
+Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
+Let $f: S \to T$ be a [[Definition:Increasing Mapping|increasing mapping]].
+Let $D \subseteq S$ such that
+:$D$ admits a [[Definition:Infimum of Set|infimum]] in $S$ and $f \sqbrk D$ admits a [[Definition:Infimum of Set|infimum]] in $T$.
+Then $\map f {\inf D} \precsim \map \inf {f \sqbrk D}$
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Infimum of Set|infimum]]:
+:$\inf D$ is [[Definition:Lower Bound of Set|lower bound]] for $D$.
+By [[Increasing Mapping Preserves Lower Bounds]]:
+:$\map f {\inf D}$ is a [[Definition:Lower Bound of Set|lower bound]] for $f \sqbrk D$.
+Thus by definition of [[Definition:Infimum of Set|infimum]]:
+:$\map f {\inf D} \precsim \map \inf {f \sqbrk D}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Finite Galois Extension}
+Tags: Galois Theory
+
+\begin{theorem}
+Let $L/K$ be a [[Definition:Finite Field Extension|finite field extension]].
+{{TFAE|def = Finite Galois Extension}}
+\end{theorem}
+
+\begin{proof}
+=== 1 implies 2 ===
+Note that by [[Finite Field Extension has Finite Galois Group]], $G = \Aut {L / K}$ is [[Definition:Finite Group|finite]].
+Let $\alpha\in L$.
+Then its [[Definition:Orbit under Group of Permutations|orbit]] under $G$ is [[Definition:Finite Set|finite]].
+By:
+:[[Minimal Polynomial of Element with Finite Orbit under Group of Automorphisms over Fixed Field in terms of Orbit]]
+its [[Definition:Minimal Polynomial|minimal polynomial]] over $K$ [[Definition:Splitting Polynomial|splits completely]] in $L$.
+Thus $L/K$ is [[Definition:Normal Field Extension|normal]].
+By:
+:[[Finite Orbit under Group of Automorphisms of Field implies Separable over Fixed Field]]
+$L / K$ is [[Definition:Separable Field Extension|separable]].
+{{qed|lemma}}
+=== 2 implies 3 ===
+Since $L / K$ is [[Definition:Galois Extension|Galois]], it is [[Definition:Separable Extension|separable]].
+Thus, by the [[Primitive Element Theorem]], there exists an $\alpha \in L$ such that $L = \map K \alpha$.
+Let $m_\alpha \in K \sqbrk x$ be the [[Definition:Minimal Polynomial|minimal polynomial]] of $\alpha$ over $K$.
+Then:
+:$\index L K = \map \deg {m_\alpha}$
+Suppose $\sigma \in \Gal {L / K}$.
+Then:
+:$\map {m_\alpha} {\map \sigma \alpha} = \map \sigma {\map {m_\alpha} \alpha} = \map \sigma 0 = 0$
+since $m_\alpha$ has coefficients in $K$.
+Therefore $\map \sigma \alpha$ must be a root of $m_\alpha$.
+Every element of $\Gal {L / K}$ is determined by its value at $\alpha$ by our assumption that $L = \map K \alpha$.
+Therefore:
+:$\order {\Gal {L / K} } \le \map \deg {m_\alpha} = \index L K$
+Next, suppose $\beta$ is a root of $m_\alpha$.
+By the [[Definition:Normal Extension|normality]] of $L / K$, we must have $\beta \in L$.
+Then, by [[Abstract Model of Algebraic Extensions]]:
+:$\map K \alpha \cong K \sqbrk x / \gen {m_\alpha} \cong \map K \beta$
+Composing [[Definition:Isomorphism|isomorphisms]] we have an [[Definition:Automorphism of Field Extension|automorphism]] of $L$ for each $\beta$.
+Thus:
+:$\order {\Gal {L / K} } \ge \map \deg {m_\alpha} = \index L K$
+from which our result follows.
+{{qed|lemma}}
+{{ProofWanted}}
+[[Category:Galois Theory]]
+nw8ll9jqppiha21abbselt53sgqx2n9
+\end{proof}<|endoftext|>
+\section{Finite Field Extension has Finite Galois Group}
+Tags: Galois Theory
+
+\begin{theorem}
+Let $E / F$ be a [[Definition:Finite Field Extension|finite field extension]].
+Then its [[Definition:Automorphism Group of Field Extension|automorphism group]] is [[Definition:Finite Group|finite]].
+\end{theorem}
+
+\begin{proof}
+Because $E / F$ is [[Definition:Finite Field Extension|finite]], it is [[Definition:Finitely Generated Field Extension|finitely generated]].
+Let $\alpha_1, \ldots, \alpha_n \in E$ with $E = \map F {\alpha_1, \ldots, \alpha_n}$.
+{{explain|notation $\map F {\alpha_1, \ldots, \alpha_n}$}}
+By [[Finite Field Extension is Algebraic]], $\alpha_1, \ldots, \alpha_n$ are [[Definition:Algebraic Element of Field Extension|algebraic]] over $F$.
+Let $f_1, \ldots, f_n$ be their [[Definition:Minimal Polynomial|minimal polynomials]].
+Let $f = f_1\dots f_n$.
+By [[Galois Group Acts Faithfully on Generating Set]], $\Gal {E / F}$ [[Definition:Faithful Group Action|acts faithfully]] on the [[Definition:Root of Polynomial|roots]] of $f$.
+Thus $\Gal {E / F}$ is [[Definition:Finite Group|finite]].
+{{qed}}
+[[Category:Galois Theory]]
+2eckmkkpx5u20bf3ukk9lt60uc2mndv
+\end{proof}<|endoftext|>
+\section{Primitive Element Theorem}
+Tags: Field Extensions
+
+\begin{theorem}
+Let $E/F$ be a [[Definition:Separable Field Extension|separable field extension]] of [[Definition:Finite Field Extension|finite degree]].
+Then $E/F$ is [[Definition:Simple Field Extension|simple]]: there exists $\alpha\in E$ such that $E=F(\alpha)$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Field Extensions]]
+qmuyhjxk05p3ffe06hzyk9oxhbac1r1
+\end{proof}<|endoftext|>
+\section{Largest Number not Expressible as Sum of Less than 32 Positive Fifth Powers}
+Tags: Fifth Powers, Hilbert-Waring Theorem, 466
+
+\begin{theorem}
+The largest [[Definition:Positive Integer|positive integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of less than $32$ [[Definition:Positive Integer|positive]] [[Definition:Fifth Power|fifth powers]] is $466$:
+:$466 = 18 \times 1^5 + 14 \times 2^5$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|It needs to be shown that there are no larger numbers with this property.}}
+\end{proof}<|endoftext|>
+\section{Smallest Square which is Sum of 3 Fourth Powers}
+Tags: Square Numbers, Fourth Powers
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|positive integer]] whose [[Definition:Square (Algebra)|square]] is the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Fourth Power|fourth powers]] is $481$:
+:$481^2 = 12^4 + 15^4 + 20^4$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 12^4 + 15^4 + 20^4
+ | r = 20 \, 736 + 50 \, 625 + 160 \, 000
+ | c =
+}}
+{{eqn | r = 231 \, 361
+ | c =
+}}
+{{eqn | r = 481^2
+ | c =
+}}
+{{end-eqn}}
+The smallest solution must be [[Definition:Fourth Power|fourth powers]] of [[Definition:Coprime Integers|coprime integers]], otherwise dividing by their [[Definition:Greatest Common Divisor|GCD]] would yield a smaller solution.
+By [[Fermat's Right Triangle Theorem]], $x^4 + y^4 = z^2$ has no solutions.
+Thus none of the [[Definition:Fourth Power|fourth powers]] is $0$.
+Consider the equation $x^4 + y^4 + z^4 = n^2$.
+By [[Square Modulo 4]], since [[Definition:Fourth Power|fourth powers]] are [[Definition:Square (Algebra)|squares]], at most one of $x, y, z$ is [[Definition:Odd Integer|odd]].
+If none of them are [[Definition:Odd Integer|odd]], the numbers are not [[Definition:Coprime Integers|coprime]].
+Therefore there is exactly one [[Definition:Odd Integer|odd]] number.
+{{qed|lemma}}
+By [[Fermat's Little Theorem]], for $3 \nmid a$:
+:$a^2 \equiv 1 \pmod 3$
+Therefore by [[Congruence of Powers]]:
+:$a^4 \equiv 1^2 \equiv 1 \pmod 3$
+For $3 \divides a$:
+:$a^2 \equiv a^4 \equiv 0 \pmod 3$
+Therefore we cannot have two of $x, y, z$ not [[Definition:Divisor of Integer|divisible]] by $3$.
+By our [[Definition:Coprime Integers|coprimality condition]] we cannot have $3 \divides x, y, z$.
+Hence either $3 \nmid x, y, z$ or exactly one of $x, y, z$ is not [[Definition:Divisor of Integer|divisible]] by $3$.
+{{qed|lemma}}
+Similarly, by [[Fermat's Little Theorem]], for $5 \nmid a$:
+:$a^4 \equiv 1 \pmod 5$
+For $5 \divides a$:
+:$a^4 \equiv 0 \pmod 5$
+By [[Square Modulo 5]]:
+:$a^2 \equiv 0, 1, 4 \pmod 5$
+Therefore there is exactly one number not [[Definition:Divisor of Integer|divisible]] by $5$.
+{{qed|lemma}}
+Since $\sqrt {481} < 22$, we only need to check [[Definition:Fourth Power|fourth powers]] of numbers less than $22$, satisfying the criteria above.
+Since there are exactly two multiples of $2$ and $5$ among $x, y, z$, at least one of them must be a multiple of $10$.
+Suppose $5 \nmid x$, $5 \divides y$ and $10 \divides z$.
+=== Case $1$: $z = 10$ ===
+The case $y = 20$ will be equivalent to Case $2$.
+Suppose $y = 15$.
+By the criteria above, $x$ is [[Definition:Divisor of Integer|divisible]] by $2$ and $3$.
+Also $x < 22$, so we only need to check $x = 6, 12, 18$.
+{{qed|lemma}}
+Suppose $y = 10$.
+By the criteria above, $x$ is not [[Definition:Divisor of Integer|divisible]] by $2$ or $3$.
+Also $x < 22$, so we only need to check $x = 1, 7, 11, 13, 17, 19$.
+{{qed|lemma}}
+Suppose $y = 5$.
+By the criteria above, $x$ is [[Definition:Divisor of Integer|divisible]] by $2$ but not $3$.
+Also $x < 22$, so we only need to check $x = 2, 4, 8, 14, 16$.
+{{qed|lemma}}
+=== Case $2$: $z = 20$ ===
+Suppose $y = 20$. Then $x^4 + y^4 + z^4 > 20^4 + 20^4 = 320 \, 000 > 481^2$.
+Suppose $y = 15$.
+By the criteria above, $x$ is [[Definition:Divisor of Integer|divisible]] by $2$ and $3$.
+Also $x < \sqrt [4] {481^2 - 20^4 - 15^4} = 12$, so we only need to check $x = 6$.
+{{qed|lemma}}
+Suppose $y = 10$.
+By the criteria above, $x$ is not [[Definition:Divisor of Integer|divisible]] by $2$ or $3$.
+Also $x < \sqrt [4] {481^2 - 20^4} < 17$, so we only need to check $x = 1, 7, 11, 13$.
+{{qed|lemma}}
+Suppose $y = 5$.
+By the criteria above, $x$ is [[Definition:Divisor of Integer|divisible]] by $2$ but not $3$.
+Also $x < \sqrt [4] {481^2 - 20^4} < 17$, so we only need to check $x = 2, 4, 8, 14, 16$.
+{{qed|lemma}}
+And we see that none of the above cases yield any smaller solutions.
+Hence the result.
+{{finish|Check the remaining $24$ candidates}}
+\end{proof}<|endoftext|>
+\section{Solutions to p^2 Divides 10^p - 10}
+Tags: Prime Numbers, 10
+
+\begin{theorem}
+The known [[Definition:Prime Number|prime numbers]] $p$ which satisfy the equation:
+:$p^2 \divides \paren {10^p - 10}$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]], are:
+:$3, 487, 56 \, 598 \, 313$
+{{OEIS|A045616}}
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Some sort of computer program can be implemented, I suppose}}
+\end{proof}<|endoftext|>
+\section{Kaprekar's Process on 3 Digit Number ends in 495}
+Tags: Kaprekar's Process, 495
+
+\begin{theorem}
+Let $n$ be a $3$-[[Definition:Digit|digit]] [[Definition:Positive Integer|integer]] whose [[Definition:Digit|digits]] are not all the same.
+[[Definition:Kaprekar's Process|Kaprekar's process]], when applied to $n$, results in $495$ after no more than $6$ iterations.
+\end{theorem}
+
+\begin{proof}
+Let $n = \sqbrk {abc}_{10}$ denote a $3$-[[Definition:Digit|digit]] [[Definition:Positive Integer|integer]] whose [[Definition:Digit|digits]] are $a, b, c$.
+If $a = b = c$ then [[Definition:Kaprekar's Process|Kaprekar's process]] trivially results in $0$ after the first iteration.
+{{WLOG}}, let $a \ge b \ge c$ but such that $a \ne c$.
+By the [[Basis Representation Theorem]]:
+:$n = 10^2 a + 10 b + c$
+Let $n' = 10^2 a' + 10 b' + c'$ be the result of [[Definition:Kaprekar's Process|Kaprekar's process]] after the $1$st iteration on $n$.
+We have:
+{{begin-eqn}}
+{{eqn | l = n'
+ | r = \paren {100 a + 10 b + c} - \paren {100 c + 10 b + a}
+ | c =
+}}
+{{eqn | r = 100 \paren {a - c} + \paren {c - a}
+ | c =
+}}
+{{eqn | r = 100 \paren {a - c} - 10 + \paren {10 + c - a}
+ | c = as $c < a$
+}}
+{{eqn | r = 100 \paren {a - c - 1} + \paren {100 - 10} + \paren {10 + c - a}
+ | c = as $c < a$
+}}
+{{eqn | r = 10^2 \paren {a - c - 1} + 10 \times 9 + \paren {10 + c - a}
+ | c = as $c < a$
+}}
+{{end-eqn}}
+Thus we have that:
+{{begin-eqn}}
+{{eqn | l = a'
+ | r = a - c - 1
+}}
+{{eqn | l = b'
+ | r = 9
+}}
+{{eqn | l = c'
+ | r = 10 + c - a
+}}
+{{end-eqn}}
+Hence we have that:
+{{begin-eqn}}
+{{eqn | l = b'
+ | o = \ge
+ | r = a'
+}}
+{{eqn | l = b'
+ | o = \ge
+ | r = c'
+}}
+{{eqn | l = a' + c'
+ | r = 9
+}}
+{{end-eqn}}
+There are in fact only the following possibilities for $n'$:
+{{begin-eqn}}
+{{eqn | l = n'
+ | r = 099
+ | c = note that [[Definition:Kaprekar's Process|Kaprekar's process]] retains leading [[Definition:Zero Digit|zeroes]]
+}}
+{{eqn | l = n'
+ | r = 198
+}}
+{{eqn | l = n'
+ | r = 297
+}}
+{{eqn | l = n'
+ | r = 396
+}}
+{{eqn | l = n'
+ | r = 495
+}}
+{{eqn | l = n'
+ | r = 594
+}}
+{{eqn | l = n'
+ | r = 693
+}}
+{{eqn | l = n'
+ | r = 792
+}}
+{{eqn | l = n'
+ | r = 891
+}}
+{{end-eqn}}
+{{WLOG}} we inspect [[Definition:Kaprekar's Process|Kaprekar's process]] on $198$:
+{{begin-eqn}}
+{{eqn | l = 099
+ | o = \to
+ | r = 990 - 099
+ | c =
+}}
+{{eqn | l = 891
+ | o = \to
+ | r = 981 - 189
+ | c =
+}}
+{{eqn | o = \to
+ | r = 792
+ | c =
+}}
+{{eqn | o = \to
+ | r = 972 - 279
+ | c =
+}}
+{{eqn | o = \to
+ | r = 693
+ | c =
+}}
+{{eqn | o = \to
+ | r = 963 - 369
+ | c =
+}}
+{{eqn | o = \to
+ | r = 594
+ | c =
+}}
+{{eqn | o = \to
+ | r = 954 - 459
+ | c =
+}}
+{{eqn | o = \to
+ | r = 495
+ | c =
+}}
+{{end-eqn}}
+and the result is seen to follow.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Nilpotent Elements of Commutative Ring form Ideal}
+Tags: Nilpotent Ring Elements, Commutative Rings, Ideal Theory
+
+\begin{theorem}
+Let $\struct {R, +, \circ}$ be a [[Definition:Commutative Ring|commutative ring]] whose [[Definition:Ring Zero|zero]] is $0_R$ and whose [[Definition:Unity of Ring|unity]] is $1_R$.
+The [[Definition:Subset|subset]] of [[Definition:Nilpotent Ring Element|nilpotent elements]] of $R$ form an [[Definition:Ideal of Ring|ideal]] of $R$.
+\end{theorem}
+
+\begin{proof}
+Let $N$ be the [[Definition:Subset|subset]] of [[Definition:Nilpotent Ring Element|nilpotent elements]].
+Because $0_R$ is [[Definition:Nilpotent Ring Element|nilpotent]], $0_R \in N$ and so $N$ is [[Definition:Non-Empty Set|non-empty]].
+Let $x \in N$ and $a \in R$.
+We have:
+{{begin-eqn}}
+{{eqn | lo= \exists n \in \Z_{>0}:
+ | l = x^n
+ | r = 0_R
+ | c = {{Defof|Nilpotent Ring Element}}
+}}
+{{eqn | ll= \leadsto
+ | l = a^n \circ x^n
+ | r = 0_R
+ | c = {{Defof|Ring Zero}}
+}}
+{{eqn | ll= \leadsto
+ | l = \paren {a \circ x}^n
+ | r = 0_R
+ | c = [[Power of Product of Commutative Elements in Semigroup]]
+}}
+{{eqn | ll= \leadsto
+ | l = a x
+ | o = \in
+ | r = N
+ | c = Definition of $N$
+}}
+{{end-eqn}}
+Let $x, y \in N$.
+Let $x^n = 0$ and $y^m = 0$.
+By the [[Binomial Theorem]], $\paren {x - y}^p = 0$ for $p \ge n + m - 1$.
+Thus $x - y \in N$.
+Thus from [[Test for Ideal]], $N$ is an [[Definition:Ideal of Ring|ideal]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Nilradical of Ring}
+Tags: Ring Theory
+
+\begin{theorem}
+{{TFAE|def = Nilradical of Ring}}
+Let $A$ be a [[Definition:Commutative Ring|commutative ring]].
+\end{theorem}
+
+\begin{proof}
+By [[Nilpotent Element is Contained in Prime Ideals]], $\Nil A$ is contained in the [[Definition:Set Intersection|intersection]] of all [[Definition:Prime Ideal of Ring|prime ideals]].
+It remains to prove the other inclusion.
+Let $f \in A$ be not [[Definition:Nilpotent Ring Element|nilpotent]].
+Let $S$ be the [[Definition:Set|set]] of [[Definition:Ideal of Ring|ideals]] of $A$ that are [[Definition:Disjoint Sets|disjoint]] from $\set {f^n: n \in \N}$.
+By [[Zorn's Lemma]], $S$ has a [[Definition:Maximal Element|maximal element]] $P$.
+In particular, $f \notin P$.
+We want to show that $P$ is [[Definition:Prime Ideal of Ring|prime]].
+Let $a, b \in A$ with $a, b \notin P$.
+Then the [[Definition:Sum of Ideals of Ring|sums of ideals]] $\ideal a + P$ and $\ideal b + P$ contain $P$ strictly.
+By the maximality of $P$, there exist $n, m \in \N$ with $f^n \in \ideal a + P$ and $f^m \in \ideal b + P$.
+Then:
+:$f^{m + n} \in \ideal {a b} + P$
+Thus:
+:$\ideal {a b} + P \notin S$
+In particular:
+:$a b \notin P$
+Thus $P$ is [[Definition:Prime Ideal of Ring|prime]], and so:
+:$\displaystyle f \notin \bigcap_{\mathfrak p \mathop \in \Spec A} \mathfrak p$
+{{qed}}
+{{AoC|Zorn's Lemma}}
+\end{proof}<|endoftext|>
+\section{Algebraic Closure of Field is Unique}
+Tags: Field Extensions
+
+\begin{theorem}
+Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
+Let $K$ and $L$ be [[Definition:Algebraic Closure|algebraic closures]] of $F$.
+Then $K$ and $L$ are [[Definition:F-Isomorphism|$F$-isomorphic]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+{{AoC}}
+[[Category:Field Extensions]]
+scjc721k2mx9ewxnymtjblmujpjd2u2
+\end{proof}<|endoftext|>
+\section{Multiplicative Group of Galois Field is Cyclic}
+Tags: Galois Fields
+
+\begin{theorem}
+Let $\GF$ be a [[Definition:Galois Field|Galois field]] of [[Definition:Order of Structure|order]] $q$.
+Then its [[Definition:Multiplicative Group|multiplicative group]] is [[Definition:Cyclic Group|cyclic]] of [[Definition:Order of Structure|order]] $q-1$:
+:$\GF^\times \cong C_{q - 1}$
+\end{theorem}
+
+\begin{proof}
+Follows immediately from [[Finite Multiplicative Subgroup of Field is Cyclic]].
+{{qed}}
+[[Category:Galois Fields]]
+jjwa16ge80xdni2x02rvbql2smj1753
+\end{proof}<|endoftext|>
+\section{Automorphism Group of C Over R}
+Tags: Field Extensions
+
+\begin{theorem}
+The [[Definition:Field Extension|field extension]] $\C / \R$ of [[Definition:Field of Complex Numbers|complex numbers]] $\C$ over [[Definition:Field of Real Numbers|real numbers]] $\R$ has [[Definition:Automorphism Group of Field Extension|automorphism group]] $\operatorname{Aut}$:
+:$\operatorname{Aut} \paren {\C / \R} = \set {\operatorname{id}, \sigma}$
+where:
+:$\operatorname{id}$ denotes [[Definition:Identity Mapping|the identity mapping]]
+:$\sigma$ denotes [[Definition:Complex Conjugation|complex conjugation]]
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Field Extensions]]
+9jybayvthhxugdi6d6hhfbu882e3rdr
+\end{proof}<|endoftext|>
+\section{Image under Inclusion Mapping}
+Tags: Inclusion Mappings
+
+\begin{theorem}
+Let $X$ be a [[Definition:Set|set]].
+Let $S \subseteq X$, $Z \subseteq S$.
+Then $i_S\left[{Z}\right] = Z$
+where
+:$i_S$ denotes the [[Definition:Inclusion Mapping|inclusion mapping]] of $S$,
+:$i_S\left[{Z}\right]$ denotes the [[Definition:Image of Subset under Mapping|image]] of $Z$ under $i_S$.
+\end{theorem}
+
+\begin{proof}
+Thus
+{{begin-eqn}}
+{{eqn | l = i_S\left[{Z}\right]
+| r = \left\{ {i_S\left({z}\right): z \in Z}\right\}
+| c = definition of [[Definition:Image of Subset under Mapping|image of set]]
+}}
+{{eqn | r = \left\{ {z: z \in Z}\right\}
+| c = definition of [[Definition:Inclusion Mapping|inclusion mapping]]
+}}
+{{eqn | r = Z
+| c = definition of [[Definition:Set Equality|set equality]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Limit Inferior of Inclusion Moore-Smith Sequence is Supremum of Directed Subset}
+Tags: Inclusion Mappings
+
+\begin{theorem}
+Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Lattice|lattice]].
+Let $D \subseteq S$ be a [[Definition:Directed Subset|directed subset]] of $S$.
+Let $\left({D, \preceq'}\right)$ be a [[Definition:Directed Set|directed]] [[Definition:Ordered Subset|ordered subset]] of $L$.
+Let $i_D: D \to S$, the [[Definition:Inclusion Mapping|inclusion mapping]], be a [[Definition:Moore-Smith Sequence|Moore-Smith sequence]] in $S$.
+Then $\liminf i_D = \sup D$
+\end{theorem}
+
+\begin{proof}
+We will prove that
+:(lemma): $\forall j \in D: \inf_L\left({\preceq'\left({j}\right)}\right) = j$
+Let $j \in D$.
+By definitions of [[Definition:Image of Element under Relation|image of element]] and [[Definition:Upper Closure of Element|upper closure of element]]:
+:$\preceq'\left({j}\right) = j^{\succeq'}$
+By [[Upper Closure in Ordered Subset is Intersection of Subset and Upper Closure]]:
+:$j^{\succeq'} = D \cap j^\succeq$
+By [[Intersection is Subset]]:
+:$j^{\succeq'} \subseteq j^\succeq$
+By [[Infimum of Subset]] and [[Infimum of Upper Closure of Element]]:
+:$j \preceq \inf_L\left({\preceq'\left({j}\right)}\right)$
+By definition of [[Definition:Reflexivity|reflexivity]]:
+:$j \preceq' j$
+By definition of [[Definition:Image of Element under Relation|image of element]]:
+:$j \in \preceq'\left({j}\right)$
+By definitions of [[Definition:Infimum of Set|infimum]] and [[Definition:Lower Bound of Set|lower bound]]:
+:$\inf_L\left({\preceq'\left({j}\right)}\right) \preceq j$
+Thus by definition of [[Definition:Antisymmetric Relation|antisymmetry]]:
+:$\inf_L\left({\preceq'\left({j}\right)}\right) = j$
+{{qed|lemma}}
+Thus
+{{begin-eqn}}
+{{eqn | l = \liminf i_D
+| r = {\sup}_L \left\{ { {\inf}_L\left({i_D\left[{\preceq'\left({j}\right)}\right]}\right): j \in D}\right\}
+| c = definition of [[Definition:Limit Inferior of Moore-Smith Sequence|limit inferior of Moore-Smith sequence]]
+}}
+{{eqn | r = {\sup}_L \left\{ { {\inf}_L\left({\preceq'\left({j}\right)}\right): j \in D}\right\}
+| c = [[Image under Inclusion Mapping]]
+}}
+{{eqn | r = {\sup}_L \left\{ { j: j \in D}\right\}
+| c = (lemma)
+}}
+{{eqn | r = {\sup}_L D
+| c = definition of [[Definition:Set Equality|set equality]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Correspondence Between Group Actions and Permutation Representations}
+Tags: Group Actions
+
+\begin{theorem}
+Let $G$ be a [[Definition:Group|group]].
+Let $X$ be a [[Definition:Set|set]].
+There is a one-to-one correspondence between [[Definition:Group Action|group actions]] of $G$ on $X$ and [[Definition:Permutation Representation|permutation representations]] of $G$ in $X$, as follows:
+Let $\phi : G \times X \to X$ be a [[Definition:Group Action|group action]].
+Let $\rho : G \to \struct {\map \Gamma X, \circ}$ be a [[Definition:Permutation Representation|permutation representation]].
+The following are [[Definition:Logically Equivalent|equivalent]]:
+:$(1): \quad \rho$ is the [[Definition:Permutation Representation/Group Action|permutation representation associated to]] $\phi$
+:$(2): \quad \phi$ is the [[Definition:Group Action/Permutation Representation|group action associated to]] $\rho$
+\end{theorem}
+
+\begin{proof}
+For $g\in G$, define the mapping $\phi_g : X \to X$ as:
+:$\map {\phi_g} x = \map \phi {g, x}$
+Then $\rho$ is the [[Definition:Permutation Representation/Group Action|permutation representation associated to]] $\phi$ {{iff}}:
+:$\forall g \in G : \map \rho g = \phi_g$
+By [[Equality of Mappings]], this is [[Definition:Logically Equivalent|equivalent]] to:
+:$\forall g \in G : \forall x \in X : \map {\map \rho g} x = \map {\phi_g} x$
+$\phi$ is the [[Definition:Group Action/Permutation Representation|group action associated to]] $\rho$ {{iff}}:
+:$\forall g \in G : \forall x \in X : \map \phi {g, x} = \map {\map \rho g} x$.
+Because $\map {\phi_g} x = \map \phi {g, x}$, they are equivalent.
+{{qed}}
+[[Category:Group Actions]]
+hdpiy278iyjot15nkzz1r1ebt9q1c89
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Field of Quotients}
+Tags: Fields of Quotients
+
+\begin{theorem}
+Let $D$ be an [[Definition:Integral Domain|integral domain]].
+Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
+{{TFAE|def = Field of Quotients}}
+\end{theorem}
+
+\begin{proof}
+=== 1 implies 2 ===
+Let $K$ be a [[Definition:Field (Abstract Algebra)|field]] such that:
+:$\iota \sqbrk D \subseteq K \subseteq F$
+We show that $F \subseteq K$.
+Let $f \in F$.
+By assumption, there exist $x, y \in D$ with $y \ne 0$ such that $f = \dfrac {\map \iota x} {\map \iota y}$.
+Because $K$ is a [[Definition:Field (Abstract Algebra)|field]] containing $\iota \sqbrk D$, $K$ also contains $f = \dfrac {\map \iota x} {\map \iota y}$.
+Thus $F \subseteq K$.
+{{qed|lemma}}
+=== 1 implies 3 ===
+Let $E$ be a [[Definition:Field (Abstract Algebra)|field]] and $\phi: D \to E$ a [[Definition:Ring Monomorphism|ring monomorphism]].
+Let:
+:$\bar \phi: F \to E$ is such that $\phi = \bar \phi \circ \iota$
+and:
+:$f \in F$ such that $f = \dfrac {\map \iota x} {\map \iota y}$ with $x, y \in D$.
+Then:
+:$\map {\bar \phi} f = \dfrac {\map {\bar \phi} {\map \iota x} } {\map {\bar \phi} {\map \iota y} } = \dfrac {\map \phi x} {\map \phi y}$
+Thus there is only one option for $\bar \phi$.
+It remains to verify that the [[Definition:Mapping|mapping]] which sends $f = \dfrac {\map \iota x} {\map \iota y}$ to $\dfrac {\map \phi x} {\map \phi y}$ is:
+:well-defined
+:a [[Definition:Field Homomorphism|field homomorphism]]
+{{finish}}
+=== 2 implies 1 ===
+Let $K$ be the [[Definition:Subset|subset]] of elements of $F$ that are of the form $\dfrac {\map \iota x} {\map \iota y}$.
+We show that $K$ is a [[Definition:Field (Abstract Algebra)|field]] containing $\iota \sqbrk D$, which by assumption implies $K = F$.
+Because:
+:$\map \iota x = \dfrac {\map \iota x} {\map \iota 1} \in K$ for $x \in D$
+we have that:
+:$\iota \sqbrk D \subseteq H$
+We use [[Subfield Test]] to show that $K$ is a [[Definition:Field (Abstract Algebra)|field]]:
+Let $\dfrac {\map \iota x} {\map \iota y}, \dfrac {\map \iota z} {\map \iota w} \in K$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \frac {\map \iota x} {\map \iota y} \cdot \frac {\map \iota z} {\map \iota w}
+ | r = \frac {\map \iota x \map \iota z} {\map \iota y \map \iota w}
+ | c =
+}}
+{{eqn | r = \frac {\map \iota {x z} } {\map \iota {y w} }
+ | c = $\iota$ is a [[Definition:Ring Homomorphism|ring homomorphism]]
+}}
+{{eqn | r = K
+ | o = \in
+}}
+{{end-eqn}}
+and:
+{{begin-eqn}}
+{{eqn | l = \frac {\map \iota x} {\map \iota y} - \frac {\map \iota z} {\map \iota w}
+ | r = \frac {\map \iota x \map \iota w - \map \iota z \map \iota y} {\map \iota y \map \iota w}
+ | c =
+}}
+{{eqn | r = \frac {\map \iota {x w - z y} } {\map \iota {y w} }
+ | c = $\iota$ is a [[Definition:Ring Homomorphism|ring homomorphism]]
+}}
+{{eqn | r = K
+ | o = \in
+}}
+{{end-eqn}}
+Let $\dfrac {\map \iota x} {\map \iota y} \in K^\times$.
+Then:
+:$x \ne 0$
+so:
+{{begin-eqn}}
+{{eqn | l = \paren {\frac {\map \iota x} {\map \iota y} }^{-1}
+ | r = \frac {\map \iota y} {\map \iota x}
+ | c =
+}}
+{{eqn | o = \in
+ | r = K
+}}
+{{end-eqn}}
+By [[Subfield Test]], $K$ is a [[Definition:Field (Abstract Algebra)|field]].
+By assumption, $K = F$.
+{{qed|lemma}}
+=== 3 implies 2 ===
+Let $K$ be a [[Definition:Field (Abstract Algebra)|field]] such that:
+:$\iota \sqbrk D \subseteq K \subseteq F$
+We show that $F \subseteq K$.
+We apply the [[Definition:Universal Property|universal property]] to $\iota: D \to K$ and $\iota: D \to F$.
+By assumption, there exists:
+:a [[Definition:Unique|unique]] [[Definition:Field Homomorphism|field homomorphism]] $\bar \iota_1 : F \to K$ such that $\iota = \bar {\iota_1} \circ \iota$
+:a [[Definition:Unique|unique]] [[Definition:Field Homomorphism|field homomorphism]] $\bar \iota_2 : F \to F$ such that $\iota = \bar {\iota_2} \circ \iota$.
+By uniqueness, $\bar {\iota_2} = I_F$ is the [[Definition:Identity Mapping|identity mapping]] on $F$.
+Because $K\subset F$, $\iota_1$ fulfills the second condition as well.
+By uniqueness, $\iota_1 = \iota_2$.
+Because $F = \Img {\iota_2} = \Img {\iota_1} \subseteq K$, we have $F \subseteq K$.
+{{qed|lemma}}
+=== 3 implies 4 ===
+Let $S = D_{\ne 0}$.
+Because $\iota$ is a [[Definition:Ring Monomorphism|monomorphism]]:
+:$\iota \sqbrk S \subseteq F_{\ne 0}$
+Because $F$ is a [[Definition:Field (Abstract Algebra)|field]]:
+:$\iota \sqbrk S \subset F^\times$
+It remains to verify the [[Definition:Universal Property|universal property]] of the [[Definition:Localization of Ring|localization]].
+Let $B$ be a [[Definition:Ring with Unity|ring with unity]]
+Let $g: D \to B$ be a [[Definition:Ring Homomorphism|ring homomorphism]] such that $g \sqbrk S \subseteq B^\times$.
+We show that there is a unique [[Definition:Ring Homomorphism|ring homomorphism]] $h: F \to B$ such that $g = h \circ \iota$.
+This is done in exactly the same way as in the implication $1$ implies $3$.
+{{qed|lemma}}
+{{finish|maybe prove diretly that 1 implies 4 instead?}}
+=== 4 implies 3 ===
+Follows immediately from the definition of [[Definition:Localization of Ring|localization]].
+{{qed|lemma}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Trivial Field Extension is Galois}
+Tags: Galois Theory
+
+\begin{theorem}
+Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
+The [[Definition:Trivial Field Extension|trivial field extension]] $F/F$ is [[Definition:Galois Extension|Galois]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Galois Theory]]
+bdti46zzjxchrqggus9s4mcigrvnc8i
+\end{proof}<|endoftext|>
+\section{Field is Galois over Fixed Field of Automorphism Group}
+Tags: Galois Theory
+
+\begin{theorem}
+Let $E/F$ be a [[Definition:Finite Field Extension|finite field extension]].
+Let $K = \operatorname{Fix}_E(\operatorname{Aut}(E/F))$ be the [[Definition:Fixed Field|fixed field]] of the [[Definition:Automorphism Group of Field Extension|automorphism group]] of $E/F$.
+Then $E/K$ is [[Definition:Galois Extension|Galois]].
+\end{theorem}
+
+\begin{proof}
+Follows from [[Closed Fields in Galois Connection for Field Extension]] (and does not use [[Fundamental Theorem of Galois Theory]]).
+{{proof wanted}}
+[[Category:Galois Theory]]
+5f8c1ibg4vz9kywca8kf6h0d0lu6e4m
+\end{proof}<|endoftext|>
+\section{Automorphism Group Acts Faithfully on Generating Set}
+Tags: Field Extensions
+
+\begin{theorem}
+Let $E/F$ be a [[Definition:Field Extension|field extension]].
+Let $\operatorname{Aut}(E/F)$ be its [[Definition:Automorphism Group of Field Extension|automorphism group]].
+Let $S\subset E$ be a [[Definition:Generator of Field Extension|generating set]] of the extension.
+Let $S$ be [[Definition:Stable Under Group Action|stable]] under the [[Definition:Group Action of Automorphism Group|group action of]] $\operatorname{Aut}(E/F)$.
+Then the induced [[Definition:Group Action|group action]] on $S$ is [[Definition:Faithful Group Action|faithful]].
+\end{theorem}
+
+\begin{proof}
+Let $\sigma \in \operatorname{Aut}(E/F)$ stabilize $S$.
+Then $S$ is contained in the [[Definition:Fixed Field|fixed field]] of $\sigma$.
+By definition of [[Definition:Generator of Field Extension|generating set]], $\sigma$ fixes $E$.
+Thus $\operatorname{Aut}(E/F)$ [[Definition:Faithful Group Action|acts faithfully]].
+{{qed}}
+[[Category:Field Extensions]]
+g4ymi5pk0s0rofvoxzbkfzzn6cods98
+\end{proof}<|endoftext|>
+\section{Upper Closure in Ordered Subset is Intersection of Subset and Upper Closure}
+Tags: Upper Closures
+
+\begin{theorem}
+Let $L = \left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
+Let $\left({T, \precsim}\right)$ be an [[Definition:Ordered Subset|ordered subset]] of $L$.
+Let $t \in T$.
+Then $t^\succsim = T \cap t^\succeq$
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Ordered Subset|ordered subset]]:
+:$T \subseteq S$
+We will prove that
+:$t^\succsim \subseteq T \cap t^\succeq$
+Let $x \in t^\succsim$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$x \in T$ and $t \precsim x$
+By definition of [[Definition:Ordered Subset|ordered subset]]:
+:$t \preceq x$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$x \in t^\succeq$
+Thus by definition of [[Definition:Set Intersection|intersection]]:
+:$x \in T \cap t^\succeq$
+{{qed|lemma}}
+We will prove that
+:$T \cap t^\succeq \subseteq t^\succsim$
+Let $x \in T \cap t^\succeq$
+By definition of [[Definition:Set Intersection|intersection]]:
+:$x \in T$ and $x \in t^\succeq$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$t \preceq x$
+By definition of [[Definition:Ordered Subset|ordered subset]]:
+:$t \precsim x$
+Thus by definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$x \in t^\succsim$
+{{qed|lemma}}
+By definition of [[Definition:Set Equality|set equality]]:
+:$t^\succsim = T \cap t^\succeq$
+{{qed}}
+[[Category:Upper Closures]]
+kb3ltlc37a3kdtoi35lw7nsmwueduh2
+\end{proof}<|endoftext|>
+\section{Products of 2-Digit Pairs which Reversed reveal Same Product}
+Tags: Recreational Mathematics, Reversals
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integers]] can be expressed as the [[Definition:Integer Multiplication|product]] of $2$ [[Definition:Digit|two-digit]] numbers in $2$ ways such that the [[Definition:Divisor of Integer|factors]] in one of those pairs is the [[Definition:Reversal|reversal]] of each of the [[Definition:Divisor of Integer|factors]] in the other:
+:$504, 756, 806, 1008, 1148, 1209, 1472, 1512, 2016, 2208, 2418, 2924, 3024, 4416$
+
+\end{theorem}
+
+\begin{proof}
+Let $n \in \Z_{>0}$ such that:
+:$n = \sqbrk {a b} \times \sqbrk {c d} = \sqbrk {b a} \times \sqbrk {d c}$
+where $\sqbrk {a b}$ denotes the [[Definition:Digit|two-digit]] [[Definition:Positive Integer|positive integer]]:
+: $10 a + b$ for $0 \le a, b \le 9$
+from the [[Basis Representation Theorem]].
+We have:
+{{begin-eqn}}
+{{eqn | l = \paren {10 a + b} \paren {10 c + d}
+ | r = \paren {10 b + a} \paren {10 d + c}
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 100 a c + 10 \paren {a d + b c} + b d
+ | r = 100 b d + 10 \paren {b c + a d} + a c
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 99 a c
+ | r = 99 b d
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = a c
+ | r = b d
+ | c =
+}}
+{{end-eqn}}
+Thus the problem boils down to finding all the [[Definition:Set|sets]] of [[Definition:Digit|one-digit]] [[Definition:Integer|integers]] $\set {a, b, c, d}$ such that $a c = b d$, and so that:
+:$n = \sqbrk {a b} \times \sqbrk {c d} = \sqbrk {b a} \times \sqbrk {d c}$
+and also:
+:$n = \sqbrk {a d} \times \sqbrk {b c} = \sqbrk {d a} \times \sqbrk {c b}$
+Thus we investigate all [[Definition:Integer|integers]] whose [[Definition:Tau Function|$\tau$ value]] is $3$ or more, and find all those which have the [[Definition:Integer Multiplication|product]] of [[Definition:Digit|single-digit]] [[Definition:Integer|integers]] in $2$ ways, as follows:
+{{begin-eqn}}
+{{eqn | l = \map \tau 4
+ | r = 3
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 4
+ | r = 1 \times 4
+ | c =
+}}
+{{eqn | r = 2 \times 2
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 12 \times 42
+ | r = 21 \times 24
+ | c =
+}}
+{{eqn | r = 504
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \tau 6
+ | r = 4
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 6
+ | r = 1 \times 6
+ | c =
+}}
+{{eqn | r = 2 \times 3
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 12 \times 63
+ | r = 21 \times 36
+ | c =
+}}
+{{eqn | r = 756
+ | c =
+}}
+{{eqn | l = 13 \times 62
+ | r = 31 \times 26
+ | c =
+}}
+{{eqn | r = 806
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \tau 8
+ | r = 4
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 8
+ | r = 1 \times 8
+ | c =
+}}
+{{eqn | r = 2 \times 4
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 12 \times 84
+ | r = 21 \times 48
+ | c =
+}}
+{{eqn | r = 1008
+ | c =
+}}
+{{eqn | l = 14 \times 82
+ | r = 41 \times 28
+ | c =
+}}
+{{eqn | r = 1148
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \tau 9
+ | r = 3
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 9
+ | r = 1 \times 9
+ | c =
+}}
+{{eqn | r = 3 \times 3
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 13 \times 93
+ | r = 31 \times 39
+ | c =
+}}
+{{eqn | r = 1209
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \tau {10}
+ | r = 4
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 10
+ | r = 1 \times 10
+ | c = and so does not lead to a solution
+}}
+{{eqn | r = 2 \times 5
+ | c =
+}}
+{{end-eqn}}
+Further integers $n$ such that $\map \tau n \le 4$ need not be investigated, as one of the pairs of [[Definition:Divisor of Integer|factors]] will be greater than $9$.
+{{begin-eqn}}
+{{eqn | l = \map \tau {12}
+ | r = 6
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 12
+ | r = 1 \times 12
+ | c = which does not lead to a solution
+}}
+{{eqn | r = 2 \times 6
+ | c =
+}}
+{{eqn | r = 3 \times 4
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 23 \times 64
+ | r = 32 \times 46
+ | c =
+}}
+{{eqn | r = 1472
+ | c =
+}}
+{{eqn | l = 24 \times 63
+ | r = 42 \times 36
+ | c =
+}}
+{{eqn | r = 1512
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \tau {16}
+ | r = 5
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 16
+ | r = 1 \times 16
+ | c = which does not lead to a solution
+}}
+{{eqn | r = 2 \times 8
+ | c =
+}}
+{{eqn | r = 4 \times 4
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 24 \times 84
+ | r = 42 \times 48
+ | c =
+}}
+{{eqn | r = 2016
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \tau {18}
+ | r = 6
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 18
+ | r = 1 \times 18
+ | c = which does not lead to a solution
+}}
+{{eqn | r = 2 \times 9
+ | c =
+}}
+{{eqn | r = 3 \times 6
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 23 \times 96
+ | r = 32 \times 69
+ | c =
+}}
+{{eqn | r = 2208
+ | c =
+}}
+{{eqn | l = 26 \times 93
+ | r = 62 \times 39
+ | c =
+}}
+{{eqn | r = 2418
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \tau {20}
+ | r = 6
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 20
+ | r = 1 \times 20
+ | c = which does not lead to a solution
+}}
+{{eqn | r = 2 \times 10
+ | c = which does not lead to a solution
+}}
+{{eqn | r = 4 \times 5
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \tau {24}
+ | r = 8
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 24
+ | r = 1 \times 24
+ | c = which does not lead to a solution
+}}
+{{eqn | r = 2 \times 12
+ | c = which does not lead to a solution
+}}
+{{eqn | r = 3 \times 8
+ | c =
+}}
+{{eqn | r = 4 \times 6
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 34 \times 86
+ | r = 43 \times 68
+ | c =
+}}
+{{eqn | r = 2924
+ | c =
+}}
+{{eqn | l = 36 \times 84
+ | r = 63 \times 48
+ | c =
+}}
+{{eqn | r = 3024
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \tau {36}
+ | r = 9
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 36
+ | r = 1 \times 36
+ | c = which does not lead to a solution
+}}
+{{eqn | r = 2 \times 18
+ | c = which does not lead to a solution
+}}
+{{eqn | r = 3 \times 12
+ | c = which does not lead to a solution
+}}
+{{eqn | r = 4 \times 9
+ | c =
+}}
+{{eqn | r = 6 \times 6
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 46 \times 96
+ | r = 64 \times 69
+ | c =
+}}
+{{eqn | r = 4416
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|The challenge remains to prove, without going through all cases exhaustively, that there are no further pairs.}}
+{{Improve|Why not write out the upper triangular part of the multiplication table for $1 \le n \le 9$, $n \ne 5, 7$, and find the common numbers in the table? $5, 7$ can be eliminated via a simple argument $p x \ge 10$, and there are only $28$ numbers left to compare}}
+\end{proof}<|endoftext|>
+\section{Approximation to Power of 7 by Power of 10}
+Tags: Powers of 7, Powers of 10
+
+\begin{theorem}
+:$7^{510} \approx 1 \cdotp 00000 \, 09377 \, 76536 \ldots \times 10^{431}$
+This is the closest known approximation of a [[Definition:Integer Power|power of $7$]] by a [[Definition:Integer Power|power of $10$]].
+\end{theorem}
+
+\begin{proof}
+An intuition is given as follows:
+Suppose for some $m, n \in \N$:
+:$7^m = c \cdot 10^n$, where $c$ is very close to $1$.
+Taking [[Definition:Common Logarithm|common logarithm]]:
+:$m \log 7 = \log c + n$
+Which leads to:
+:$\log 7 = \dfrac n m + \dfrac {\log c} m$
+where $\dfrac {\log c} m$ is very close to $0$.
+To make a good approximation is to minimize $\size {\log c}$.
+One may calculate the [[Definition:Continued Fraction|continued fraction]] of $\log 7$.
+We use [[Definition:Continued Fraction|continued fractions]] because [[Convergents are Best Approximations]].
+One obtains $\sqbrk {0; 1, 5, 2, 5, 6, 1, 4813, 1, 1, 2, 2, \dots}$.
+{{OEIS|A082571}}
+Let $\dfrac {p_n} {q_n}$ be the [[Definition:Convergent of Continued Fraction|convergent]] $\sqbrk {0; a_1, \dots, a_n}$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \size {\log c}
+ | r = q_k \size {\log 7 - \frac {p_k} {q_k} }
+}}
+{{eqn | o = <
+ | r = q_k \paren {\frac 1 {q_k q_{k + 1} } }
+ | c = [[Accuracy of Convergents of Continued Fraction]]
+}}
+{{eqn | r = \frac 1 {q_{k + 1} }
+}}
+{{eqn | r = \frac 1 {a_{k + 1} q_k + q_{k - 1} }
+ | c = [[Definition:Numerators and Denominators of Continued Fraction|Recursive Definition of Denominators of Convergents]]
+}}
+{{end-eqn}}
+Thus, at least intuitively, the larger the value of the next $a_i$, the smaller $\size {\log c}$ is.
+{{finish|I am not familiar with this, this is just some justification}}
+$4813$ is the largest number in the expansion among at least the first $100$ numbers.
+Note that $\sqbrk {0; 1, 5, 2, 5, 6, 1} = \dfrac {431} {510}$
+This gives the estimate $7^{510} \approx 10^{431}$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Limit Inferior of Repetition Moore-Smith Sequence}
+Tags: Mapping Theory
+
+\begin{theorem}
+Let $L = \left({S, \wedge, \preceq}\right)$ be a [[Definition:Meet Semilattice|meet semilattice]].
+Let $N = \left({\N, \le}\right)$ be a [[Definition:Directed Set|directed]] [[Definition:Ordered Set|ordered set]].
+Let $a, b \in S$.
+Let $f = \left({c_i}\right)_{i \in \N} = \left({a, b, a, b, \dots}\right):\N \to S$ be a [[Definition:Moore-Smith Sequence|Moore-Smith sequence]].
+Then $\liminf \left({c_i}\right)_{i \in \N} = a \wedge b$
+\end{theorem}
+
+\begin{proof}
+We will prove that
+:(lemma): $\forall j \in \N: f\left[{\le\left({j}\right)}\right] = \left\{ {a, b}\right\}$
+Let $j \in \N$.
+Let $x \in S$.
+Assume that
+:$x \in f\left[{\le\left({j}\right)}\right]$
+By definition of [[Definition:Image of Subset under Mapping|image of set]]:
+:$\exists i \in \le\left({j}\right): x = f\left({i}\right)$
+By definition of $f$:
+:$x = a$ or $x = b$
+Thus by definition of [[Definition:Unordered Tuple|unordered tuple]]:
+:$x \in \left\{ {a, b}\right\}$
+{{qed|lemma}}
+Assume that
+:$x \in \left\{ {a, b}\right\}$
+We have:
+:$j \le j$ and $j \le j+1$
+By definition of [[Definition:Image of Element under Relation|image of element]]:
+:$j, j+1 \in \le\left({j}\right)$
+By definition of $f$:
+:$\left\{ {f\left({j}\right), f\left({j+1}\right)}\right\} = \left\{ {a, b}\right\}$
+By definition of [[Definition:Image of Subset under Mapping|image of set]]:
+:$f\left({j}\right), f\left({j+1}\right) \in f\left[{\le\left({j}\right)}\right]$
+Thus by definition of [[Definition:Unordered Tuple|unordered tuple]]:
+$x \in f\left[{\le\left({j}\right)}\right]$
+{{qed|lemma}}
+Thus by definition of [[Definition:Set Equality|set equality]]:
+:$f\left[{\le\left({j}\right)}\right] = \left\{ {a, b}\right\}$
+{{qed|lemma}}
+Thus
+{{begin-eqn}}
+{{eqn | l = \liminf \left({c_i}\right)_{i \mathop\in \N}
+| r = \sup \left\{ {\inf \left({f\left[{\mathord\le\left({j}\right)}\right]}\right): j \in \N}\right\}
+| c = definition of [[Definition:Limit Inferior of Moore-Smith Sequence|limit inferior]]
+}}
+{{eqn | r = \sup \left\{ {\inf \left\{ {a, b}\right\}: j \in \N}\right\}
+| c = (lemma)
+}}
+{{eqn | r = \sup \left\{ {a \wedge b: j \in \N}\right\}
+| c = definition of [[Definition:Meet (Order Theory)|meet]]
+}}
+{{eqn | r = a \wedge b
+| c = [[Supremum of Singleton]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping is Increasing}
+Tags: Increasing Mappings
+
+\begin{theorem}
+Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be [[Definition:Lattice|lattices]].
+Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that
+:for all [[Definition:Directed Set|directed set]] $\left({D, \precsim}\right)$ and [[Definition:Moore-Smith Sequence|Moore-Smith sequence]] $N:D \to S$ in $S$: $f\left({\liminf N}\right) \preceq_2 \liminf\left({f \circ N}\right)$
+Then $f$ is an [[Definition:Increasing Mapping|increasing mapping]].
+\end{theorem}
+
+\begin{proof}
+Let $a, b \in S$ such that
+:$a \preceq_1 b$
+Define $M = \left({\N, \le}\right)$ being an [[Definition:Ordered Set|ordered set]].
+We will prove that
+:$M$ is a [[Definition:Directed Set|directed set]].
+Let $x, y \in \N$.
+Thus by definition of [[Definition:Max Operation|max operation]]:
+:$\max\left({x, y}\right) \in \N$
+Thus by definition of [[Definition:Max Operation|max operation]]:
+:$x \le \max\left({x, y}\right)$ and $y \le \max\left({x, y}\right)$
+{{qed|lemma}}
+Define $g = \left({c_i}\right)_{i \in \N} = \left({a, b, a, b, \dots}\right):\N \to S$ being a [[Definition:Moore-Smith Sequence|Moore-Smith sequence]] in $S$.
+By [[Limit Inferior of Repetition Moore-Smith Sequence]]:
+:$\liminf \left({c_i}\right)_{i \in \N} = a \wedge_1 b$
+By [[Preceding iff Meet equals Less Operand]]:
+:$f\left({\liminf \left({c_i}\right)_{i \in \N} }\right) = f\left({a}\right)$
+By definition of [[Definition:Composition of Mappings|composition of mappings]]:
+:$f \circ g = \left({f\left({c_i}\right)}\right)_{i \in \N} = \left({f\left({a}\right), f\left({b}\right), f\left({a}\right), f\left({b}\right), \dots}\right)$
+By [[Limit Inferior of Repetition Moore-Smith Sequence]]:
+:$\liminf \left({f \circ g}\right) = f\left({a}\right) \wedge_2 f\left({b}\right)$
+By assumption:
+:$f\left({a}\right) \preceq_2 f\left({a}\right) \wedge_2 f\left({b}\right)$
+By [[Meet Precedes Operands]]:
+:$f\left({a}\right) \wedge_2 f\left({b}\right) \preceq_2 f\left({a}\right)$
+By definition of [[Definition:Antisymmetric Relation|antisymmetry]]:
+:$f\left({a}\right) = f\left({a}\right) \wedge_2 f\left({b}\right)$
+Thus by [[Preceding iff Meet equals Less Operand]]:
+:$f\left({a}\right) \preceq_2 f\left({b}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset}
+Tags: Order Theory
+
+\begin{theorem}
+Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be [[Definition:Up-Complete|up-complete]] [[Definition:Lattice|lattices]].
+Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that
+:for all [[Definition:Directed Set|directed set]] $\left({D, \precsim}\right)$ and [[Definition:Moore-Smith Sequence|Moore-Smith sequence]] $N:D \to S$ in $S$: $f\left({\liminf N}\right) \preceq_2 \liminf\left({f \circ N}\right)$
+Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$.
+Then $\sup \left({f\left[{D}\right]}\right) = f\left({\sup D}\right)$
+where $f\left[{D}\right]$ denotes the [[Definition:Image of Subset under Mapping|image]] of $D$ under $f$.
+\end{theorem}
+
+\begin{proof}
+By [[Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping is Increasing]]:
+:$f$ is an [[Definition:Increasing Mapping|increasing mapping]].
+By [[Image of Directed Subset under Increasing Mapping is Directed]]:
+:$f\left[{D}\right]$ is [[Definition:Directed Subset|directed]].
+By definition of [[Definition:Up-Complete|up-complete]]:
+:$D$ and $f\left[{D}\right]$ admit [[Definition:Supremum of Set|suprema]].
+By [[Subset and Image Admit Suprema and Mapping is Increasing implies Supremum of Image Precedes Mapping at Supremum]]:
+:$\sup \left({f\left[{D}\right]}\right) \preceq_2 f\left({\sup D}\right)$
+By [[Limit Inferior of Inclusion Moore-Smith Sequence is Supremum of Directed Subset]]:
+:$\sup D = \liminf i_D$
+By assumption:
+:$f\left({\sup D}\right) \preceq_2 \liminf \left({f \circ i_D}\right)$
+By [[Composition of Mapping and Inclusion is Restriction of Mapping]]:
+:$f \circ i_D = f\restriction{D}$
+By [[Limit Inferior of Restriction Moore-Smith Sequence is Supremum of Image of Directed Subset]]:
+:$\sup \left({f\left[{D}\right]}\right) = \liminf \left({f \circ i_D}\right)$
+Then
+:$f\left({\sup D}\right) \preceq_2 \sup \left({f\left[{D}\right]}\right)$
+Thus by definition of [[Definition:Antisymmetric Relation|antisymmetry]]:
+:$\sup \left({f\left[{D}\right]}\right) = f\left({\sup D}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Prime Values of Double Factorial plus 1}
+Tags: Double Factorials
+
+\begin{theorem}
+Let $n!!$ denote the [[Definition:Double Factorial|double factorial]] function.
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] $n$ such that $n!! + 1$ is [[Definition:Prime Number|prime]] begins:
+:$0, 1, 2, 518, 33 \, 416, 37 \, 310, 52 \, 608, 123 \, 998, 220 \, 502, \ldots$
+{{OEIS|A080778}}
+\end{theorem}
+
+\begin{proof}
+We have that:
+{{begin-eqn}}
+{{eqn | l = 0!! + 1
+ | r = 1 + 1
+ | c = {{Defof|Double Factorial}}
+}}
+{{eqn | r = 2
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 1!! + 1
+ | r = 1 + 1
+ | c = {{Defof|Double Factorial}}
+}}
+{{eqn | r = 2
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 2!! + 1
+ | r = 2 \times 0!! + 1
+ | c = {{Defof|Double Factorial}}
+}}
+{{eqn | r = 2 \times 1 + 1
+ | c = {{Defof|Double Factorial}}
+}}
+{{eqn | r = 3
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that the other integers are the only other ones with this property. Have fun.}}
+\end{proof}<|endoftext|>
+\section{Limit Inferior of Restriction Moore-Smith Sequence is Supremum of Image of Directed Subset}
+Tags: Restrictions
+
+\begin{theorem}
+Let $L = \left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be [[Definition:Up-Complete|up-complete]] [[Definition:Lattice|lattices]].
+Let $f:S \to T$ be an [[Definition:Increasing Mapping|increasing mapping]].
+Let $D \subseteq S$ be a [[Definition:Directed Subset|directed subset]] of $S$.
+Let $\left({D, \preceq'}\right)$ be a [[Definition:Directed Set|directed]] [[Definition:Ordered Subset|ordered subset]] of $L$.
+Let $f \restriction D: D \to T$, the [[Definition:Restriction of Mapping|restriction of mapping]], be a [[Definition:Moore-Smith Sequence|Moore-Smith sequence]] in $T$.
+Then $\liminf \left({f \restriction D}\right) = \sup \left({f\left[{D}\right]}\right)$
+\end{theorem}
+
+\begin{proof}
+We will prove that
+:(lemma): $\forall j \in D: \inf_L\left({\left({f \restriction D}\right)\left[{\preceq'\left({j}\right)}\right]}\right) = f\left({j}\right)$
+Let $j \in D$.
+By definitions of [[Definition:Image of Element under Relation|image of element]] and [[Definition:Upper Closure of Element|upper closure of element]]:
+:$\preceq'\left({j}\right) = j^{\succeq'}$
+By [[Upper Closure in Ordered Subset is Intersection of Subset and Upper Closure]]:
+:$j^{\succeq'} = D \cap j^{\succeq_1}$
+By [[Intersection is Subset]]:
+:$j^{\succeq'} \subseteq j^{\succeq_1}$
+By [[Image of Subset under Relation is Subset of Image/Corollary 2]]:
+:$\left({f \restriction D}\right)\left[{j^{\succeq'} }\right] \subseteq \left({f \restriction D}\right)\left[{j^{\succeq_1} }\right]$
+By [[Infimum of Subset]] and [[Infimum of Image of Upper Closure of Element under Increasing Mapping]]:
+:$f\left({j}\right) \preceq_2 \inf_L\left({\left({f \restriction D}\right)\left[{\preceq'\left({j}\right)}\right]}\right)$
+By definition of [[Definition:Reflexivity|reflexivity]]:
+:$j \preceq' j$
+By definition of [[Definition:Image of Element under Relation|image of element]]:
+:$j \in \preceq'\left({j}\right)$
+By definition of [[Definition:Image of Subset under Mapping|image of set]]:
+:$f\left({j}\right) \in \left({f \restriction D}\right)\left[{\preceq'\left({j}\right)}\right]$
+By definitions of [[Definition:Infimum of Set|infimum]] and [[Definition:Lower Bound of Set|lower bound]]:
+:$\inf_L\left({\left({f \restriction D}\right)\left[{\preceq'\left({j}\right)}\right]}\right) \preceq_2 f\left({j}\right)$
+Thus by definition of [[Definition:Antisymmetric Relation|antisymmetry]]:
+:$\inf_L\left({\left({f \restriction D}\right)\left[{\preceq'\left({j}\right)}\right]}\right) = f\left({j}\right)$
+{{qed|lemma}}
+Thus
+{{begin-eqn}}
+{{eqn | l = \liminf \left({f \restriction D}\right)
+| r = {\sup} \left\{ { {\inf}\left({\left({f \restriction D}\right)\left[{\preceq'\left({j}\right)}\right]}\right): j \in D}\right\}
+| c = definition of [[Definition:Limit Inferior of Moore-Smith Sequence|limit inferior of Moore-Smith sequence]]
+}}
+{{eqn | r = {\sup} \left\{ { f\left({j}\right): j \in D}\right\}
+| c = (lemma)
+}}
+{{eqn | r = {\sup} \left({f\left[{D}\right]}\right)
+| c = definition of [[Definition:Image of Subset under Mapping|image of set]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Infimum of Image of Upper Closure of Element under Increasing Mapping}
+Tags: Upper Closures
+
+\begin{theorem}
+Let $\left({S, \preceq}\right)$ and $\left({T, \precsim}\right)$ be [[Definition:Ordered Set|ordered set]].
+Let $f:S \to T$ be an [[Definition:Increasing Mapping|increasing mapping]].
+Let $x \in S$.
+Then $\inf \left({f\left[{x^\succeq}\right]}\right) = f\left({x}\right)$
+\end{theorem}
+
+\begin{proof}
+By [[Infimum of Upper Closure of Element]]:
+:$\inf x^\succeq = x$
+By definition of [[Definition:Infimum of Set|infimum]]:
+:$x$ is [[Definition:Lower Bound of Set|lower bound]] for $x^\succeq$
+Thus by [[Increasing Mapping Preserves Lower Bounds]]:
+:$f\left({x}\right)$ is [[Definition:Lower Bound of Set|lower bound]] for $f\left[{x^\succeq}\right]$
+By definition of [[Definition:Reflexivity|reflexivity]]:
+:$x \preceq x$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$x \in x^\succeq$
+By definition of [[Definition:Image of Subset under Mapping|image of set]]:
+:$f\left({x}\right) \in f\left[{x^\succeq}\right]$
+Thus by definition:
+:$\forall y \in T: y$ is [[Definition:Lower Bound of Set|lower bound]] for $f\left[{x^\succeq}\right] \implies y \precsim f\left({x}\right)$
+Thus by definition of [[Definition:Infimum of Set|infimum]]:
+:$\inf \left({f\left[{x^\succeq}\right]}\right) = f\left({x}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Poulet Numbers which are also Magic Constant for Magic Square}
+Tags: Poulet Numbers, Magic Squares
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Poulet Number|Poulet numbers]] which are also the [[Definition:Magic Constant|magic constant]] of a [[Definition:Magic Square|magic square]] begins:
+:$1105, 2465, \ldots$
+\end{theorem}
+
+\begin{proof}
+From the [[Definition:Poulet Number/Sequence|sequence of Poulet numbers]], these are [[Definition:Poulet Number|Poulet numbers]]:
+:$1105, 2465, \ldots$
+Then we have:
+{{begin-eqn}}
+{{eqn | l = 1105
+ | r = \dfrac {13 \paren {13^2 + 1} } 2
+ | c = so $1105$ is the [[Definition:Magic Constant|magic constant]] of the [[Definition:Order of Magic Square|order $13$]] [[Definition:Magic Square|magic square]]
+}}
+{{eqn | l = 2465
+ | r = \dfrac {17 \paren {17^2 + 1} } 2
+ | c = so $2465$ is the [[Definition:Magic Constant|magic constant]] of the [[Definition:Order of Magic Square|order $17$]] [[Definition:Magic Square|magic square]]
+}}
+{{end-eqn}}
+{{expand|Further investigation needed. I have not seen this correspondence explored on the internet. I note though that both $1105$ and $2465$ are also [[Definition:Carmichael Number|Carmichael]].}}
+[[Category:Poulet Numbers]]
+[[Category:Magic Squares]]
+ahytsnqo4qufaah5amkvd28zlhygl7u
+\end{proof}<|endoftext|>
+\section{Composition of Mapping and Inclusion is Restriction of Mapping}
+Tags: Restrictions
+
+\begin{theorem}
+Let $S, T$ be [[Definition:Set|sets]].
+Let $f:S \to T$ be a [[Definition:Mapping|mapping]].
+Let $A \subseteq S$.
+Then $f \circ i_A = f \restriction A$
+where
+:$i_A$ denotes the [[Definition:Inclusion Mapping|inclusion mapping]] of $A$,
+:$f \restriction A$ denotes the [[Definition:Restriction of Mapping|restriction]] of $f$ to $A$.
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Inclusion Mapping|inclusion mapping]]:
+:$i_A: A \to S$
+By definitions of [[Definition:Composition of Mappings|composition of mappings]] and [[Definition:Restriction of Mapping|restriction of mapping]]:
+:$f \circ i_A: A \to T$ and $f \restriction A: A \to T$
+Let $a \in A$.
+Thus
+{{begin-eqn}}
+{{eqn | l = \left({f \circ i_A}\right)\left({a}\right)
+| r = f \left({i_A \left({a}\right)}\right)
+| c = definition of [[Definition:Composition of Mappings|composition of mappings]]
+}}
+{{eqn | r = f \left({a}\right)
+| c = definition of [[Definition:Inclusion Mapping|inclusion mapping]]
+}}
+{{eqn | r = \left({f \restriction A}\right)\left({a}\right)
+| c = definition of [[Definition:Restriction of Mapping|restriction of mapping]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema}
+Tags: Order Theory
+
+\begin{theorem}
+Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Complete Lattice|complete lattices]].
+Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that
+:for all [[Definition:Directed Set|directed sets]] $\struct {D, \precsim}$ and [[Definition:Moore-Smith Sequence|Moore-Smith sequences]] $N:D \to S$ in $S$: $\map f {\liminf N} \preceq_2 \map \liminf {f \circ N}$
+Then $f$ [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
+\end{theorem}
+
+\begin{proof}
+Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$.
+Assume that
+:$D$ admits a [[Definition:Supremum of Set|supremum]].
+Thus by definition of [[Definition:Complete Lattice|complete lattice]]:
+:$f \sqbrk D$ admits a [[Definition:Supremum of Set|supremum]].
+Thus by [[Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset]]:
+:$\map \sup {f \sqbrk D} = \map f {\sup D}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Thurston's Geometrization Conjecture}
+Tags: Topology
+
+\begin{theorem}
+When a [[Definition:Topological Manifold|topological manifold]] of [[Definition:Dimension of Locally Euclidean Space|dimension $3$]] has been split into its [[Definition:Connected Sum|connected sum]] and the [[Definition:Jaco-Shalen-Johannson Torus Decomposition|Jaco-Shalen-Johannson torus decomposition]], the remaining [[Definition:Component of Decomposed Manifold|components]] each admit exactly one of the following [[Definition:Geometry of Manifold|geometries]]:
+:$(1): \quad$ [[Definition:Euclidean Geometry|Euclidean geometry]]
+:$(2): \quad$ [[Definition:Hyperbolic Geometry|Hyperbolic geometry]]
+:$(3): \quad$ [[Definition:Spherical Geometry|Spherical geometry]]
+:$(4): \quad$ The [[Definition:Geometry of Manifold|geometry]] of $\mathbb S^2 \times \R$
+:$(5): \quad$ The [[Definition:Geometry of Manifold|geometry]] of $\mathbb H^2 \times \R$
+:$(6): \quad$ The [[Definition:Geometry of Manifold|geometry]] of the [[Definition:Universal Cover|universal cover]] $S L_2 \R^~$ of the [[Definition:Lie Group|Lie group $S L_2 \R$]]
+:$(7): \quad$ [[Definition:Nil Geometry|Nil geometry]]
+:$(8): \quad$ [[Definition:Sol Geometry|Sol geometry]]
+where:
+:$\mathbb S^2$ is the [[Definition:Sphere (Topology)|2-sphere]]
+:$\mathbb H^2$ is the [[Definition:Hyperbolic Plane|hyperbolic plane]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Discuss {{AuthorRef|Grigori Perelman}}'s work -- it is generally believed his proof of this is valid, in which case this needs to be renamed to Theorem.}}
+{{Namedfor|William Paul Thurston|cat = Thurston}}
+\end{proof}<|endoftext|>
+\section{Poincaré Conjecture/Historical Note}
+Tags: Historical Notes
+
+\begin{theorem}
+The [[Poincaré Conjecture]] was first posed in $1904$ by {{AuthorRef|Jules Henri Poincaré}}.
+It was finally resolved by the work of {{AuthorRef|Grigori Perelman}}, who solved [[Thurston's Geometrization Conjecture]] in $2003$.
+\end{theorem}<|endoftext|>
+\section{Definition:Stabilizer/Linguistic Note}
+Tags: Linguistic Notes, Linguistic Notes
+
+\begin{theorem}
+The British English spelling for '''[[Definition:Stabilizer|stabilizer]]''' is '''stabiliser'''.
+\end{theorem}<|endoftext|>
+\section{Mapping Preserves Directed Suprema implies Mapping is Continuous}
+Tags: Topological Order Theory
+
+\begin{theorem}
+Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be [[Definition:Up-Complete|up-complete]] [[Definition:Ordered Set|ordered sets]] with [[Definition:Scott Topology|Scott topologies]].
+Let $f: S \to T$ be a [[Definition:Mapping Preserves Supremum/Directed|directed suprema preserving]] [[Definition:Mapping|mapping]].
+Then $f$ is [[Definition:Continuous (Topology)|continuous]].
+\end{theorem}
+
+\begin{proof}
+Let $P$ be a [[Definition:Closed Set (Topology)|closed]] [[Definition:Subset|subset]] of $T$.
+By [[Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set]]:
+:$P$ is [[Definition:Lower Set|lower]] and [[Definition:Closed under Directed Suprema|closed under directed suprema]].
+We will prove that
+:for all [[Definition:Directed Subset|directed subset]] $D$ of $S$: $D \subseteq f^{-1}\left[{P}\right] \implies \sup D \in f^{-1}\left[{P}\right]$
+Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$ such that
+:$D \subseteq f^{-1}\left[{P}\right]$
+By definition of [[Definition:Mapping Preserves Supremum/Directed|mapping preserves directed suprema]]:
+:$f$ [[Definition:Mapping Preserves Supremum/Subset|preserves the supremum]] of $D$.
+By definition of [[Definition:Up-Complete|up-complete]]:
+:$D$ admits a [[Definition:Supremum of Set|supremum]].
+By definition of [[Definition:Mapping Preserves Supremum/Subset|mapping preserves the supremum]]:
+:$ \sup \left({f\left[{D}\right]}\right) = f \left({\sup D}\right)$
+By [[Directed Suprema Preserving Mapping is Increasing]]:
+:$f$ is [[Definition:Increasing Mapping|increasing]].
+By [[Image of Directed Subset under Increasing Mapping is Directed]]:
+:$f\left[{D}\right]$ is [[Definition:Directed Subset|directed]].
+By [[Preimage is Subset of Set implies Image of Set is Subset]]:
+:$f\left[{D}\right] \subseteq P$
+By definition of [[Definition:Closed under Directed Suprema|closed under directed suprema]]:
+:$\sup \left({f\left[{D}\right]}\right) \in P$
+Thus by definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
+:$\sup D \in f^{-1}\left[{P}\right]$
+{{qed|lemma}}
+By definition:
+:$f^{-1}\left[{P}\right]$ is [[Definition:Closed under Directed Suprema|closed under directed suprema]].
+By [[Preimage of Lower Set under Increasing Mapping is Lower]]:
+:$f^{-1}\left[{P}\right]$ is [[Definition:Lower Set|lower]].
+Thus by [[Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set]]:
+:$f^{-1}\left[{P}\right]$ is [[Definition:Closed Set (Topology)|closed]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Secant of Complex Number}
+Tags: Secant Function, Complex Numbers
+
+\begin{theorem}
+Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
+Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
+Then:
+:$\sec \paren {a + b i} = \dfrac {\cos a \cosh b + i \sin a \sinh b} {\cos^2 a \cosh^2 b + \sin^2 a \sinh^2 b}$
+where:
+:$\sec$ denotes the [[Definition:Complex Secant Function|complex secant function]].
+:$\sin$ denotes the [[Definition:Real Sine Function|real sine function]]
+:$\cos$ denotes the [[Definition:Real Cosine Function|real cosine function]]
+:$\sinh$ denotes the [[Definition:Hyperbolic Sine|hyperbolic sine function]]
+:$\cosh$ denotes the [[Definition:Hyperbolic Cosine|hyperbolic cosine function]]
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sec \paren {a + b i}
+ | r = \frac 1 {\cos \paren {a + b i} }
+ | c = {{Defof|Complex Secant Function}}
+}}
+{{eqn | r = \dfrac 1 {\cos a \cosh b - i \sin a \sinh b}
+ | c = [[Cosine of Complex Number]]
+}}
+{{eqn | r = \dfrac {\cos a \cosh b + i \sin a \sinh b} {\paren {\cos a \cosh b + i \sin a \sinh b} \paren {\cos a \cosh b - i \sin a \sinh b} }
+ | c = multiplying [[Definition:Denominator|denominator]] and [[Definition:Numerator|numerator]] by $\cos a \cosh b + i \sin a \sinh b$
+}}
+{{eqn | r = \dfrac {\cos a \cosh b + i \sin a \sinh b} {\cos^2 a \cosh^2 b - i^2 \sin^2 a \sinh^2 b}
+ | c = [[Difference of Two Squares]]
+}}
+{{eqn | r = \dfrac {\cos a \cosh b + i \sin a \sinh b} {\cos^2 a \cosh^2 b + \sin^2 a \sinh^2 b}
+ | c = {{Defof|Imaginary Unit}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Tangent of Complex Number/Formulation 1}
+Tags: Tangent of Complex Number
+
+\begin{theorem}
+:$\tan \paren {a + b i} = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \tan \paren {a + b i}
+ | r = \frac {\sin \paren {a + b i} } {\cos \paren {a + b i} }
+ | c = {{Defof|Complex Tangent Function}}
+}}
+{{eqn | r = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}
+ | c = [[Sine of Complex Number]] and [[Cosine of Complex Number]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Tangent of Complex Number/Formulation 2}
+Tags: Tangent of Complex Number
+
+\begin{theorem}
+:$\tan \paren {a + b i} = \dfrac {\tan a + i \tanh b} {1 - i \tan a \tanh b}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \tan \paren {a + b i}
+ | r = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}
+ | c = [[Tangent of Complex Number/Formulation 1|Tangent of Complex Number: Formulation 1]]
+}}
+{{eqn | r = \dfrac {\tan a \cosh b + i \sinh b} {\cosh b - i \tan a \sinh b}
+ | c = multiplying [[Definition:Denominator|denominator]] and [[Definition:Numerator|numerator]] by $\dfrac 1 {\cos a}$
+}}
+{{eqn | r = \frac {\tan a + i \tanh b} {1 - i \tan a \tanh b}
+ | c = multiplying [[Definition:Denominator|denominator]] and [[Definition:Numerator|numerator]] by $\dfrac 1 {\cosh b}$
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Tangent of Complex Number/Formulation 3}
+Tags: Tangent of Complex Number
+
+\begin{theorem}
+:$\tan \paren {a + b i} = \dfrac {\tan a - \tan a \tanh ^2 b} {1 + \tan ^2 a \tanh ^2 b} + \dfrac {\tanh b + \tan ^2 a \tanh b} {1 + \tan ^2 a \tanh ^2 b} i$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \tan \paren {a + b i}
+ | r = \frac {\tan a + i \tanh b} {1 - i \tan a \tanh b}
+ | c = [[Tangent of Complex Number/Formulation 2|Tangent of Complex Number: Formulation 2]]
+}}
+{{eqn | r = \frac {\paren {\tan a + i \tanh b} \paren {1 + i \tan a \tanh b} } {1 + \tan ^2 a \tanh ^2 b}
+ | c = multiplying [[Definition:Denominator|denominator]] and [[Definition:Numerator|numerator]] by $1 + i \tan a \tanh b$
+}}
+{{eqn | r = \frac {\tan a + i \tanh b + i \tan ^2 a \tanh b - \tan a \tanh ^2 b} {1 + \tan ^2 a \tanh ^2 b}
+ | c =
+}}
+{{eqn | r = \frac {\tan a - \tan a \tanh ^2 b} {1 + \tan ^2 a \tanh ^2 b} + \frac {\tanh b + \tan ^2 a \tanh b} {1 + \tan ^2 a \tanh ^2 b} i
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cosecant of Complex Number}
+Tags: Cosecant Function, Complex Numbers
+
+\begin{theorem}
+Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
+Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
+Then:
+:$\csc \paren {a + b i} = \dfrac {\sin a \cosh b - i \cos a \sinh b} {\sin^2 a \cosh^2 b + \cos^2 a \sinh^2 b}$
+where:
+:$\csc$ denotes the [[Definition:Complex Cosecant Function|complex cosecant function]].
+:$\sin$ denotes the [[Definition:Real Sine Function|real sine function]]
+:$\cos$ denotes the [[Definition:Real Cosine Function|real cosine function]]
+:$\sinh$ denotes the [[Definition:Hyperbolic Sine|hyperbolic sine function]]
+:$\cosh$ denotes the [[Definition:Hyperbolic Cosine|hyperbolic cosine function]]
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \csc \paren {a + b i}
+ | r = \frac 1 {\sin \paren {a + b i} }
+ | c = {{Defof|Complex Cosecant Function}}
+}}
+{{eqn | r = \dfrac 1 {\sin a \cosh b + i \cos a \sinh b}
+ | c = [[Sine of Complex Number]]
+}}
+{{eqn | r = \dfrac {\sin a \cosh b - i \cos a \sinh b} {\paren {\sin a \cosh b - i \cos a \sinh b} \paren {\sin a \cosh b + i \cos a \sinh b} }
+ | c = multiplying [[Definition:Denominator|denominator]] and [[Definition:Numerator|numerator]] by $\sin a \cosh b - i \cos a \sinh b$
+}}
+{{eqn | r = \dfrac {\sin a \cosh b - i \cos a \sinh b} {\sin^2 a \cosh^2 b - i^2 \cos^2 a \sinh^2 b}
+ | c = [[Difference of Two Squares]]
+}}
+{{eqn | r = \dfrac {\sin a \cosh b - i \cos a \sinh b} {\sin^2 a \cosh^2 b + \cos^2 a \sinh^2 b}
+ | c = {{Defof|Imaginary Unit}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cotangent of Complex Number}
+Tags: Cotangent Function, Complex Numbers, Cotangent of Complex Number
+
+\begin{theorem}
+Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
+Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
+Then:
+\end{theorem}<|endoftext|>
+\section{Cotangent of Complex Number/Formulation 1}
+Tags: Cotangent of Complex Number
+
+\begin{theorem}
+:$\cot \paren {a + b i} = \dfrac {\cos a \cosh b - i \sin a \sinh b} {\sin a \cosh b + i \cos a \sinh b}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \cot \paren {a + b i}
+ | r = \frac {\cos \paren {a + b i} } {\sin \paren {a + b i} }
+ | c = {{Defof|Complex Cotangent Function}}
+}}
+{{eqn | r = \dfrac {\cos a \cosh b - i \sin a \sinh b} {\sin a \cosh b + i \cos a \sinh b}
+ | c = [[Sine of Complex Number]] and [[Cosine of Complex Number]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cotangent of Complex Number/Formulation 2}
+Tags: Cotangent of Complex Number
+
+\begin{theorem}
+:$\map \cot {a + b i} = \dfrac {i \cot a \coth b - 1} {\cot a - i \coth b}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \map \cot {a + b i}
+ | r = \dfrac {\cos a \cosh b - i \sin a \sinh b} {\sin a \cosh b + i \cos a \sinh b}
+ | c = [[Cotangent of Complex Number/Formulation 1|Cotangent of Complex Number: Formulation 1]]
+}}
+{{eqn | r = \dfrac {\cot a \cosh b - i \sinh b} {\cosh b + i \cot a \sinh b}
+ | c = multiplying [[Definition:Denominator|denominator]] and [[Definition:Numerator|numerator]] by $\dfrac 1 {\sin a}$
+}}
+{{eqn | r = \dfrac {\cot a \coth b - i} {\coth b + i \cot a}
+ | c = multiplying [[Definition:Denominator|denominator]] and [[Definition:Numerator|numerator]] by $\dfrac 1 {\sinh b}$
+}}
+{{eqn | r = \dfrac {i \cot a \coth b - 1} {\cot a - i \coth b}
+ | c = multiplying [[Definition:Denominator|denominator]] and [[Definition:Numerator|numerator]] by $-i$
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cotangent of Complex Number/Formulation 3}
+Tags: Cotangent of Complex Number
+
+\begin{theorem}
+:$\cot \paren {a + b i} = \dfrac {- \cot a - \cot a \coth^2 b} {\cot^2 a + \coth^2 b} + \dfrac {\cot^2 a \coth b - \coth b} {\cot^2 a + \coth^2 b} i$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \cot \paren {a + b i}
+ | r = \dfrac {i \cot a \coth b - 1} {\cot a - i \coth b}
+ | c = [[Cotangent of Complex Number/Formulation 2|Cotangent of Complex Number: Formulation 2]]
+}}
+{{eqn | r = \dfrac {\paren {i \cot a \coth b - 1} \paren {\cot a + i \coth b} } {\paren {\cot a - i \coth b} \paren {\cot a + i \coth b} }
+ | c = multiplying [[Definition:Denominator|denominator]] and [[Definition:Numerator|numerator]] by $\cot a + i \coth b$
+}}
+{{eqn | r = \dfrac {\paren {i \cot a \coth b - 1} \paren {\cot a + i \coth b} } {\cot^2 a + \coth^2 b}
+ | c = [[Difference of Two Squares]]
+}}
+{{eqn | r = \dfrac {i \cot^2 a \coth b - \cot a - \cot a \coth^2 b - i \coth b} {\cot^2 a + \coth^2 b}
+ | c =
+}}
+{{eqn | r = \dfrac {- \cot a - \cot a \coth^2 b} {\cot^2 a + \coth^2 b} + \dfrac {\cot^2 a \coth b - \coth b} {\cot^2 a + \coth^2 b} i
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Continuous iff Mapping at Limit Inferior Precedes Limit Inferior of Composition of Mapping and Sequence}
+Tags: Topological Order Theory
+
+\begin{theorem}
+Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Scott Topology|Scott topologies]].
+Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
+Then $f$ is [[Definition:Continuous (Topology)|continuous]] {{iff}}
+:for all [[Definition:Directed Set|directed set]] $\left({D, \precsim}\right)$ and [[Definition:Moore-Smith Sequence|Moore-Smith sequence]] $N:D \to S$ in $S$: $f\left({\liminf N}\right) \preceq_2 \liminf\left({f \circ N}\right)$
+\end{theorem}
+
+\begin{proof}
+=== Sufficient Condition ===
+Assume that
+:$f$ is [[Definition:Continuous (Topology)|continuous]].
+Let $\left({D, \precsim}\right)$ be a [[Definition:Directed Set|directed set]].
+Let $N:D \to S$ be a [[Definition:Moore-Smith Sequence|Moore-Smith sequence]] in $S$.
+Aiming for a [[Definition:Contradiction|contradiction]] suppose that
+:$f\left({\liminf N}\right) \npreceq_2 \liminf\left({f \circ N}\right)$
+By definition of [[Definition:Lower Closure of Element|lower closure of element]]:
+:$f\left({\liminf N}\right) \notin \left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2}$
+By definition of [[Definition:Relative Complement|relative complement]]:
+:$f\left({\liminf N}\right) \in \complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)$
+By [[Lower Closure of Element is Topologically Closed in Scott Topological Ordered Set]]:
+:$\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2}$ is [[Definition:Closed Set (Topology)|closed]].
+By definition of [[Definition:Closed Set (Topology)|closed set]]:
+:$\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)$ is [[Definition:Open Set (Topology)|open]].
+By definition of [[Definition:Continuous (Topology)|continuous]]:
+:$f^{-1}\left[{\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)}\right]$ is [[Definition:Open Set (Topology)|open]].
+By [[Open iff Upper and with Property (S) in Scott Topological Lattice]]:
+:$f^{-1}\left[{\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)}\right]$ has [[Definition:Property (S)|property (S)]].
+By [[Set of Infima for Sequence is Directed]]:
+:$X := \left\{ {\inf \left({N\left[{\precsim\left({j}\right)}\right]}\right): j \in D}\right\}$ is [[Definition:Directed Subset|directed]].
+By definition of [[Definition:Limit Inferior of Moore-Smith Sequence|limit inferior]]:
+:$\liminf N = \sup X$
+By definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
+:$\liminf N \in f^{-1}\left[{\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)}\right]$
+By definition of [[Definition:Property (S)|property (S)]]:
+:$\exists y \in X:\forall x \in X: y \preceq_1 x \implies x \in f^{-1}\left[{\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)}\right]$
+We have:
+:$\exists j \in D: y = \inf \left({N\left[{\precsim\left({j}\right)}\right]}\right)$
+By definition of [[Definition:Reflexivity|reflexivity]]:
+:$y \preceq_1 y$
+Then:
+:$y \in f^{-1}\left[{\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)}\right]$
+Define $y' := \inf \left({\left({f \circ N}\right)\left[{\precsim\left({j}\right)}\right]}\right)$
+Define $X' := \left\{ {\inf \left({\left({f \circ N}\right)\left[{\precsim\left({j}\right)}\right]}\right): j \in D}\right\}$
+By [[Mapping is Increasing implies Mapping at Infimum for Sequence Precedes Infimum for Composition of Mapping and Sequence]]:
+:$f\left({y}\right) \preceq_2 y'$
+By definition of $X'$:
+:$y' \in X'$
+By definitions of [[Definition:Supremum of Set|supremum]] and [[Definition:Upper Bound of Set|upper bound]]:
+:$y' \preceq_2 \sup X'$
+By definition of [[Definition:Limit Inferior of Moore-Smith Sequence|limit inferior]]:
+:$\liminf \left({f \circ N}\right) = \sup X'$
+By definition of [[Definition:Transitivity|transitivity]]:
+:$f\left({y}\right) \preceq_2 \liminf \left({f \circ N}\right)$
+By definition of [[Definition:Lower Closure of Element|lower closure of element]]:
+:$f\left({y}\right) \in \left({\liminf \left({f \circ N}\right)}\right)^{\preceq_2}$
+By definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
+:$y \in f^{-1}\left[{\left({\liminf \left({f \circ N}\right)}\right)^{\preceq_2} }\right]$
+By [[Complement of Preimage equals Preimage of Complement]]:
+:$f^{-1}\left[{\complement_T\left({\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right)}\right] = \complement_S\left({f^{-1}\left[{\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right]}\right)$
+Thus this contradicts $y \in \complement_S\left({f^{-1}\left[{\left({ \liminf\left({f \circ N}\right) }\right)^{\preceq_2} }\right]}\right)$
+{{qed|lemma}}
+=== Necessary Condition ===
+Assume that
+:for all [[Definition:Directed Set|directed set]] $\left({D, \precsim}\right)$ and [[Definition:Moore-Smith Sequence|Moore-Smith sequence]] $N:D \to S$ in $S$: $f\left({\liminf N}\right) \preceq_2 \liminf\left({f \circ N}\right)$
+By [[Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema]]:
+:$f$ [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
+Thus by [[Mapping Preserves Directed Suprema implies Mapping is Continuous]]:
+:$f$ is [[Definition:Continuous (Topology)|continuous]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Mapping is Increasing implies Mapping at Infimum for Sequence Precedes Infimum for Composition of Mapping and Sequence}
+Tags: Order Theory
+
+\begin{theorem}
+Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be [[Definition:Complete Lattice|complete lattices]].
+Let $f: S \to T$ be an [[Definition:Increasing Mapping|increasing mapping]].
+Let $\left({D, \precsim}\right)$ be a [[Definition:Directed Set|directed set]].
+Let $N: D \to S$ be a [[Definition:Moore-Smith Sequence|Moore-Smith sequence]] in $S$.
+Let $j \in D$.
+Then $f\left({\inf\left({N\left[{\precsim \left({j}\right)}\right]}\right)}\right) \preceq_2 \inf \left({\left({f \circ N}\right)\left[{\precsim \left({j}\right)}\right]}\right)$
+\end{theorem}
+
+\begin{proof}
+By definitions of [[Definition:Image of Subset under Mapping|image of set]] and [[Definition:Composition of Mappings|composition of mappings]]:
+:$f\left[{N\left[{\precsim \left({j}\right)}\right]}\right] = \left({f \circ N}\right)\left[{\precsim \left({j}\right)}\right]$
+By definition of [[Definition:Complete Lattice|complete lattice]]:
+:$f\left[{N\left[{\precsim \left({j}\right)}\right]}\right]$ and $N\left[{\precsim \left({j}\right)}\right]$ admit [[Definition:Infimum of Set|infima]].
+Thus by [[Subset and Image Admit Infima and Mapping is Increasing implies Infimum of Image Succeeds Mapping at Infimum]]:
+:$f\left({\inf\left({N\left[{\precsim \left({j}\right)}\right]}\right)}\right) \preceq_2 \inf \left({\left({f \circ N}\right)\left[{\precsim \left({j}\right)}\right]}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Set of Infima for Sequence is Directed}
+Tags: Order Theory
+
+\begin{theorem}
+Let $\left({S, \vee, \wedge, \preceq}\right)$ be a [[Definition:Complete Lattice|complete lattice]].
+Let $\left({A, \precsim}\right)$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Directed Set|directed set]].
+Let $Z: A \to S$ be a [[Definition:Moore-Smith Sequence|Moore-Smith sequence]].
+Let $D = \left\{ {\inf \left({Z\left[{\precsim \left({j}\right)}\right]}\right): j \in A}\right\}$ be a [[Definition:Subset|subset]] of $S$.
+Then $D$ is [[Definition:Directed Subset|directed]].
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Non-Empty Set|non-empty set]]:
+:$\exists j: j \in A$
+By definition of $D$:
+:$\inf \left({Z\left[{\precsim \left({j}\right)}\right]}\right) \in D$
+Hence by definition:
+:$D$ is a [[Definition:Non-Empty Set|non-empty set]].
+Let $x, y \in D$.
+By definition of $D$:
+:$\exists j_1 \in A: x = \inf \left({Z\left[{\precsim \left({j_1}\right)}\right]}\right)$
+and
+:$\exists j_2 \in A: y = \inf \left({Z\left[{\precsim \left({j_2}\right)}\right]}\right)$
+By definition of [[Definition:Directed Set|directed set]]:
+:$\exists j \in A: j_1 \precsim j \land j_2 \precsim j$
+By [[Preceding implies Image is Subset of Image]]:
+:$\precsim\left({j}\right) \subseteq \mathord\precsim\left({j_1}\right)$ and $\precsim\left({j}\right) \subseteq \mathord\precsim\left({j_2}\right)$
+By [[Image of Subset under Relation is Subset of Image/Corollary 2]]
+:$Z\left[{\precsim\left({j}\right)}\right] \subseteq Z\left[{\precsim\left({j_1}\right)}\right]$ and $Z\left[{\precsim\left({j}\right)}\right] \subseteq Z\left[{\precsim\left({j_2}\right)}\right]$
+Thus by definition of $D$:
+:$z := \inf \left({Z\left[{\precsim \left({j}\right)}\right]}\right) \in D$
+Thus by [[Infimum of Subset]]:
+:$x \preceq z$ and $y \preceq z$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Preceding implies Image is Subset of Image}
+Tags: Preorder Theory
+
+\begin{theorem}
+Let $\left({S, \precsim}\right)$ be a [[Definition:Preordered Set|preordered set]].
+Let $x, y \in S$ such that
+:$x \precsim y$
+Then $\precsim\left({y}\right) \subseteq \mathord\precsim\left({x}\right)$
+where $\precsim\left({y}\right)$ denotes the [[Definition:Image of Element under Relation|image]] of $y$ under $\precsim$.
+\end{theorem}
+
+\begin{proof}
+Let $z \in \mathord\precsim\left({y}\right)$
+By definition of [[Definition:Image of Element under Relation|image of element]]:
+:$y \precsim z$
+By definition of [[Definition:Transitivity|transitivity]]:
+:$x \precsim z$
+Thus by definition of [[Definition:Image of Element under Relation|image of element]]:
+:$z \in \mathord\precsim\left({x}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Universal Property of Group Ring}
+Tags: Universal Properties, Group Rings
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+Let $G$ be a [[Definition:Group|group]].
+Let $R \sqbrk G$ be the corresponding [[Definition:Group Ring|group ring]].
+Let $S$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+Let $\phi: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
+Let $\beta : G \to R^\times$ be a [[Definition:Group Homomorphism|group homomorphism]], where $R^\times$ is the [[Definition:Multiplicative Group of Ring|multiplicative group]] of $R$.
+Then there exists a unique [[Definition:Ring Homomorphism|ring homomorphism]] from $R \sqbrk G$ to $S$ which extends $\phi$ and $\beta$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Universal Properties]]
+[[Category:Group Rings]]
+7hshi30ywgnzgihw6y2dzbb4y6hox22
+\end{proof}<|endoftext|>
+\section{Continuous iff Directed Suprema Preserving}
+Tags: Topological Order Theory
+
+\begin{theorem}
+Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Scott Topology|Scott topologies]].
+Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
+Then $f$ is [[Definition:Continuous (Topology)|continuous]] {{iff}} $f$ [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
+\end{theorem}
+
+\begin{proof}
+=== Sufficient Condition ===
+Assume that
+:$f$ is [[Definition:Continuous (Topology)|continuous]].
+By [[Continuous iff Mapping at Limit Inferior Precedes Limit Inferior of Composition of Mapping and Sequence]]:
+:for all [[Definition:Directed Set|directed set]] $\left({D, \precsim}\right)$ and [[Definition:Moore-Smith Sequence|Moore-Smith sequence]] $N:D \to S$ in $S$: $f\left({\liminf N}\right) \preceq_2 \liminf\left({f \circ N}\right)$
+Thus by [[Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema]]:
+:$f$ [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
+{{qed|lemma}}
+=== Necessary Condition ===
+Thus by [[Mapping Preserves Directed Suprema implies Mapping is Continuous]]:
+:if $f$ [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]], then $f$ is [[Definition:Continuous (Topology)|continuous]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Universal Property of Quotient of Topological Group}
+Tags: Topological Groups, Universal Properties
+
+\begin{theorem}
+Let $G$ and $H$ be [[Definition:Topological Group|topological groups]].
+Let $N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
+Let $\pi : G \to G/N$ be the [[Definition:Quotient Mapping|quotient mapping]].
+Let $f : G \to H$ be a [[Definition:Continuous Mapping (Topology)|continuous]] [[Definition:Group Homomorphism|group homomorphism]] whose [[Definition:Kernel of Group Homomorphism|kernel]] contains $N$.
+Then there exists a unique [[Definition:Continuous Mapping (Topology)|continuous]] [[Definition:Group Homomorphism|group homomorphism]] $\overline f : G/N \to H$ such that $f = \overline f \circ \pi$.
+\end{theorem}
+
+\begin{proof}
+Because $N\subset \operatorname{ker} f$, $f$ is constant on the [[Definition:Coset|cosets]] of $N$.
+Thus $f$ is [[Definition:Invariant Under Equivalence Relation|invariant under]] [[Definition:Congruence Modulo Subgroup|congruence modulo]] $N$.
+By [[Universal Property of Quotient Set]], there exists a unique [[Definition:Mapping|mapping]] $\overline f : G/N \to H$ such that $f = \overline f \circ \pi$.
+It suffices to verify that it is a [[Definition:Continuous Mapping (Topology)|continuous]] [[Definition:Group Homomorphism|group homomorphism]].
+By [[Universal Property of Quotient Space]], there exists a [[Definition:Continuous Mapping (Topology)|continuous mapping]] $\overline g : G/N \to H$ such that $f = \overline g \circ \pi$.
+By uniqueness of $f$, $g=f$.
+By [[Universal Property of Quotient Group]], there exists a [[Definition:Group Homomorphism|group homomorphism]] $\overline h : G/N \to H$ such that $f = \overline h \circ \pi$.
+By uniqueness of $f$, $h=f$.
+{{qed}}
+[[Category:Topological Groups]]
+[[Category:Universal Properties]]
+k5zu4pdsabc95m0js903eifs2c2vpgq
+\end{proof}<|endoftext|>
+\section{Universal Property of Quotient Set}
+Tags: Equivalence Relations, Universal Properties
+
+\begin{theorem}
+Let $X$ and $Y$ be [[Definition:Set|sets]].
+Let $\sim$ be an [[Definition:Equivalence Relation|equivalence relation]] on $X$.
+Let $\pi : X \to X/\sim$ be the [[Definition:Quotient Mapping|quotient mapping]] to the [[Definition:Quotient Set|quotient set]].
+Let $f : X \to Y$ be [[Definition:Invariant Mapping Under Equivalence Relation|$\sim$-invariant]].
+Then there exists a unique [[Definition:Mapping|mapping]] $\overline f : X/\!\sim \to Y$ such that $f = \overline f \circ \pi$.
+:$\xymatrix{
+X \ar[d]_\pi \ar[r]^f & Y\\
+X/\sim \ar@{.>}[ru]_{\overline f} }$
+\end{theorem}
+
+\begin{proof}
+=== Existence ===
+For $x \in X$, let $[x] \in X/\sim$ denote its [[Definition:Equivalence Class|equivalence class]] under $\sim$.
+Define the [[Definition:Relation|relation]] $\overline f : X/\sim \to Y$ by:
+:$([x], y) \in \overline f \iff f(x) = y$.
+We check that this is a [[Definition:Well-Defined Mapping|well-defined mapping]].
+By [[Quotient Mapping is Surjection]], $f$ is [[Definition:Left-Total Relation|left-total]].
+Let $[x] \in X/\sim$ and $y, z \in Y$ with:
+:$([x], y) \in \overline f$
+:$([x], z) \in \overline f$
+We show that $y = z$.
+By definition of $\overline f$, there exist $u, v \in X$ with $[u] = [v] = [x]$ and $f(u) = y$ and $f(v) = z$.
+By definition of [[Definition:Equivalence Class|equivalence class]], $u \sim v$.
+Because $f$ is [[Definition:Invariant Mapping Under Equivalence Relation|$\sim$-invariant]], $y = z$.
+Thus $\overline f$ is [[Definition:Many-to-One Relation|many-to-one]].
+Thus $\overline f$ is a [[Definition:Well-Defined Mapping|well-defined mapping]].
+{{qed|lemma}}
+=== Uniqueness ===
+Follows from:
+* [[Quotient Mapping is Surjection]]
+* [[Surjection iff Epimorphism in Category of Sets]]
+{{qed}}
+[[Category:Equivalence Relations]]
+[[Category:Universal Properties]]
+0pajq8g4foujgbs0tcgtmdtt5vow2hf
+\end{proof}<|endoftext|>
+\section{Continuous iff Way Below iff There Exists Element that Way Below and Way Below}
+Tags: Topological Order Theory, Way Below Relation, Continuous Lattices
+
+\begin{theorem}
+Let $\struct {S, \preceq_1, \tau_1}$ and $\struct {T, \preceq_2, \tau_2}$ be [[Definition:Complete Lattice|complete]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Scott Topology|Scott topologies]].
+Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
+Then $f$ is [[Definition:Continuous (Topology)|continuous]] {{iff}}
+:$\forall x \in S, y \in T: y \ll \map f x \iff \exists w \in S: w \ll x \land y \ll \map f w$
+{{explain|link to definition of $\ll$ and $\gg$}}
+\end{theorem}
+
+\begin{proof}
+=== Sufficient Condition ===
+Assume that
+:$f$ is [[Definition:Continuous (Topology)|continuous]].
+By [[Continuous iff Directed Suprema Preserving]]:
+:$f$ [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
+By [[Directed Suprema Preserving Mapping at Element is Supremum]]:
+:$\forall x \in S: \map f x = \sup \set {\map f w: w \in S \land w \ll x}$
+Thus by [[Mapping at Element is Supremum implies Way Below iff There Exists Element that Way Below and Way Below]]:
+:$\forall x \in S, y \in T: y \ll \map f x \iff \exists w \in S: w \ll x \land y \ll \map f w$
+{{qed|lemma}}
+=== Necessary Condition ===
+Assume that
+:$\forall x \in S, y \in T: y \ll \map f x \iff \exists w \in S: w \ll x \land y \ll \map f w$
+Let $U$ be an [[Definition:Open Set (Topology)|open]] [[Definition:Subset|subset]] of $T$.
+By definition of [[Definition:Scott Topology|Scott topology]]:
+:$U$ is [[Definition:Upper Set|upper]].
+By definition of [[Definition:Interior (Topology)|interior]]:
+:$\paren {f^{-1} \sqbrk U}^\circ \subseteq f^{-1} \sqbrk U$
+We will prove that
+:$f^{-1} \sqbrk U \subseteq \paren {f^{-1} \sqbrk U}^\circ$
+Let $x \in f^{-1} \sqbrk U$
+By definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
+:$p := \map f x \in U$
+By [[Open implies There Exists Way Below Element]]:
+:$\exists u \in T: u \ll p \land u \in U$
+By assumption:
+:$\exists w \in S: w \ll x \land u \ll \map f w$
+We will prove that
+:$f \sqbrk {w^\gg} \subseteq U$
+Let $h \in f \sqbrk {w^\gg}$
+By definition of [[Definition:Image of Subset under Mapping|image of set]]:
+:$\exists z \in w^\gg: h = \map f z$
+By definition of [[Definition:Way Above Closure|way above closure]]:
+:$w \ll z$
+By assumption:
+:$u \ll h$
+By [[Way Below implies Preceding]]:
+:$u \preceq_2 h$
+Thus by definition of [[Definition:Upper Set|upper set]]:
+:$h \in U$
+{{qed|lemma}}
+By [[Image of Subset under Relation is Subset of Image/Corollary 3]]:
+:$f^{-1} \sqbrk {f \sqbrk {w^\gg} } \subseteq f^{-1} \sqbrk U$
+By [[Preimage of Image under Left-Total Relation is Superset]]:
+:$w^\gg \subseteq \paren {f^{-1} \circ f} \sqbrk {w^\gg}$
+By definition of [[Definition:Composition of Relations|composition of relations]]:
+:$w^\gg \subseteq f^{-1} \sqbrk {f \sqbrk {w^\gg} }$
+By [[Subset Relation is Transitive]]:
+:$w^\gg \subseteq f^{-1} \sqbrk U$
+By [[Interior is Union of Way Above Closures]]:
+:$\paren {f^{-1} \sqbrk U}^\circ = \bigcup \set {g^\gg: g \in S \land g^\gg \subseteq f^{-1} \sqbrk U}$
+By definition of [[Definition:Way Above Closure|way above closure]]:
+:$x \in w^\gg$
+We have:
+:$w^\gg \in \set {g^\gg: g \in S \land g^\gg \subseteq f^{-1} \sqbrk U}$
+Thus by definition of [[Definition:Set Union/General Definition|union]]:
+:$x \in \paren {f^{-1} \sqbrk U}^\circ$
+{{qed|lemma}}
+By definition of [[Definition:Set Equality|set equality]]:
+:$\paren {f^{-1} \sqbrk U}^\circ = f^{-1} \sqbrk U$
+Thus by definition of [[Definition:Interior (Topology)|interior]]:
+:$f^{-1} \sqbrk U$ is [[Definition:Open Set (Topology)|open]].
+Hence $f$ is [[Definition:Continuous (Topology)|continuous]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Finite Symmetric Group is Ambivalent}
+Tags: Ambivalent Groups
+
+\begin{theorem}
+Let $n$ be a [[Definition:Natural Number|natural number]].
+Let $S_n$ be a [[Definition:Symmetric Group|symmetric group]] of [[Definition:Order of Structure|order $n$]].
+Then $S_n$ is [[Definition:Ambivalent Group|ambivalent]].
+\end{theorem}
+
+\begin{proof}
+The [[Conjugacy Classes of Symmetric Group]] are determined uniquely by the [[Definition:Cycle Type|cycle type]] of the [[Definition:Element|elements]].
+Since any element in $S_n$ is of the same [[Definition:Cycle Type|cycle type]] with its [[Definition:Inverse Element|inverse]], they are in the same [[Definition:Conjugacy Class|conjugacy class]].
+Hence they are [[Definition:Conjugate of Group Element|conjugates]] of each other.
+This implies that $S_n$ is [[Definition:Ambivalent Group|ambivalent]].
+{{qed}}
+[[Category:Ambivalent Groups]]
+8dh89ilrc2bqnqj6sgtu3z57lhhvwpt
+\end{proof}<|endoftext|>
+\section{Alternating Groups that are Ambivalent}
+Tags: Alternating Groups, Ambivalent Groups
+
+\begin{theorem}
+Let $n$ be a [[Definition:Natural Number|natural number]].
+Then the $n$th [[Definition:Alternating Group|alternating group]] $A_n$ is [[Definition:Ambivalent Group|ambivalent]] {{iff}} $n\in \{1, 2, 5, 6, 10, 14\}$.
+{{OEIS|A115200}}
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Seems related to the representation theory of alternating group.}}
+[[Category:Alternating Groups]]
+[[Category:Ambivalent Groups]]
+ewk30s3y2q0fjb0jzhk6sjyyw9szdng
+\end{proof}<|endoftext|>
+\section{Transitivity of Big-O Estimates/General}
+Tags: Asymptotic Notation
+
+\begin{theorem}
+Let $X$ be a [[Definition:Topological Space|topological space]].
+Let $V$ be a [[Definition:Normed Vector Space|normed vector space]] over $\R$ or $\C$ with [[Definition:Norm on Vector Space|norm]] $\norm {\,\cdot\,}$.
+Let $f, g, h: X \to V$ be functions.
+Let $x_0 \in X$.
+Let $f = \map \OO g$ and $g = \map \OO h$ as $x \to x_0$, where $\OO$ denotes [[Definition:Big-O Notation|big-O notation]].
+Then $f = \map \OO h$ as $x \to x_0$.
+\end{theorem}
+
+\begin{proof}
+Because $f = \map \OO g$ and $g = \map \OO h$, there exist [[Definition:Neighborhood of Point in Topological Space|neighborhoods]] $U$ and $V$ of $x_0$ and [[Definition:Real Number|real numbers]] $c, d \ge 0$ such that:
+:$\norm {\map f x} \le c \cdot \norm {\map g x}$ for all $x \in U$
+:$\norm {\map g x} \le d \cdot \norm {\map h x}$ for all $x \in V$.
+By [[Intersection of Neighborhoods in Topological Space is Neighborhood]], $U\cap V$ is a [[Definition:Neighborhood of Point in Topological Space|neighborhood]] of $x_0$.
+For $x \in U \cap V$, we have:
+:$\norm {\map f x} \le c \cdot \norm {\map g x} \le c d \cdot \norm {\map h x}$
+Thus $f = \map \OO h$ for $x \to x_0$.
+{{qed}}
+[[Category:Asymptotic Notation]]
+2q0vllwo8bhm4dwet2s8v688obgsd8e
+\end{proof}<|endoftext|>
+\section{Transitivity of Big-O Estimates/Sequences}
+Tags: Asymptotic Notation
+
+\begin{theorem}
+Let $\sequence {a_n}$, $\sequence {b_n}$ and $\sequence {c_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
+Let $a_n = \map \OO {\sequence {b_n} }$ and $b_n = \map \OO {\sequence {c_n} }$, where $O$ denotes [[Definition:Big-O Notation for Sequences|big-O notation]].
+Then $a_n = \map \OO {\sequence {c_n} }$.
+\end{theorem}
+
+\begin{proof}
+Because $a_n = \map \OO {\sequence {b_n} }$, there exists $K \ge 0$ and $n_0 \in \N$ such that $\size {a_n} \le K \cdot \size {b_n}$ for $n \ge n_0$.
+Because $b_n = \map \OO {\sequence {c_n} }$, there exists $L \ge 0$ and $n_1 \in \N$ such that $\size {b_n} \le L \cdot \size {c_n}$ for $n \ge n_1$.
+Then $\size {a_n} \le K L \cdot \size {c_n}$ for $n \ge \max \set {n_0, n_1}$.
+Thus $a_n = \map \OO {\sequence {c_n} }$.
+{{qed}}
+[[Category:Asymptotic Notation]]
+irejv5z2pk0odh7n08v6of2ee5kohph
+\end{proof}<|endoftext|>
+\section{Conjugate of Cycle}
+Tags: Cyclic Permutations, Conjugacy
+
+\begin{theorem}
+Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
+Let $S_n$ be the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
+Let $\pi, \sigma \in S_n$.
+Let $\sigma$ be a [[Definition:Cyclic Permutation|cycle]] of [[Definition:Length of Cyclic Permutation|length]] $k$.
+Then the [[Definition:Conjugate of Group Element|conjugate]] $\pi \sigma \pi^{-1}$ is a [[Definition:Cyclic Permutation|cycle]] of [[Definition:Length of Cyclic Permutation|length]] $k$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Cyclic Permutations]]
+[[Category:Conjugacy]]
+kb93zsmofjdom32g3wdt12j1pdxo3hx
+\end{proof}<|endoftext|>
+\section{Center of Group is Kernel of Conjugacy Action}
+Tags: Conjugacy Action, Centers of Groups
+
+\begin{theorem}
+Let $G$ be a [[Definition:Group|group]].
+Let $Z$ be the [[Definition:Kernel of Group Action|kernel]] of the [[Definition:Conjugacy Action|conjugacy action]].
+Then $Z$ is the [[Definition:Center of Group|center]] of $G$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | o =
+ | r = x \text { is in the kernel of the conjugacy action}
+}}
+{{eqn | ll= \leadstoandfrom
+ | lo= \forall y \in G:
+ | o =
+ | r = x y x^{-1} = y
+}}
+{{eqn | ll= \leadstoandfrom
+ | lo= \forall y \in G:
+ | o =
+ | r = x y = y x
+}}
+{{eqn | ll= \leadstoandfrom
+ | o =
+ | r = x \text { is in the center of } G
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Conjugacy Action]]
+[[Category:Centers of Groups]]
+14ofzoc6s3hossn3pl04dwpwwa2qytd
+\end{proof}<|endoftext|>
+\section{Little-O Implies Big-O/General Result}
+Tags: Asymptotic Notation, Order Notation
+
+\begin{theorem}
+Let $X$ be a [[Definition:Topological Space|topological space]].
+Let $V$ be a [[Definition:Normed Vector Space|normed vector space]] over $\R$ or $\C$ with [[Definition:Norm on Vector Space|norm]] $\norm {\,\cdot\,}$
+Let $f, g: X \to V$ be [[Definition:Mapping|mappings]].
+Let $x_0 \in X$.
+Let $f = \map o g$ as $x \to x_0$, where $o$ denotes [[Definition:Little-O|little-O notation]].
+Then $f = \map \OO g$ as $x \to x_0$, where $\OO$ denotes [[Definition:Big-O|big-$\OO$ notation]].
+\end{theorem}
+
+\begin{proof}
+From the definition of [[Definition:Little-O|little-O notation]]:
+:there exists a [[Definition:Neighborhood of Point in Topological Space|neighborhood]] $U$ of $x_0$ such that $\norm {\map f x} \le \norm {\map g x}$ for all $x \in U$.
+By definition of [[Definition:Big-O|big-$\OO$ notation]], $f = \map \OO g$ as $x \to x_0$.
+{{qed}}
+[[Category:Asymptotic Notation]]
+[[Category:Order Notation]]
+m96cmklqesjtl62cy9wwf2wxtn5d0wc
+\end{proof}<|endoftext|>
+\section{Little-O Implies Big-O/Sequences}
+Tags: Asymptotic Notation
+
+\begin{theorem}
+Let $\sequence {a_n}$ and $\sequence {b_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
+Let $a_n = \map o {b_n}$ where $o$ denotes [[Definition:Little-O Notation|little-O notation]].
+Then $a_n = \map \OO {b_n}$ where $\OO$ denotes [[Definition:Big-O Notation|big-$\OO$ notation]].
+\end{theorem}
+
+\begin{proof}
+Because $a_n = \map o {b_n}$, there exists $n_0 \in \N$ such that $\size {a_n} \le 1 \cdot \size {b_n}$ for $n \ge n_0$.
+Thus $a_n = \map \OO {b_n}$.
+{{qed}}
+[[Category:Asymptotic Notation]]
+sihfpytj7955n6e5181hd2fyhtved9i
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Change of Basis Matrix}
+Tags: Linear Algebra, Matrix Theory
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $G$ be a [[Definition:Dimension (Linear Algebra)|finite-dimensional]] [[Definition:Unitary Module|unitary $R$-module]].
+Let $A = \sequence {a_n}$ and $B = \sequence {b_n}$ be [[Definition:Ordered Basis|ordered bases]] of $G$.
+{{TFAE|def = Change of Basis Matrix}}
+\end{theorem}
+
+\begin{proof}
+It will be shown that the two [[Definition:Matrix|matrices]] defined are equal [[Definition:Column of Matrix|column-wise]].
+Let $\displaystyle b_i = \sum_{j \mathop = 1}^n c_{i j} a_j$ for $i$ ranging from $1$ to $n$, where $c_{i j}$'s are [[Definition:Scalar (Module)|scalars]].
+The [[Definition:Unique|uniqueness]] of the above expression is justified by [[Expression of Vector as Linear Combination from Basis is Unique]].
+Then by definition of [[Definition:Coordinate Vector|coordinate vector]], the $i$th [[Definition:Column of Matrix|column of the matrix]] defined in [[Definition:Change of Basis Matrix/Definition 1|definition 1]] is:
+:$\begin {bmatrix} c_{i 1} & c_{i 2} & \ldots & c_{in} \end {bmatrix}^T$
+We also have:
+:$\displaystyle \map {I_G} {b_i} = b_i = \sum_{j \mathop = 1}^n c_{i j} a_j$
+So by definition of [[Definition:Relative Matrix|relative matrix]], the $i$th [[Definition:Column of Matrix|column of the matrix]] defined in [[Definition:Change of Basis Matrix/Definition 2|definition 2]] is:
+:$\begin {bmatrix} c_{i 1} & c_{i 2} & \ldots & c_{i n} \end {bmatrix}^T$
+The two [[Definition:Matrix|matrices]] are equal, so the two definitions are [[Definition:Logical Equivalence|equivalent]].
+{{qed}}
+[[Category:Linear Algebra]]
+[[Category:Matrix Theory]]
+18lwx8an6rap6pmfy0ne495dys719ed
+\end{proof}<|endoftext|>
+\section{Substitution in Big-O Estimate/General Result}
+Tags: Asymptotic Notation
+
+\begin{theorem}
+Let $X$ and $Y$ be [[Definition:Topological Space|topological spaces]].
+Let $V$ be a [[Definition:Normed Vector Space|normed vector space]] over $\R$ or $\C$ with [[Definition:Norm on Vector Space|norm]] $\norm {\,\cdot\,}$.
+Let $x_0 \in X$ and $y_0 \in Y$.
+Let $f: X \to Y$ be a function with $\map f {x_0} = y_0$ that is [[Definition:Continuous Mapping at Point (Topology)|continuous]] at $x_0$.
+Let $g, h: Y \to V$ be functions.
+Suppose $\map g y = \map O {\map h y}$ as $y \to y_0$, where $O$ denotes [[Definition:Big-O Notation|big-O notation]].
+Then $\map {\paren {g \circ f} } x = \map O {\map {\paren {h \circ f} } x}$ as $x \to x_0$.
+\end{theorem}
+
+\begin{proof}
+Because $g = \map O h$, there exists a [[Definition:Neighborhood of Point in Topological Space|neighborhood]] $V$ of $y_0$ and a [[Definition:Real Number|real number]] $c$ such that:
+:$\norm {\map g x} \le c \cdot \norm {\map h x}$ for all $y \in V$.
+By definition of [[Definition:Continuous Mapping at Point (Topology)|continuity]], there exists a [[Definition:Neighborhood of Point in Topological Space|neighborhood]] $U$ of $x_0$ with $\map f U \subset V$.
+For $x \in U$, we have:
+:$\norm {\map g {\map f x} } \le c \cdot \norm {\map h {\map f x} }$
+Thus $g \circ f = \map O {h \circ f}$ as $x \to x_0$.
+{{qed}}
+[[Category:Asymptotic Notation]]
+9s3z4pj7o5jlquyp13g5r1u6ap5193l
+\end{proof}<|endoftext|>
+\section{Substitution in Big-O Estimate/Sequences}
+Tags: Asymptotic Notation
+
+\begin{theorem}
+Let $(a_n)$ and $(b_n)$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
+Let $a_n = O(b_n)$ where $O$ denotes [[Definition:Big-O Notation|big-O notation]].
+Let $(n_k)$ be a [[Definition:Divergent Sequence|diverging sequence]] of [[Definition:Natural Number|natural numbers]].
+Then $a_{n_k} = O(b_{n_k})$.
+\end{theorem}
+
+\begin{proof}
+Because $a_n = O(b_n)$, there exists $M\geq0$ and $n_0 \in\N$ such that $|a_n| \leq M \cdot |b_n|$ for $n\geq n_0$.
+Because $n_k$ [[Definition:Divergent Sequence|diverges]], there exists $k_0\in\N$ such that $n_k\geq n_0$ for $k\geq k_0$.
+Then $|a_{n_k}| \leq M\cdot |b_{n_k}|$ for $k\geq k_0$.
+Thus $a_{n_k} = O(b_{n_k})$.
+{{qed}}
+[[Category:Asymptotic Notation]]
+q917p4cyyh16q4kjx1g18nd1hkq82oy
+\end{proof}<|endoftext|>
+\section{First Sequence of Three Consecutive Strictly Decreasing Euler Phi Values}
+Tags: Euler Phi Function
+
+\begin{theorem}
+The first [[Definition:Integer Sequence|sequence]] of $3$ consecutive [[Definition:Positive Integer|positive integers]] whose [[Definition:Euler Phi Function|Euler $\phi$ values]] are [[Definition:Strictly Decreasing Sequence|strictly decreasing]] is:
+:$\map \phi {523} > \map \phi {524} > \map \phi {525}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \map \phi {523}
+ | r = 522
+ | c = [[Euler Phi Function of Prime]]
+}}
+{{eqn | l = \map \phi {524}
+ | r = 260
+ | c = {{EulerPhiLink|524}}
+}}
+{{eqn | l = \map \phi {525}
+ | r = 240
+ | c = {{EulerPhiLink|525}}
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that this is the first such.}}
+\end{proof}<|endoftext|>
+\section{Bases of Finitely Generated Free Module have Equal Cardinality}
+Tags: Module Theory
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+Let $M$ be a [[Definition:Free Module|free]] $R$-[[Definition:Module|module]].
+Let $M$ be [[Definition:Finitely Generated Module|finitely generated]].
+Let $B$ and $C$ be [[Definition:Basis of Module|bases]] of $M$.
+Then $B$ and $C$ are [[Definition:Finite|finite]] and have the same [[Definition:Cardinality of Finite Set|cardinality]].
+\end{theorem}<|endoftext|>
+\section{Bases of Free Module have Equal Cardinality}
+Tags: Free Modules
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+Let $M$ be a [[Definition:Free Module|free]] $R$-[[Definition:Module|module]].
+Let $B$ and $C$ be [[Definition:Basis of Module|bases]] of $M$.
+Then $B$ and $C$ are [[Definition:Equinumerous|equinumerous]].
+\end{theorem}
+
+\begin{proof}
+By definition, a [[Definition:Basis of Module|basis]] is a [[Definition:Generator of Module|generator]].
+By [[Basis of Free Module is No Greater than Generator]], there exist:
+:an [[Definition:Injection|injection]] $f : B \to C$
+:an [[Definition:Injection|injection]] $g : C \to B$
+By the [[Cantor-Bernstein-Schröder Theorem]], $B$ and $C$ are [[Definition:Set Equivalence|equivalent]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sequences of Three Consecutive Strictly Increasing Euler Phi Values}
+Tags: Euler Phi Function
+
+\begin{theorem}
+The following [[Definition:Integer Sequence|sequences]] of $3$ consecutive [[Definition:Positive Integer|positive integers]] have the property that their [[Definition:Euler Phi Function|Euler $\phi$ values]] are [[Definition:Strictly Increasing Sequence|strictly increasing]]:
+:$\tuple {105, 106, 107}, \tuple {165, 166, 167}, \tuple {315, 316, 317}, \tuple {525, 526, 527}, \dots$
+{{expand|Related sequences: A161962 (superset), A161963}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \map \phi {525}
+ | r = 240
+ | c = {{EulerPhiLink|525}}
+}}
+{{eqn | l = \map \phi {526}
+ | r = 262
+ | c = {{EulerPhiLink|526}}
+}}
+{{eqn | l = \map \phi {527}
+ | r = 480
+ | c = {{EulerPhiLink|527}}
+}}
+{{end-eqn}}
+{{finish}}
+\end{proof}<|endoftext|>
+\section{Basis of Vector Space Injects into Generator}
+Tags: Vector Spaces
+
+\begin{theorem}
+Let $K$ be a [[Definition:Division Ring|division ring]].
+Let $V$ be a [[Definition:Vector Space|vector space]] over $K$.
+Let $B$ be a [[Definition:Basis of Vector Space|basis]] of $V$.
+Let $G$ be a [[Definition:Generator of Vector Space|generator]] of $V$.
+Then there exists an [[Definition:Injection|injection]] from $B$ to $G$.
+\end{theorem}
+
+\begin{proof}
+By [[Vector Space has Basis between Linearly Independent Set and Spanning Set]], there exists a [[Definition:Basis of Vector Space|basis]] $C \subset G$.
+By [[Bases of Vector Space have Equal Cardinality]], there exists a [[Definition:Bijection|bijection]] between $B$ and $C$.
+By [[Composite of Injections is Injection]], composing this [[Definition:Bijection|bijection]] with the [[Definition:Inclusion Mapping|inclusion]] of $C$ in $G$, we obtain an [[Definition:Injection|injection]] from $B$ to $G$.
+{{qed}}
+[[Category:Vector Spaces]]
+o50qamytl06tbrbu0zgnt3j68uncmuj
+\end{proof}<|endoftext|>
+\section{Increasing Union of Ideals is Ideal/Sequence}
+Tags: Set Union, Ideal Theory
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
+Let $S_0 \subseteq S_1 \subseteq S_2 \subseteq \dotsb \subseteq S_i \subseteq \dotsb$ be [[Definition:Ideal of Ring|ideals]] of $R$.
+Then the [[Definition:Increasing Union|increasing union]] $S$:
+:$\displaystyle S = \bigcup_{i \mathop \in \N} S_i$
+is an [[Definition:Ideal of Ring|ideal]] of $R$.
+\end{theorem}
+
+\begin{proof}
+Let $\displaystyle S = \bigcup_{i \mathop \in \N} S_i$.
+From [[Increasing Union of Subrings is Subring]], we have that $S$ is a [[Definition:Subring|subring]] of $R$.
+Now we need to show that it is an [[Definition:Ideal of Ring|ideal]] of $R$.
+Let $a \in S$.
+Then $\exists i \in \N: a \in S_i$.
+Let $b \in R$.
+Then $a b \in S_i$ and $b a \in S_i$, as $S_i$ is an [[Definition:Ideal of Ring|ideal]] of $R$.
+Thus $a b \in S$ and $b a \in S$.
+So $S$ is an [[Definition:Ideal of Ring|ideal]] of $R$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Increasing Union of Ideals is Ideal/Chain}
+Tags: Ring Theory, Ideal Theory
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
+Let $\left({P, \subseteq}\right)$ be the [[Definition:Ordered Set|ordered set]] consisting of all [[Definition:Ideal of Ring|ideals]] of $R$, ordered by [[Definition:Subset|inclusion]].
+Let $\left\{{I_\alpha}\right\}_{\alpha \in A}$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Chain (Set Theory)|chain]] of ideals in $P$.
+Let $\displaystyle I = \bigcup_{\alpha \in A} I_\alpha$ be their [[Definition:Set Union|union]].
+Then $I$ is an [[Definition:Ideal of Ring|ideal]] of $R$.
+\end{theorem}
+
+\begin{proof}
+===== Property 1: $0 \in I$ =====
+Since $\left\{{I_\alpha}\right\}_{\alpha \in A}$ is [[Definition:Non-Empty Set|non-empty]] [[Definition:Chain (Set Theory)|chain]], it must contain some [[Definition:Ideal of Ring|ideal]] $I_\beta$
+Since $I_\beta$ is an [[Definition:Ideal of Ring|ideal]], $0 \in I_\beta$.
+Thus $0 \in I$.
+===== Property 2: $x \in I \implies -x \in I$ =====
+If $x \in I$, then $x \in I_\beta$ for some $\beta\in A$.
+Since $I_\beta$ is an [[Definition:Ideal of Ring|ideal]], $-x \in I_\beta$.
+Thus $-x \in I$.
+===== Property 3: $x, y \in I \implies x + y \in I$ =====
+If $x,y \in I$, then $x \in I_\beta$ for some $\beta\in A$, and $y \in I_\gamma$ for some $\gamma \in A$.
+Since $\left\{{I_\alpha}\right\}_{\alpha \in A}$ is [[Definition:Total Ordering|totally ordered]], $I_\beta \subseteq I_\gamma$ or $I_\gamma\subseteq I_\beta$.
+{{WLOG}}, we can assume $I_\beta \subseteq I_\gamma$, which gives us $x,y \in I_\gamma$.
+Since $I_\gamma$ is an [[Definition:Ideal of Ring|ideal]], $x + y \in I_\gamma$.
+Thus $x + y \in I$.
+===== Property 4: $x \in I \land r \in R \implies rx, xr \in I$ =====
+If $x \in I$, then $x \in I_\beta$ for some $\beta\in A$.
+If $r \in R$, then since $I_\beta$ is an [[Definition:Ideal of Ring|ideal]], $rx, xr \in I_\beta$.
+Thus $rx, xr \in I$.
+Since $I$ satisfies these 4 properties, it is a [[Definition:Ideal of Ring|ideal]] of $R$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Union of Chain of Proper Ideals is Proper Ideal}
+Tags: Ring Theory, Ideal Theory
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $\struct {P, \subseteq}$ be the [[Definition:Ordered Set|ordered set]] consisting of all [[Definition:Ideal of Ring|ideals]] of $R$, ordered by [[Definition:Subset|inclusion]].
+Let $\sequence {I_\alpha}_{\alpha \mathop \in A}$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Chain (Set Theory)|chain]] of [[Definition:Proper Ideal of Ring|proper ideals]] in $P$.
+Let $\displaystyle I = \bigcup_{\alpha \mathop \in A} I_\alpha$ be their [[Definition:Set Union|union]].
+Then $I$ is a [[Definition:Proper Ideal of Ring|proper ideal]] of $R$.
+\end{theorem}
+
+\begin{proof}
+By [[Union of Chain of Ideals is Ideal]], $I$ is an [[Definition:Ideal of Ring|ideal]].
+It remains to show that $I \subsetneq R$.
+The [[Definition:Ideal of Ring|ideals]] $I_\alpha$ are all [[Definition:Proper Ideal of Ring|proper]], so none of them contain the [[Definition:Unity of Ring|unity]].
+Thus $I$ does not contain [[Definition:Unity of Ring|$1$]], which means $I \subsetneq R$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Dimension of Free Vector Space on Set}
+Tags: Vector Spaces
+
+\begin{theorem}
+Let $k$ be a [[Definition:Division Ring|division ring]].
+Let $X$ be a [[Definition:Set|set]].
+Let $k^{\paren X}$ be the [[Definition:Free Vector Space on Set|free vector space]] on $X$.
+The [[Definition:Vector Space|vector space]] $k^{\paren X}$ has [[Definition:Dimension of Vector Space|dimension]] the [[Definition:Cardinality|cardinality]] of $X$.
+\end{theorem}
+
+\begin{proof}
+Follows from:
+:[[Canonical Basis of Free Module on Set is Basis]]
+:[[Cardinality of Canonical Basis of Free Module on Set]]
+{{qed}}
+[[Category:Vector Spaces]]
+tfppnzuta22hqh19b9ep3xu1k0mgyru
+\end{proof}<|endoftext|>
+\section{Numbers not Expressible as Sum of Fewer than 19 Fourth Powers}
+Tags: Fourth Powers, Hilbert-Waring Theorem
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integer]] are the only ones which cannot be expressed as the [[Definition:Integer Addition|sum]] of fewer than $19$ [[Definition:Fourth Power|fourth powers]]:
+:$79, 159, 239, 319, 399, 479, 559$
+{{OEIS|A046050}}
+\end{theorem}
+
+\begin{proof}
+On a case-by-case basis:
+From [[Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers]]:
+{{:Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers}}
+From [[159 is not Expressible as Sum of Fewer than 19 Fourth Powers]]:
+{{:159 is not Expressible as Sum of Fewer than 19 Fourth Powers}}
+From [[239 is not Expressible as Sum of Fewer than 19 Fourth Powers]]:
+{{:239 is not Expressible as Sum of Fewer than 19 Fourth Powers}}
+From [[319 is not Expressible as Sum of Fewer than 19 Fourth Powers]]
+{{:319 is not Expressible as Sum of Fewer than 19 Fourth Powers}}
+From [[399 is not Expressible as Sum of Fewer than 19 Fourth Powers]]
+{{:399 is not Expressible as Sum of Fewer than 19 Fourth Powers}}
+From [[479 is not Expressible as Sum of Fewer than 19 Fourth Powers]]
+{{:479 is not Expressible as Sum of Fewer than 19 Fourth Powers}}
+From [[559 is not Expressible as Sum of Fewer than 19 Fourth Powers]]
+{{:559 is not Expressible as Sum of Fewer than 19 Fourth Powers}}
+{{ProofWanted|It needs to be shown that there are no other numbers known with this property.}}
+\end{proof}<|endoftext|>
+\section{239 is not Expressible as Sum of Fewer than 19 Fourth Powers}
+Tags: Fourth Powers, Hilbert-Waring Theorem, 239
+
+\begin{theorem}
+:$239 = 13 \times 1^4 + 4 \times 2^4 + 2 \times 3^4$
+\end{theorem}
+
+\begin{proof}
+First note that $4^4 = 256 > 239$.
+Then note that $3 \times 3^4 = 243 > 239$.
+Hence any expression of $239$ as [[Definition:Fourth Power|fourth powers]] uses no $n^4$ for $n \ge 4$, and uses not more than $2$ instances $3^4$.
+For the remainder, using $2^4$ uses fewer [[Definition:Fourth Power|fourth powers]] than $16$ instances $1^4$ does.
+Now we have:
+{{begin-eqn}}
+{{eqn | l = 239
+ | r = 2 \times 3^4 + 4 \times 2^4 + 13 \times 1^4
+}}
+{{eqn | r = 1 \times 3^4 + 9 \times 2^4 + 14 \times 1^4
+}}
+{{eqn | r = 0 \times 3^4 + 13 \times 2^4 + 15 \times 1^4
+}}
+{{end-eqn}}
+and it is apparent the first one uses the least number of [[Definition:Fourth Power|fourth powers]], at $19$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{479 is not Expressible as Sum of Fewer than 19 Fourth Powers}
+Tags: Fourth Powers, Hilbert-Waring Theorem, 479
+
+\begin{theorem}
+:$479 = 13 \times 1^4 + 3 \times 2^4 + 2 \times 3^4 + 4^4$
+or:
+:$479 = 10 \times 1^4 + 4 \times 2^4 + 5 \times 3^4$
+\end{theorem}
+
+\begin{proof}
+First note that $5^4 = 625 > 479$.
+Then note that $2 \times 4^4 = 512 > 479$.
+Hence any expression of $479$ as [[Definition:Fourth Power|fourth powers]] uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.
+For the remainder, using $2^4$ uses fewer [[Definition:Fourth Power|fourth powers]] than $16$ instances of $1^4$ does.
+Now we have:
+{{begin-eqn}}
+{{eqn | l = 479
+ | r = 4^4 + 2 \times 3^4 + 3 \times 2^4 + 13 \times 1^4
+}}
+{{eqn | r = 4^4 + 1 \times 3^4 + 8 \times 2^4 + 14 \times 1^4
+}}
+{{eqn | r = 4^4 + 0 \times 3^4 + 13 \times 2^4 + 15 \times 1^4
+}}
+{{eqn | r = 5 \times 3^4 + 4 \times 2^4 + 10 \times 1^4
+}}
+{{eqn | r = 4 \times 3^4 + 9 \times 2^4 + 11 \times 1^4
+}}
+{{eqn | r = 3 \times 3^4 + 14 \times 2^4 + 12 \times 1^4
+}}
+{{eqn | r = 2 \times 3^4 + 19 \times 2^4 + 13 \times 1^4
+}}
+{{eqn | r = 1 \times 3^4 + 24 \times 2^4 + 14 \times 1^4
+}}
+{{eqn | r = 0 \times 3^4 + 29 \times 2^4 + 15 \times 1^4
+}}
+{{end-eqn}}
+and it can be seen that the first and the fourth uses the least number of [[Definition:Fourth Power|fourth powers]], at $19$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{559 is not Expressible as Sum of Fewer than 19 Fourth Powers}
+Tags: Fourth Powers, Hilbert-Waring Theorem, 559
+
+\begin{theorem}
+:$559 = 15 \times 1^4 + 2 \times 2^4 + 2 \times 4^4$
+or:
+:$559 = 9 \times 1^4 + 4 \times 2^4 + 6 \times 3^4$
+\end{theorem}
+
+\begin{proof}
+First note that $5^4 = 625 > 559$.
+Then note that $3 \times 4^4 = 768 > 559$.
+Hence any expression of $559$ as [[Definition:Fourth Power|fourth powers]] uses no $n^4$ for $n \ge 5$, and uses not more than $2$ instances of $4^4$.
+For the remainder, using $2^4$ uses fewer [[Definition:Fourth Power|fourth powers]] than $16$ instances of $1^4$ does.
+Now we have:
+{{begin-eqn}}
+{{eqn | l = 559
+ | r = 2 \times 4^4 + 0 \times 3^4 + 2 \times 2^4 + 15 \times 1^4
+}}
+{{eqn | r = 1 \times 4^4 + 3 \times 3^4 + 3 \times 2^4 + 12 \times 1^4
+}}
+{{eqn | r = 1 \times 4^4 + 2 \times 3^4 + 8 \times 2^4 + 13 \times 1^4
+}}
+{{eqn | r = 1 \times 4^4 + 1 \times 3^4 + 13 \times 2^4 + 14 \times 1^4
+}}
+{{eqn | r = 1 \times 4^4 + 0 \times 3^4 + 18 \times 2^4 + 15 \times 1^4
+}}
+{{eqn | r = 6 \times 3^4 + 4 \times 2^4 + 9 \times 1^4
+}}
+{{eqn | r = 5 \times 3^4 + 9 \times 2^4 + 10 \times 1^4
+}}
+{{eqn | r = 4 \times 3^4 + 14 \times 2^4 + 11 \times 1^4
+}}
+{{eqn | r = 3 \times 3^4 + 19 \times 2^4 + 12 \times 1^4
+}}
+{{eqn | r = 2 \times 3^4 + 24 \times 2^4 + 13 \times 1^4
+}}
+{{eqn | r = 1 \times 3^4 + 29 \times 2^4 + 14 \times 1^4
+}}
+{{eqn | r = 0 \times 3^4 + 34 \times 2^4 + 15 \times 1^4
+}}
+{{end-eqn}}
+and it can be seen that the first and the sixth use the least number of [[Definition:Fourth Power|fourth powers]], at $19$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Free Module on Set is Free}
+Tags: Free Modules
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $I$ be a [[Definition:Set|set]].
+Let $R^{\paren I}$ be the [[Definition:Free Module on Set|free $R$-module on $I$]].
+Then $R^{\paren I}$ is a [[Definition:Free Module|free $R$-module]].
+\end{theorem}
+
+\begin{proof}
+From [[Canonical Basis of Free Module on Set is Basis]], $R^{\paren I}$ has a [[Definition:Basis of Module|basis]].
+{{qed}}
+[[Category:Free Modules]]
+09a67r1n3513hjmtc6sbhi8db56nhxo
+\end{proof}<|endoftext|>
+\section{Canonical Basis of Free Module on Set is Basis}
+Tags: Free Modules
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $I$ be a [[Definition:Set|set]].
+Let $R^{(I)}$ be the [[Definition:Free Module on Set|free $R$-module on $I$]].
+Let $B$ be its [[Definition:Canonical Basis of Free Module on Set|canonical basis]].
+Then $B$ is a [[Definition:Basis of Module|basis]] of $R^{(I)}$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Free Modules]]
+ngdn9dpydyzxfulu90d31j8vry2dutc
+\end{proof}<|endoftext|>
+\section{Korselt's Theorem}
+Tags: Carmichael Numbers
+
+\begin{theorem}
+Let $n \ge 2$ be an [[Definition:Integer|integer]].
+Then $n$ is a '''[[Definition:Carmichael Number|Carmichael number]]''' {{iff}}:
+: $(1): \quad n$ is [[Definition:Odd Integer|odd]]
+and the following conditions hold for every [[Definition:Prime Factor|prime factor]] $p$ of $n$:
+: $(2): \quad p^2 \nmid n$
+: $(3): \quad \paren {p - 1} \divides \paren {n - 1}$
+where:
+:$\divides$ denotes [[Definition:Divisor of Integer|divisibility]]
+:$\nmid$ denotes non-[[Definition:Divisor of Integer|divisibility]].
+\end{theorem}
+
+\begin{proof}
+=== Sufficient Condition ===
+Let $n$ be a [[Definition:Carmichael Number|Carmichael number]]:
+:$(4): \quad \forall a \in \Z: a \perp n: a^n \equiv a \pmod n$
+where $\perp$ denotes [[Definition:Coprime Integers|coprimality]].
+Suppose $n$ is [[Definition:Even Integer|even]].
+Set $a = -1$ in $(4)$.
+Then $\paren {-1}^n = 1$ and so:
+:$1 \equiv -1 \pmod n$
+resulting in $n = 2$.
+But as $2$ is not a [[Definition:Carmichael Number|Carmichael number]], it follows that for $n$ to be [[Definition:Carmichael Number|Carmichael]] it must be [[Definition:Odd Integer|odd]].
+Thus:
+: $(1): n$ is [[Definition:Odd Integer|odd]]
+holds.
+From the [[Fundamental Theorem of Arithmetic]], let $n$ be expressed as:
+:$n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$
+where:
+:the $p_i$'s are [[Definition:Distinct|distinct]] [[Definition:Odd Prime|odd primes]]
+:for all $i$, $k_i \ge 1$.
+By [[Integers with Primitive Roots]] there exists a [[Definition:Primitive Root|primitive root]] $a_i$ modulo $p_i^{k_i}$ for each $i$.
+In particular, we have that $a_i \perp p_i$.
+By the [[Chinese Remainder Theorem]]:
+:$\exists a: \forall i: a \equiv a_i \pmod {p_i^{k_i} }, a \perp n$
+Consider a particular $i$.
+We have:
+{{begin-eqn}}
+{{eqn | l = a^n
+ | o = \equiv
+ | r = a \pmod n
+ | c = from $(4)$
+}}
+{{eqn | ll= \leadsto
+ | l = a^{n - 1}
+ | o = \equiv
+ | r = 1 \pmod n
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = a^{n - 1}
+ | o = \equiv
+ | r = 1 \pmod {p_i^{k_i} }
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = {a_i}^{n - 1}
+ | o = \equiv
+ | r = 1 \pmod {p_i^{k_i} }
+ | c =
+}}
+{{end-eqn}}
+We have that $a_i$ is a [[Definition:Primitive Root|primitive root]] modulo $p_i^{k_i}$.
+Thus:
+{{begin-eqn}}
+{{eqn | l = \map {\operatorname {ord}_{p_i^{k_i} } } {a_i}
+ | r = \map \phi {p_i^{k_i} }
+ | c = where $\phi$ is the [[Definition:Euler Phi Function|Euler $\phi$ function]]
+}}
+{{eqn | r = p_i^{k_i - 1} \paren {p_i - 1}
+ | c =
+}}
+{{eqn | o = \divides
+ | r = n - 1
+ | c =
+}}
+{{end-eqn}}
+But because $p_i \divides n$:
+:$p_i \perp n - 1$
+Thus:
+:$k_i = 1$
+giving that $n$ is [[Definition:Square-Free|square-free]], and:
+:$p_i - 1 \divides n - 1$
+As $i$ is arbitrary, both conditions $(1)$ and $(2)$ of the statement of the theorem are thus fulfilled:
+That is:
+: $(1): n$ is [[Definition:Odd Integer|odd]]
+and for all [[Definition:Prime Number|primes]] $p$ such that $p \divides n$:
+: $(2): \quad p^2 \nmid n$
+: $(3): \quad \paren {p - 1} \divides \paren {n - 1}$
+{{qed|lemma}}
+=== Necessary Condition ===
+Suppose $n$ is such that:
+: $(1): n$ is [[Definition:Odd Integer|odd]]
+and for all [[Definition:Prime Number|primes]] $p$ such that $p \divides n$:
+: $(2): \quad p^2 \nmid n$
+: $(3): \quad \paren {p - 1} \divides \paren {n - 1}$
+Thus:
+: $n = p_1 p_2 \cdots p_r$
+is the [[Definition:Integer Multiplication|product]] of $r$ [[Definition:Distinct|distinct]] [[Definition:Odd Prime|odd primes]] such that:
+:$\paren {p_i - 1} \divides \paren {n - 1}$
+for all $i$.
+Let $a \in \Z$.
+Let $a \perp p_i$.
+Then by [[Fermat's Little Theorem]]:
+:$a^{p_i - 1} \equiv 1 \pmod {p_i}$
+Thus:
+:$a^{n - 1} \equiv 1 \pmod {p_i}$
+and so:
+:$a^n \equiv a \pmod {p_i}$
+Let $\gcd \set {a, p_i} > 1$.
+Then as $p_i$ is [[Definition:Prime Number|prime]]:
+:$p_i \divides a$
+and so:
+:$a^n \equiv a \equiv 0 \pmod {p_i}$
+Thus for all $i$ and for all $a \in \Z$:
+:$a^n \equiv a \pmod {p_i}$
+From the [[Chinese Remainder Theorem]]:
+:$a^n \equiv a \pmod n$
+for all $a$.
+Thus, by definition, $n$ is a [[Definition:Carmichael Number|Carmichael number]].
+{{qed}}
+{{Proofread}}
+\end{proof}<|endoftext|>
+\section{Carmichael Number has 3 Odd Prime Factors}
+Tags: Carmichael Numbers
+
+\begin{theorem}
+Let $n$ be a [[Definition:Carmichael Number|Carmichael number]].
+Then $n$ has at least $3$ [[Definition:Distinct|distinct]] [[Definition:Odd Prime|odd]] [[Definition:Prime Factor|prime factors]].
+\end{theorem}
+
+\begin{proof}
+By [[Korselt's Theorem]], $n$ is [[Definition:Odd Integer|odd]].
+Therefore $n$ has at least $1$ [[Definition:Odd Prime|odd]] [[Definition:Prime Factor|prime factor]].
+By [[Korselt's Theorem]], for each [[Definition:Prime Factor|prime factor]] of $n$:
+:$p^2 \nmid n$
+:$\paren {p - 1} \divides \paren {n - 1}$
+Suppose $n = p^k$ for some [[Definition:Odd Prime|odd]] [[Definition:Prime Number|prime]] $p$.
+By [[Korselt's Theorem]], $k = 1$.
+However by definition of a [[Definition:Carmichael Number|Carmichael Number]], $n$ cannot be [[Definition:Prime Number|prime]].
+Therefore $n$ has at least $2$ [[Definition:Distinct|distinct]] [[Definition:Odd Prime|odd]] [[Definition:Prime Factor|prime factors]].
+Suppose $n = p^a q^b$ for [[Definition:Distinct|distinct]] [[Definition:Odd Prime|odd]] [[Definition:Prime Number|primes]] $p$ and $q$.
+By [[Korselt's Theorem]], the following holds:
+:$a = b = 1$
+:$n = p q$
+:$\paren {p - 1} \divides \paren {n - 1}$
+:$\paren {q - 1} \divides \paren {n - 1}$
+Hence:
+{{begin-eqn}}
+{{eqn | l = \paren {p - 1}
+ | o = \divides
+ | r = \paren {n - 1 - q \paren {p - 1} }
+}}
+{{eqn | r = p q - 1 - p q + q
+}}
+{{eqn | r = q - 1
+}}
+{{end-eqn}}
+Swapping $p$ and $q$ yields $\paren {q - 1} \divides \paren {p - 1}$.
+Hence $p - 1 = q - 1$ and $p = q$, which is a [[Definition:Contradiction|contradiction]].
+Therefore $n$ has at least $3$ [[Definition:Distinct|distinct]] [[Definition:Odd Prime|odd]] [[Definition:Prime Factor|prime factors]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Intersection of Submodules is Submodule}
+Tags: Module Theory
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
+Let $M$ be an $R$-[[Definition:Module|module]].
+Let $S$ be a [[Definition:Set|set]] of [[Definition:Submodule|submodules]] of $M$.
+Then the [[Definition:Intersection|intersection]] $\bigcap S$ is a [[Definition:Submodule|submodule]] of $M$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Module Theory]]
+9awn5pi2z2b6ojl6mw92w24270sxhgs
+\end{proof}<|endoftext|>
+\section{There are Infinitely Many Carmichael Numbers}
+Tags: Carmichael Numbers
+
+\begin{theorem}
+There are [[Definition:Infinite Set|infinitely many]] [[Definition:Carmichael Number|Carmichael numbers]].
+\end{theorem}<|endoftext|>
+\section{Product of Big-O Estimates/Sequences}
+Tags: Asymptotic Notation
+
+\begin{theorem}
+Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
+Let:
+:$a_n = \map \OO {b_n}$
+:$c_n = \map \OO {d_n}$
+where $\OO$ denotes [[Definition:Big-O Notation|big-O notation]].
+Then:
+:$a_n c_n = \map \OO {b_n d_n}$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Asymptotic Notation]]
+gv2fsl4w3ps2083791hiislfjhiw1zw
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Order of Entire Function}
+Tags: Entire Functions
+
+\begin{theorem}
+Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]].
+Let $\ln$ denote the [[Definition:Natural Logarithm|natural logarithm]].
+{{TFAE| def = Order of Entire Function}}
+\end{theorem}
+
+\begin{proof}
+Let:
+:$\alpha_1 = \displaystyle \limsup_{R \mathop \to \infty} \frac {\displaystyle \ln \ln \max_{\cmod z \mathop \le R} \cmod f} {\ln R}$
+:$\alpha_2 = \inf \set {\beta \ge 0: \displaystyle \map \ln {\max_{\cmod z \mathop \le R} \cmod {\map f z} } = \map \OO {R^\beta} }$
+:$\alpha_3 = \inf \set {\beta \ge 0: \map f z = \map \OO {\map \exp {\cmod z^\beta} } }$
+=== 3 less than 2 ===
+Let $\beta > \alpha_2$.
+Then there exists $K > 0$ such that:
+:$\displaystyle \ln \max_{\cmod z \mathop \le R} \cmod f \le K \cdot R^\beta$
+for $R$ sufficiently large.
+That is:
+:$\dfrac {\ln \ln \max_{\cmod z \le R} \cmod f} {\ln R} \le \frac K {\ln R} + \beta$
+for $R$ sufficiently large.
+Taking $\limsup$, we get $\alpha_1 \le \beta$.
+Taking limits, $\alpha_2 \ge \alpha_1$.
+{{qed|lemma}}
+=== 2 less than 3 ===
+Let $\beta > \alpha_1$.
+We have $\displaystyle \ln \max_{\cmod z \mathop \le R} \cmod f \le R^\beta$ for $R$ sufficiently large.
+Thus $\beta \ge \alpha_2$.
+Taking limits, $\alpha_1 \ge \alpha_2$.
+{{qed|lemma}}
+=== 1 less than 2 ===
+Let $\beta > \alpha_2$.
+Let $\epsilon > 0$ be such that $\beta - \epsilon > \alpha_1$.
+Then:
+:$\displaystyle \map \ln {\max_{\cmod z \mathop \le R} \cmod {\map f z} } = \map \OO {R^{\beta - \epsilon} }$
+Then there exists $R_0 > 0$ such that:
+:$\displaystyle \map \ln {\max_{\cmod z \mathop \le R} \cmod {\map f z} } \le R^\beta$
+for $R \ge R_0$.
+By [[Exponential is Strictly Increasing]], this implies:
+:$\displaystyle \max_{\cmod z \mathop \le R} \cmod {\map f z} \le \map \exp {R^\beta}$
+Let $z \in \C$ with $\cmod z \ge R_0$.
+Then:
+:$\cmod {\map f z} \le \max_{\cmod w \le \cmod z} \cmod {\map f w} \le \map \exp {\cmod z^\beta}$
+Thus:
+:$\map f z = \map \OO {\map \exp {\cmod z^\beta} }$
+Thus $\beta \ge \alpha_3$.
+Taking the limit $\beta \to \alpha_2$, we obtain:
+:$\alpha_2 \ge \alpha_3$
+{{qed|lemma}}
+{{ProofWanted}}
+[[Category:Entire Functions]]
+l8pol3ujopmruaz3voa85cyvlb36j7f
+\end{proof}<|endoftext|>
+\section{Little-O Times Big-O is Little-O/Sequences}
+Tags: Asymptotic Notation
+
+\begin{theorem}
+Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
+Let:
+:$a_n = \map \OO {b_n}$
+:$c_n = \map o {d_n}$
+where:
+:$\OO$ denotes [[Definition:Big-O Notation|big-O notation]]
+:$o$ denotes [[Definition:Little-O Notation|little-o notation]].
+Then:
+:$a_n c_n = \map o {b_n d_n}$
+\end{theorem}
+
+\begin{proof}
+Let $\epsilon \in \R_{> 0}$.
+Since $a_n = \map \OO {b_n}$:
+:$\exists c \in \R: c \ge 0: \exists n_0 \in \N: \paren {n \ge n_0 \implies \size {a_n} \le c \cdot \size {b_n} }$
+Since $c_n = \map o {d_n}$:
+:$\exists n_1 \in \N: \paren {n \ge n_1 \implies \size {c_n} \le \dfrac \epsilon {c + 1} \cdot \size {d_n} }$
+Thus for $n \ge \max \set {n_0, n_1}$:
+{{begin-eqn}}
+{{eqn | o =
+ | r = \size {a_n c_n}
+}}
+{{eqn | r = \size {a_n} \size {c_n}
+}}
+{{eqn | o = \le
+ | r = \paren {c \cdot \size {b_n} } \paren {\frac \epsilon {c + 1} \cdot \size {d_n} }
+}}
+{{eqn | o = \le
+ | r = \size {b_n} \paren {\epsilon \cdot \size {d_n} }
+}}
+{{eqn | r = \epsilon \cdot \size {b_n d_n}
+}}
+{{end-eqn}}
+Thus $a_n c_n = \map o {b_n d_n}$.
+{{qed}}
+[[Category:Asymptotic Notation]]
+azifd1smnfq9aprpnmw1nhgif54bazq
+\end{proof}<|endoftext|>
+\section{Field Norm of Complex Number Equals Field Norm}
+Tags: Field Extensions, Field Norm of Complex Number
+
+\begin{theorem}
+Let $z = a + i b$ be a [[Definition:Complex Number|complex number]], where $a, b \in \R$.
+Then the [[Definition:Field Norm of Complex Number|field norm]] of $z$ is the [[Definition:Field Norm|field norm]] with respect to the [[Definition:Field Extension|field extension]] $\C / \R$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Field Extensions]]
+[[Category:Field Norm of Complex Number]]
+0i5v8vch6mvpd11ja0d3gdrgpmfcvbj
+\end{proof}<|endoftext|>
+\section{Big-O Notation for Sequences Coincides with General Definition}
+Tags: Asymptotic Notation
+
+\begin{theorem}
+Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
+Let $\N$ be given the [[Definition:Discrete Topology|discrete topology]].
+{{TFAE}}
+:$(1): \quad$ $a_n = O(b_n)$, where $O$ denotes [[Definition:Big-O Notation for Sequences|big-O notation for sequences]]
+:$(2): \quad$ $a_n = O(b_n)$, where $O$ stands for the [[Definition:O Notation/Big-O Notation/General Definition|general definition of big-O notation]]
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Asymptotic Notation]]
+34yd1nfcmie7stnlonf5x1tv3syz2dh
+\end{proof}<|endoftext|>
+\section{Pandigital Square Equation}
+Tags: Square Numbers, Recreational Mathematics
+
+\begin{theorem}
+The following [[Definition:Equation|equations]], which include each [[Definition:Digit|digit]] from $1$ to $9$ inclusive, are the only ones of their kind:
+{{begin-eqn}}
+{{eqn | l = 567^2
+ | r = 321 \, 489
+}}
+{{eqn | l = 854^2
+ | r = 729 \, 316
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+The [[Definition:Square (Algebra)|square]] of a $2$-[[Definition:Digit|digit]] [[Definition:Integer|integer]] cannot have more than $4$ [[Definition:Digit|digits]]:
+:$99^2 = 9801$
+The [[Definition:Square (Algebra)|square]] of a $4$-[[Definition:Digit|digit]] [[Definition:Integer|integer]] has at least $7$ [[Definition:Digit|digits]]:
+:$1000^2 = 1 \, 000 \, 000$
+Hence we only need to inspect $3$-[[Definition:Digit|digit]] [[Definition:Integer|integers]], with a corresponding $6$-[[Definition:Digit|digit]] [[Definition:Square (Algebra)|square]].
+A lower bound is given by $\ceiling {\sqrt {123456}} = 352$, where $123456$ is the smallest $6$-[[Definition:Digit|digit]] [[Definition:Integer|integer]] without repeating [[Definition:Digit|digits]] or $0$.
+An upper bound is given by $\floor {\sqrt {876543}} = 936$, where $876543$ is the largest $6$-[[Definition:Digit|digit]] [[Definition:Integer|integer]] without repeating [[Definition:Digit|digits]] or $9$.
+It is seen that it is not necessary to investigate $6$-[[Definition:Digit|digit]] [[Definition:Square Number|squares]] beginning with $9$, because its [[Definition:Square Root|square root]] would also begin with $9$.
+It is noted that [[Definition:Integer|integers]] ending in $1$, $5$ or $6$ have [[Definition:Square (Algebra)|squares]] ending in those same [[Definition:Digit|digits]].
+Such numbers can be eliminated from our search, as they will duplicate the appearance of those [[Definition:Digit|digits]].
+We also cannot have $0$ as a [[Definition:Digit|digit]].
+Moreover, observe that an [[Definition:Integer|integer]] $\bmod 9$ and its [[Definition:Square (Algebra)|square]] $\bmod 9$ have the following pattern:
+:$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
+\hline n \bmod 9 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
+\hline n^2 \bmod 9 & 0 & 1 & 4 & 0 & 7 & 7 & 0 & 4 & 1 \\
+\hline
+\end{array}$
+By [[Congruence of Sum of Digits to Base Less 1]], the sum of digits of $n$ and $n^2 \pmod 9$ are congruent to $n$ and $n^2 \pmod 9$ respectively.
+We must require their sum to be congruent to $0 \bmod 9$ since the sum of all their digits is:
+:$1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 \equiv 0 \pmod 9$
+Hence $n \equiv 0$ or $8 \pmod 9$.
+{{finish|There are $52$ numbers left to check:}}
+Those divisible by $9$:
+:$369, 378, 387, 423, 432, 459, 468, 513, 549, 567, 594, 612, 639, 648, 657, 684, 693, 729, 738, 783, 792, 819, 837, 864, 873, 918, 927$
+Those not divisible by $9$:
+:$359, 368, 413, 458, 467, 512, 539, 548, 584, 593, 629, 638, 647, 674, 683, 692, 719, 728, 764, 782, 827, 854, 863, 872, 917$
+\end{proof}<|endoftext|>
+\section{Bounded iff Big-O of 1/Sequences}
+Tags: Asymptotic Notation
+
+\begin{theorem}
+Let $\sequence {a_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
+{{TFAE}}
+:$(1): \quad a_n$ is [[Definition:Bounded Sequence|bounded]]
+:$(2): \quad a_n = \map \OO 1$, where $\OO$ denotes [[Definition:Big-O Notation for Sequences|big-$\OO$ notation]]
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = a_n
+ | o = \text {is}
+ | r = \text {bounded}
+}}
+{{eqn | ll = \leadstoandfrom
+ | lo = \exists k \in \R:
+ | l = \size {a_n}
+ | o = \le
+ | r = k
+ | c = {{Defof|Bounded Sequence}}
+}}
+{{eqn | ll = \leadstoandfrom
+ | lo = \exists k \in \R:
+ | l = \size {a_n}
+ | o = \le
+ | r = k \cdot \size 1
+}}
+{{eqn | ll = \leadstoandfrom
+ | l = a_n
+ | r = \map \OO 1
+ | c = {{Defof|O Notation|subdef = Big-O Notation/Sequence|Big-$\OO$ Notation}}
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Asymptotic Notation]]
+t642smig8dem4j9rdauopt8o0pzunlx
+\end{proof}<|endoftext|>
+\section{Sequence of 11 Primes by Trebling and Adding 16}
+Tags: Prime Numbers, Polynomial Expressions for Primes
+
+\begin{theorem}
+The process of [[Definition:Integer Multiplication|multiplication]] by $3$ and then adding $16$ produces a [[Definition:Integer Sequence|sequence]] of $11$ [[Definition:Prime Number|primes]] when starting from $587$:
+:$587, 1777, 5347, 16 \, 057, 48 \, 187, 144 \, 577, 433 \, 747, 1 \, 301 \, 257, 3 \, 903 \, 787, 11 \, 711 \, 377, 35 \, 134 \, 147$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | o =
+ | r = 587
+ | c = is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 3 \times 587 + 16
+ | r = 1777
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 3 \times 1777 + 16
+ | r = 5347
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 3 \times 5347 + 16
+ | r = 16 \, 057
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 3 \times 16 \, 057 + 16
+ | r = 48 \, 187
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 3 \times 48 \, 187 + 16
+ | r = 144 \, 577
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 3 \times 144 \, 577 + 16
+ | r = 433 \, 747
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 3 \times 433 \, 747 + 16
+ | r = 1 \, 301 \, 257
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 3 \times 1 \, 301 \, 257 + 16
+ | r = 3 \, 903 \, 787
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 3 \times 3 \, 903 \, 787 + 16
+ | r = 11 \, 711 \, 377
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 3 \times 11 \, 711 \, 377 + 16
+ | r = 35 \, 134 \, 147
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 3 \times 35 \, 134 \, 147 + 16
+ | r = 105 \, 402 \, 457
+ | c = which is not [[Definition:Prime Number|prime]] : $105 \, 402 \, 457 = 67 \times 137 \times 11 \, 483$
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum of Little-O Estimates/Sequences}
+Tags: Asymptotic Notation
+
+\begin{theorem}
+Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real]] or [[Definition:Complex Number|complex numbers]].
+Let:
+:$a_n = \map o {b_n}$
+:$c_n = \map o {d_n}$
+where $o$ denotes [[Definition:Little-O Notation for Sequences|little-o notation]].
+Then:
+:$a_n + c_n = \map o {\size {b_n} + \size {d_n} }$
+\end{theorem}
+
+\begin{proof}
+Let $\epsilon > 0$.
+Then by definition of [[Definition:Little-O Notation for Sequences|little-o notation]]:
+:$\exists n_1 \in \N: \paren {n \ge n_1 \implies \size {a_n} \le \epsilon \cdot \size {b_n}}$
+:$\exists n_2 \in \N: \paren {n \ge n_2 \implies \size {c_n} \le \epsilon \cdot \size {d_n}}$
+For $n \ge \max \set {n_1, n_2}$:
+{{begin-eqn}}
+{{eqn | l = \size {a_n + c_n}
+ | o = \le
+ | r = \size {a_n} + \size {c_n}
+ | c = [[Triangle Inequality]]
+}}
+{{eqn | o = \le
+ | r = \epsilon \cdot \size {b_n} + \epsilon \cdot \size {d_n}
+}}
+{{eqn | r = \epsilon \cdot \size {\size {b_n} + \size {d_n} }
+ | c = [[Definition:Absolute Value|Absolute value]] is [[Definition:Positive Real Number|positive]]
+}}
+{{end-eqn}}
+Hence by definition of [[Definition:Little-O Notation for Sequences|little-o notation]]:
+:$a_n + c_n = \map o {\size {b_n} + \size {d_n} }$
+{{qed}}
+[[Category:Asymptotic Notation]]
+kospwasornr8yzsr5oo9di0ssd1ev2u
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Asymptotically Equal Sequences}
+Tags: Asymptotic Notation
+
+\begin{theorem}
+Let $\sequence {a_n}$ and $\sequence {b_n}$ be [[Definition:Sequence|sequences in $\R$]].
+{{TFAE|def = Asymptotically Equal Sequences}}
+\end{theorem}
+
+\begin{proof}
+=== 2 iff 3 ===
+Let $a_n - b_n = \map o {b_n}$.
+Let $0 < \epsilon < 1/2$.
+Then
+{{begin-eqn}}
+{{eqn | l = \epsilon \cdot \size {b_n}
+ | o = \ge
+ | r = \size {a_n - b_n}
+ | c = For $n$ [[Definition:Sufficiently Large|sufficiently large]]
+}}
+{{eqn | r = \paren {1 - \epsilon} \size {a_n - b_n} + \epsilon \cdot \size {a_n - b_n}
+ | c =
+}}
+{{eqn | o = \ge
+ | r = \paren {1 - \epsilon} \size {a_n - b_n} - \epsilon \cdot \size {a_n} + \epsilon \cdot \size {b_n}
+ | c = [[Triangle Inequality]]
+}}
+{{end-eqn}}
+So $\size {a_n - b_n} \le \dfrac {\epsilon \cdot \size {a_n} } {1 - \epsilon} \le 2 \epsilon \cdot \size {a_n}$.
+Thus $a_n - b_n = \map o {a_n}$.
+The other implication follows by symmetry.
+{{qed|lemma}}
+=== 1 iff 2 ===
+{{begin-eqn}}
+{{eqn | l = \lim_{n \to \infty} \dfrac {a_n} {b_n}
+ | o = \to
+ | r = 1
+}}
+{{eqn | ll = \leadstoandfrom
+ | l = \lim_{n \to \infty} \paren {\dfrac {a_n} {b_n} - \dfrac {b_n} {b_n} }
+ | o = \to
+ | r = 0
+}}
+{{eqn | ll = \leadstoandfrom
+ | l = \lim_{n \to \infty} \dfrac {a_n - b_n} {b_n}
+ | o = \to
+ | r = 0
+}}
+{{eqn | ll = \leadstoandfrom
+ | l = a_n - b_n
+ | r = \map o {b_n}
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Asymptotic Notation]]
+d46q51xbla75ikhfwlgi0fbzo6j4d6p
+\end{proof}<|endoftext|>
+\section{Largest Number not Expressible as Sum of Multiples of 23 and 28}
+Tags: Integer Combinations
+
+\begin{theorem}
+The largest [[Definition:Positive Integer|integer]] $n$ that cannot be expressed in the form:
+:$n = 23 x + 28 y$
+for $x, y \in \Z_{>0}$ is $593$.
+\end{theorem}
+
+\begin{proof}
+By [[Largest Number not Expressible as Sum of Multiples of Coprime Integers]], the largest such number is:
+:$\paren {23 - 1} \times \paren {28 - 1} - 1 = 593$
+{{ProofWanted|This is a specific "historical" example of a general result which I read in Polya and Szego some time back, which still needs to be added into {{ProofWiki}}. All in due course. I may already have it on here, I misremember.}}
+\end{proof}<|endoftext|>
+\section{Open implies There Exists Way Below Element}
+Tags: Topological Order Theory, Way Below Relation, Continuous Lattices
+
+\begin{theorem}
+Let $L = \left({S, \preceq, \tau}\right)$ be a [[Definition:Continuous Ordered Set|continuous]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Scott Topology|Scott topology]].
+Let $p \in S, A \subseteq S$ such that
+:$A$ is [[Definition:Open Set (Topology)|open]] and $p \in A$.
+Then
+:$\exists q \in A: q \ll p$
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Continuous Ordered Set|continuous ordered set]]:
+:$p^\ll$ is [[Definition:Directed Subset|directed]]
+and
+:$L$ satisfies [[Definition:Axiom of Approximation|axiom of approximation]].
+By [[Definition:Axiom of Approximation|axiom of approximation]]:
+:$p = \sup \left({p^\ll}\right)$
+By definition of [[Definition:Scott Topology|Scott topology]]:
+:$A$ is [[Definition:Inaccessible by Directed Suprema|inaccessible by directed suprema]].
+By definition of [[Definition:Inaccessible by Directed Suprema|inaccessible by directed suprema]]:
+:$A \cap p^\ll \ne \varnothing$
+By definition of [[Definition:Non-Empty Set|non-empty set]]:
+:$\exists q: q \in A \cap p^\ll$
+Thus by definition of [[Definition:Set Intersection|intersection]]:
+:$q \in A$
+By definition of [[Definition:Set Intersection|intersection]]:
+:$q \in p^\ll$
+By definition of [[Definition:Way Below Closure|way below closure]]:
+:$q \ll p$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Fourth Power which is Sum of 5 Fourth Powers}
+Tags: Fourth Powers, 625
+
+\begin{theorem}
+$625$ is the smallest [[Definition:Fourth Power|fourth power]] which is the [[Definition:Integer Addition|sum]] of $5$ [[Definition:Fourth Power|fourth powers]]:
+:$625 = 5^4 = 2^4 + 2^4 + 3^4 + 4^4 + 4^4$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|To be proved that there are no smaller such $4$th powers.}}
+\end{proof}<|endoftext|>
+\section{Interior is Union of Way Above Closures}
+Tags: Topological Order Theory, Continuous Lattices
+
+\begin{theorem}
+Let $\left({S, \preceq, \tau}\right)$ be a [[Definition:Complete Lattice|complete]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Scott Topology|Scott topology]].
+Let $X \subseteq S$.
+Then $X^\circ = \bigcup \left\{ {x^\gg: x \in S \land x^\gg \subseteq X}\right\}$
+where
+:$X^\circ$ denotes the [[Definition:Interior (Topology)|interior]] of $X$,
+:$x^\gg$ denotes the [[Definition:Way Above Closure|way above closure]] of $x$.
+\end{theorem}
+
+\begin{proof}
+We have:
+:$\left\{ {G \in \left\{ {g^\gg: g \in S}\right\}: G \subseteq X}\right\} = \left\{ {x^\gg: x \in S \land x^\gg \subseteq X}\right\}$
+By [[Way Above Closures Form Basis]]:
+:$\left\{ {x^\gg: x \in S}\right\}$ is [[Definition:Analytic Basis|basis]] of $\left({S, \tau}\right)$.
+By [[Interior is Union of Elements of Basis]]:
+:$X^\circ = \bigcup \left\{ {x^\gg: x \in S \land x^\gg \subseteq X}\right\}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Way Above Closures Form Basis}
+Tags: Topological Order Theory, Continuous Lattices
+
+\begin{theorem}
+Let $L = \left({S, \preceq, \tau}\right)$ be a [[Definition:Complete Lattice|complete]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Scott Topology|Scott topology]].
+Then $\left\{ {x^\gg: x \in S}\right\}$ is [[Definition:Analytic Basis|basis]] of $L$.
+\end{theorem}
+
+\begin{proof}
+Define $B = \left\{ {x^\gg: x \in S}\right\}$
+Thus by [[Way Above Closure is Open]]:
+:$B \subseteq \tau$
+We will prove that
+:for all $x \in S$: there exists [[Definition:Local Basis|local basis]] $Q$ of $x$: $Q \subseteq B$
+Let $x \in S$.
+By [[Way Above Closures that Way Below Form Local Basis]]:
+:$Q := \left\{ {g^\gg: g \in S \land g \ll x}\right\}$ is [[Definition:Local Basis|local basis]] at $x$.
+Thus by definition of [[Definition:Subset|subset]]:
+:$Q \subseteq B$
+{{qed|lemma}}
+Thus by [[Characterization of Analytic Basis by Local Bases]]:
+:$B$ is [[Definition:Analytic Basis|basis]] of $L$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers whose Product with Reverse are Equal}
+Tags: Recreational Mathematics, 651, 156, 372, 273
+
+\begin{theorem}
+:$651 \times 156 = 372 \times 273$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 651 \times 156
+ | r = \paren {3 \times 7 \times 31} \times \paren {2^2 \times 3 \times 13}
+ | c =
+}}
+{{eqn | r = 2^2 \times 3^2 \times 7 \times 13 \times 31
+ | c =
+}}
+{{eqn | r = \paren {2^2 \times 3 \times 31} \times \paren {3 \times 7 \times 13}
+ | c =
+}}
+{{eqn | r = 372 \times 273
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Record Gaps between Twin Primes}
+Tags: Twin Primes
+
+\begin{theorem}
+The gaps between the following [[Definition:Ordered Pair|pairs]] of [[Definition:Twin Primes|twin primes]] are larger than those for all smaller [[Definition:Ordered Pair|pairs]]:
+{{begin-eqn}}
+{{eqn | l = \tuple {3, 5}
+ | o = \to
+ | r = \tuple {5, 7}
+ | c = a gap of $0$
+}}
+{{eqn | l = \tuple {5, 7}
+ | o = \to
+ | r = \tuple {11, 13}
+ | c = a gap of $4$
+}}
+{{eqn | l = \tuple {17, 19}
+ | o = \to
+ | r = \tuple {29, 31}
+ | c = a gap of $10$
+}}
+{{eqn | l = \tuple {41, 43}
+ | o = \to
+ | r = \tuple {59, 61}
+ | c = a gap of $16$
+}}
+{{eqn | l = \tuple {69, 73}
+ | o = \to
+ | r = \tuple {101, 103}
+ | c = a gap of $28$
+}}
+{{eqn | l = \tuple {311, 313}
+ | o = \to
+ | r = \tuple {347, 349}
+ | c = a gap of $34$
+}}
+{{eqn | l = \tuple {347, 349}
+ | o = \to
+ | r = \tuple {419, 421}
+ | c = a gap of $70$
+}}
+{{eqn | l = \tuple {659, 661}
+ | o = \to
+ | r = \tuple {809, 811}
+ | c = a gap of $148$
+}}
+{{end-eqn}}
+{{OEIS|A036061}}
+{{finish|Add this list to the Number Sequence}}
+\end{theorem}
+
+\begin{proof}
+By cases and inspection.
+\end{proof}<|endoftext|>
+\section{Consecutive Sophie Germain Primes cannot be Pair of Twin Primes}
+Tags: Twin Primes, Sophie Germain Primes
+
+\begin{theorem}
+Let $p$ and $p + 2$ be [[Definition:Twin Primes|twin primes]].
+Then unless $p = 3$ it is not possible for both $p$ and $p + 2$ to be [[Definition:Sophie Germain Prime|Sophie Germain primes]].
+\end{theorem}
+
+\begin{proof}
+First it is noted that $3$ and $5$ [[Definition:Twin Primes|twin primes]] which are both [[Definition:Sophie Germain Prime|Sophie Germain]].
+[[Definition:Prime Number|Prime numbers]] greater than $3$ are of the form $6 n - 1$ and $6 n + 1$.
+Thus a pair of [[Definition:Twin Primes|twin primes]] is of the form $\left({6 n - 1, 6 n + 1}\right)$.
+The result follows from [[Sophie Germain Prime cannot be 6n+1]].
+{{qed}}
+[[Category:Twin Primes]]
+[[Category:Sophie Germain Primes]]
+snxstobc2zqlmxfufczkwl31v2642ye
+\end{proof}<|endoftext|>
+\section{Sophie Germain Prime cannot be 6n+1}
+Tags: Sophie Germain Primes
+
+\begin{theorem}
+Let $p$ be a [[Definition:Sophie Germain Prime|Sophie Germain prime]].
+Then $p$ cannot be of the form $6 n + 1$, where $n$ is a [[Definition:Strictly Positive Integer|positive integer]].
+\end{theorem}
+
+\begin{proof}
+Let $p$ be a [[Definition:Sophie Germain Prime|Sophie Germain prime]].
+Then, by definition, $2 p + 1$ is [[Definition:Prime Number|prime]].
+{{AimForCont}} $p = 6 n + 1$ for some $n \in \Z_{>0}$.
+Then:
+{{begin-eqn}}
+{{eqn | l = 2 p + 1
+ | r = 2 \paren {6 n + 1} + 1
+ | c =
+}}
+{{eqn | r = 12 n + 3
+ | c =
+}}
+{{eqn | r = 3 \paren {4 n + 1}
+ | c =
+}}
+{{end-eqn}}
+and so $2 p + 1$ is not [[Definition:Prime Number|prime]].
+The result follows by [[Proof by Contradiction]].
+{{qed}}
+[[Category:Sophie Germain Primes]]
+runb03zh6lpwhgzstdneqn105w7lp87
+\end{proof}<|endoftext|>
+\section{Way Above Closure is Open}
+Tags: Topological Order Theory, Continuous Lattices
+
+\begin{theorem}
+Let $L = \left({S, \preceq, \tau}\right)$ be a [[Definition:Complete Lattice|complete]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Scott Topology|Scott topology]].
+Let $x \in S$.
+Then $x^\gg$ is [[Definition:Open Set (Topology)|open]]
+where $x^\gg$ denotes the [[Definition:Way Above Closure|way above closure]] of $x$.
+\end{theorem}
+
+\begin{proof}
+By [[Way Above Closure is Upper]]:
+:$x^\gg$ is [[Definition:Upper Set|upper]].
+We will prove that
+:$x^\gg$ is [[Definition:Inaccessible by Directed Suprema|inaccessible by directed suprema]].
+Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$ such that
+:$\sup D \in x^\gg$
+By definition of [[Definition:Way Above Closure|way above closure]]:
+:$x \ll \sup D$
+By [[Way Below iff Second Operand Preceding Supremum of Directed Set There Exists Element of Directed Set First Operand Way Below Element]]:
+:$\exists d \in D: x \ll d$
+By definition of [[Definition:Way Above Closure|way above closure]]:
+:$d \in x^\gg$
+By definitions of [[Definition:Set Intersection|intersection]] and [[Definition:Non-Empty Set|non-empty set]]:
+:$x^\gg \cap D \ne \varnothing$
+{{qed|lemma}}
+Thus by definition of [[Definition:Scott Topology|Scott topology]]:
+:$x^\gg$ is [[Definition:Open Set (Topology)|open]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Way Above Closure is Upper}
+Tags: Way Below Relation
+
+\begin{theorem}
+Let $\left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
+Let $x \in S$.
+Then $x^\gg$ is [[Definition:Upper Set|upper]]
+where $x^\gg$ denotes the [[Definition:Way Above Closure|way above closure]] of $x$.
+\end{theorem}
+
+\begin{proof}
+Let $y \in x^\gg$, $z \in S$ such that
+:$y \preceq z$
+By definition of [[Definition:Way Above Closure|way above closure]]:
+:$x \ll y$
+By [[Preceding and Way Below implies Way Below]]:
+:$x \ll z$
+Thus by definition of [[Definition:Way Above Closure|way above closure]]:
+:$z \in x^\gg$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Way Above Closures that Way Below Form Local Basis}
+Tags: Topological Order Theory, Way Below Relation, Continuous Lattices
+
+\begin{theorem}
+Let $L = \left({S, \preceq, \tau}\right)$ be a [[Definition:Complete Lattice|complete]] [[Definition:Continuous Ordered Set|continuous]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Scott Topology|Scott topology]].
+Let $p \in S$.
+Then $\left\{ {q^\gg: q \in S \land q \ll p}\right\}$ is [[Definition:Local Basis|local basis]] at $p$.
+\end{theorem}
+
+\begin{proof}
+Define $B := \left\{ {q^\gg: q \in S \land q \ll p}\right\}$
+By [[Way Above Closure is Open]]:
+:$B \subseteq \tau$
+By definition of [[Definition:Way Above Closure|way above closure]]:
+:$\forall X \in B: p \in X$
+Thus by definition:
+:$B$ is [[Definition:Set of Sets|set]] of [[Definition:Open Neighborhood|open neighborhoods]].
+Let $U$ be an [[Definition:Open Set (Topology)|open]] [[Definition:Subset|subset]] of $S$ such that
+:$p \in U$
+By [[Open implies There Exists Way Below Element]]:
+:$\exists u \in U: u \ll p$
+Thus by definition of $B$:
+:$u^\gg \in B$
+By definition of [[Definition:Scott Topology|Scott topology]]:
+:$U$ is [[Definition:Upper Set|upper]].
+We will prove that
+:$u^\gg \subseteq U$
+Let $z \in u^\gg$
+By definition of [[Definition:Way Above Closure|way above closure]]:
+:$u \ll z$
+By [[Way Below implies Preceding]]:
+:$u \preceq z$
+Thus by definition of [[Definition:Upper Set|upper set]]:
+:$z \in U$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Characterization of Analytic Basis by Local Bases}
+Tags: Topology
+
+\begin{theorem}
+Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
+Let $P$ be a [[Definition:Set of Sets|set]] of [[Definition:Subset|subsets]] of $S$ such that
+:$P \subseteq \tau$
+and
+:for all $p \in S$: there exists [[Definition:Local Basis|local basis]] $B$ at $p: B \subseteq P$
+Then $P$ is [[Definition:Analytic Basis|basis]] of $T$.
+\end{theorem}
+
+\begin{proof}
+By assumption:
+:$P \subseteq \tau$
+Let $U$ be an [[Definition:Open Set (Topology)|open]] [[Definition:Subset|subset]] of $S$.
+Define:
+:$X := \set {V \in P: V \subseteq U}$
+By definition of [[Definition:Subset|subset]]:
+:$X \subseteq P$
+We will prove that:
+:$\forall u \in S: u \in U \iff \exists Z \in X: u \in Z$
+Let $u \in S$.
+We will prove that:
+:$u \in U \implies \exists Z \in X: u \in Z$
+Assume that:
+:$u \in U$
+By assumption:
+:there exists [[Definition:Local Basis|local basis]] $B$ at $u: B \subseteq P$.
+By definition of [[Definition:Local Basis|local basis]]:
+:$\exists V \in B: V \subseteq U$
+Thus by definitions of [[Definition:Subset|subset]] and $X$:
+:$V \in X$
+Thus by definition of [[Definition:Local Basis|local basis]]:
+:$u \in V$
+{{qed|lemma}}
+Assume that:
+:$\exists Z \in X: u \in Z$
+By definition of $X$:
+:$Z \subseteq U$
+Thus by definition of [[Definition:Subset|subset]]:
+:$u \in U$
+{{qed|lemma}}
+Thus by definition of [[Definition:Set Union/General Definition|union]]:
+:$U = \bigcup X$
+Hence $P$ is [[Definition:Analytic Basis|basis]] of $L$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Approximation to Golden Ratio using 666}
+Tags: Approximations, 666, Golden Mean
+
+\begin{theorem}
+The [[Definition:Golden Ratio|Golden Ratio]] is approximated closely by the following formula:
+:$\phi \approx -2 \sin \left({666}\right) = 1.61803 \, 39887 \, 5 \ldots$
+\end{theorem}
+
+\begin{proof}
+By calculation.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Euler Phi Function of 666 equals Product of Digits}
+Tags: Euler Phi Function, 666
+
+\begin{theorem}
+The number $666$ has the following interesting property:
+:$\map \phi {666} = 6 \times 6 \times 6$
+where $\phi$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ function]].
+\end{theorem}
+
+\begin{proof}
+From [[Euler Phi Function of Integer]]:
+:$\displaystyle \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
+where $p \divides n$ denotes the [[Definition:Prime Number|primes]] which [[Definition:Divisor of Integer|divide]] $n$.
+We have that:
+:$666 = 2 \times 3^2 \times 37$
+Thus:
+{{begin-eqn}}
+{{eqn | l = \map \phi {666}
+ | r = 666 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 3} \paren {1 - \dfrac 1 {37} }
+ | c =
+}}
+{{eqn | r = 666 \times \frac 1 2 \times \frac 2 3 \frac {36} {37}
+ | c =
+}}
+{{eqn | r = 3 \times 1 \times 2 \times 36
+ | c =
+}}
+{{eqn | r = 3 \times 2 \times \paren {2^2 \times 3^2}
+ | c =
+}}
+{{eqn | r = 216
+ | c =
+}}
+{{eqn | r = 6 \times 6 \times 6
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum of Sequence of Squares of Primes}
+Tags: Prime Numbers, Square Numbers
+
+\begin{theorem}
+Let $S = \sequence {s_n}$ be the [[Definition:Integer Sequence|integer sequence]] defined as:
+:$\displaystyle s_n = \sum_{i \mathop = 1}^n {p_i}^2$
+where $P_i$ denotes the $i$th [[Definition:Prime Number|prime number]].
+Then $S$ begins:
+:$4, 13, 38, 87, 208, 377, 666, 1027, 1556, 2397, 3358, 4727, 6408, 8257, 10466, \ldots$
+{{OEIS|A024450}}
+\end{theorem}<|endoftext|>
+\section{Pair of Consecutive Powerful Numbers whose First is Odd}
+Tags: Powerful Numbers, 675, 676
+
+\begin{theorem}
+The only known [[Definition:Ordered Pair|pair]] of consecutive [[Definition:Integer|integers]] which are both [[Definition:Powerful Number|powerful numbers]] such that the first of the [[Definition:Ordered Pair|pair]] is [[Definition:Odd Integer|odd]] is:
+:$\tuple {675, 676}$
+\end{theorem}
+
+\begin{proof}
+By investigation:
+{{begin-eqn}}
+{{eqn | l = 675
+ | r = 3^3 \times 5^2
+}}
+{{eqn | l = 676
+ | r = 2^2 \times 13^2
+}}
+{{end-eqn}}
+That there are no smaller ones can be determined again by investigation.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Interior is Union of Elements of Basis}
+Tags: Topological Bases
+
+\begin{theorem}
+Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
+Let $B$ be a [[Definition:Analytic Basis|basis]] of $T$.
+Let $V$ be a [[Definition:Subset|subset]] of $S$.
+Then $V^\circ = \bigcup \left\{ {G \in B: G \subseteq V}\right\}$
+where $V^\circ$ denotes the [[Definition:Interior (Topology)|interior]] of $V$.
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Interior (Topology)|interior]]:
+:$\left\{ {G \in B: G \subseteq V}\right\} = \left\{ {G \in B: G \subseteq V^\circ}\right\}$
+and
+:$V^\circ$ is [[Definition:Open Set (Topology)|open]].
+Thus by [[Open Set is Union of Elements of Basis]]:
+:$V^\circ = \bigcup \left\{ {G \in B: G \subseteq V}\right\}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Palindromic Squares with Non-Palindromic Roots}
+Tags: Square Numbers, Palindromic Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Palindromic Number|palindromic]] [[Definition:Square Number|squares]] with non-[[Definition:Palindromic Number|palindromic]] [[Definition:Square Root|square roots]] begins:
+:$676, 69 \, 696, 94 \, 249, 698 \, 896, 5 \, 221 \, 225, 6 \, 948 \, 496, 522 \, 808 \, 225, \ldots$
+This sequence is not explicitly given in {{OEISLink}}.
+The [[Definition:Integer Sequence|sequence]] of those corresponding non-[[Definition:Palindromic Number|palindromic]] [[Definition:Square Root|square roots]] begins:
+:$26, 264, 307, 836, 2285, 2636, 22 \, 865, 24 \, 846, 30 \, 693, \ldots$
+{{OEIS|A251673}}
+\end{theorem}
+
+\begin{proof}
+By investigating all [[Definition:Square Number|square numbers]] which are [[Definition:Palindromic Number|palindromic]].
+{{begin-eqn}}
+{{eqn | l = 676
+ | r = 26^2
+}}
+{{eqn | l = 69 \, 696
+ | r = 264^2
+}}
+{{eqn | l = 94 \, 249
+ | r = 307^2
+}}
+{{eqn | l = 698 \, 896
+ | r = 836^2
+}}
+{{eqn | l = 5 \, 221 \, 225
+ | r = 2285^2
+}}
+{{eqn | l = 6 \, 948 \, 496
+ | r = 2636^2
+}}
+{{eqn | l = 522 \, 808 \, 225
+ | r = 22 \, 865^2
+}}
+{{end-eqn}}
+and so on.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Tetrahedral Numbers which are Sum of 2 Tetrahedral Numbers}
+Tags: Tetrahedral Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Tetrahedral Number|tetrahedral numbers]] which are the [[Definition:Integer Addition|sum]] of two other [[Definition:Tetrahedral Number|tetrahedral numbers]] begins:
+:$680, \ldots$
+{{expand|To be expanded when we have some more.}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | o =
+ | r = 680
+ | c = the $15$th [[Definition:Tetrahedral Number|tetrahedral number]]
+}}
+{{eqn | r = 120
+ | c = the $8$th [[Definition:Tetrahedral Number|tetrahedral number]]
+}}
+{{eqn | o =
+ | ro= +
+ | r = 560
+ | c = the $14$th [[Definition:Tetrahedral Number|tetrahedral number]]
+}}
+{{end-eqn}}
+{{ProofWanted|Demonstrate cases}}
+\end{proof}
\ No newline at end of file