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+\section{Open Set 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 an [[Definition:Open Set (Topology)|open]] [[Definition:Subset|subset]] of $S$.
+Then $V = \bigcup \left\{ {G \in B: G \subseteq V}\right\}$
+\end{theorem}
+
+\begin{proof}
+Let $x$ be arbitrary.
+We will prove that
+:$x \in V \implies \exists Y \in \left\{ {G \in B: G \subseteq V}\right\}: x \in Y$
+Assume that
+:$x \in V$
+By definition of [[Definition:Analytic Basis|basis]]:
+:$\exists F \subseteq B: V = \bigcup F$
+By definition of [[Definition:Set Union/General Definition|union]]:
+:$\exists Y \in F: x \in Y$
+By [[Set is Subset of Union/General Result]]:
+:$Y \subseteq V$
+Thus by definition of [[Definition:Subset|subset]]:
+:$Y \in \left\{ {G \in B: G \subseteq V}\right\}$
+Thus $x \in Y$
+{{qed|lemma}}
+Assume that
+:$\exists Y \in \left\{ {G \in B: G \subseteq V}\right\}: x \in Y$
+By assumption:
+:$Y \subseteq V$
+Thus by definition of [[Definition:Subset|subset]]:
+:$x \in V$
+Hence by definition of [[Definition:Set Union/General Definition|union]]:
+:$V = \bigcup \left\{ {G \in B: G \subseteq V}\right\}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Mapping at Element is Supremum of Compact Elements implies Mapping is Increasing}
+Tags: Way Below Relation
+
+\begin{theorem}
+Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ be a [[Definition:Lattice|lattice]].
+Let $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be a [[Definition:Complete Lattice|complete lattice]].
+Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that
+:$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \preceq_1 x \land w}\right.$ is [[Definition:Compact Element|compact]]$\left.{}\right\}$
+Then $f$ is [[Definition:Increasing Mapping|increasing]].
+\end{theorem}
+
+\begin{proof}
+Let $x, y \in S$ such that
+:$x \preceq_1 y$
+By [[Compact Closure is Increasing]]:
+:$x^{\mathrm{compact} } \subseteq y^{\mathrm{compact} }$
+By [[Image of Subset under Relation is Subset of Image/Corollary 2]]:
+:$f\left[{x^{\mathrm{compact} } }\right] \subseteq f\left[{y^{\mathrm{compact} } }\right]$
+By assumption:
+:$f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \preceq_1 x \land w}\right.$ is [[Definition:Compact Element|compact]]$\left.{}\right\}$
+and
+:$f\left({y}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \preceq_1 y \land w}\right.$ is [[Definition:Compact Element|compact]]$\left.{}\right\}$
+By definitions of [[Definition:Image of Subset under Mapping|image of set]] and [[Definition:Compact Closure|compact closure]]:
+:$f\left({x}\right) = \sup \left({f\left[{x^{\mathrm{compact} } }\right]}\right)$
+and
+:$f\left({y}\right) = \sup \left({f\left[{y^{\mathrm{compact} } }\right]}\right)$
+Thus by [[Supremum of Subset]] and definition of [[Definition:Complete Lattice|complete lattice]]:
+:$f\left({x}\right) \preceq_2 f\left({y}\right)$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Compatible Atlases}
+Tags: Manifolds
+
+\begin{theorem}
+Let $M$ be a [[Definition:Topological Space|topological space]].
+Let $\mathscr F, \mathscr G$ be $d$-[[Definition:Dimension of Atlas|dimensional]] [[Definition:Atlas|atlases]] of [[Definition:Class of Atlas|class]] $C^k$ on $M$.
+{{TFAE|def = Compatible Atlases}}
+\end{theorem}
+
+\begin{proof}
+=== 1 implies 2 ===
+Follows immediately from the definition of $C^k$-[[Definition:Atlas|atlas]].
+{{qed|lemma}}
+=== 2 implies 1 ===
+Let $(U,\phi)$ and $(V,\psi)$ be [[Definition:Chart|charts]] in $\mathscr F \cup \mathscr G$.
+If they are both in $\mathscr F$, they are [[Definition:Compatible Charts|$C^k$-compatible]] because $\mathscr F$ is a $C^k$-[[Definition:Atlas|atlas]].
+If they are both in $\mathscr G$, they are [[Definition:Compatible Charts|$C^k$-compatible]] because $\mathscr G$ is a $C^k$-[[Definition:Atlas|atlas]].
+If $(U,\phi) \in \mathscr F$ and $(V,\psi) \in \mathscr G$, they are [[Definition:Compatible Charts|$C^k$-compatible]] by assumption.
+Thus $\mathscr F \cup \mathscr G$ is a $C^k$-[[Definition:Atlas|atlas]].
+{{qed}}
+[[Category:Manifolds]]
+kbyy48ho66z0akiplglybgcbh9iadz9
+\end{proof}<|endoftext|>
+\section{Atlas is Contained in Unique Maximal Atlas}
+Tags: Manifolds
+
+\begin{theorem}
+Let $M$ be a [[Definition:Topological Space|topological space]].
+Let $A$ be a $d$-[[Definition:Dimension of Atlas|dimensional]] [[Definition:Atlas|atlas]] of [[Definition:Class of Atlas|class]] $C^k$.
+Then $A$ is contained in a [[Definition:Unique|unique]] [[Definition:Maximal Atlas|maximal atlas]] of class $C^k$.
+\end{theorem}
+
+\begin{proof}
+=== Existence ===
+=== Uniqueness ===
+{{ProofWanted}}
+[[Category:Manifolds]]
+lygxq4j52mhi1204vkwcnszqeea56pu
+\end{proof}<|endoftext|>
+\section{Locally Euclidean iff has C0-Atlas}
+Tags: Manifolds
+
+\begin{theorem}
+Let $M$ be a [[Definition:Topological Space|topological space]].
+{{TFAE}}
+:$(1):\quad$ $M$ is [[Definition:Locally Euclidean Space|locally euclidean]].
+:$(2):\quad$ There exists a [[Definition:Atlas|$C^0$-atlas]] on $M$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+use [[Transition Mapping Between Charts is Homeomorphism]]
+[[Category:Manifolds]]
+il4ktsasdf6tjtm7to57hdoa83u74ty
+\end{proof}<|endoftext|>
+\section{Differentiable Structure Contains Unique Maximal Atlas}
+Tags: Manifolds
+
+\begin{theorem}
+Let $M$ be a [[Definition:Topological Space|topological space]].
+Let $k$ and $d$ be [[Definition:Natural Number|natural numbers]].
+Let $S$ be a $d$-dimensional [[Definition:Differentiable Structure|differentiable structure]] of class $C^k$ on $M$.
+Then $S$ contains a [[Definition:Unique|unique]] [[Definition:Maximal Atlas|maximal $C^k$-atlas]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Manifolds]]
+gmyyw9oue8o4qisuj41w60g4bbo9h26
+\end{proof}<|endoftext|>
+\section{Consecutive Integers whose Product is Primorial}
+Tags: Primorials, Recreational Mathematics
+
+\begin{theorem}
+The following [[Definition:Primorial|primorials]] can be expressed as the [[Definition:Integer Multiplication|product]] of consecutive [[Definition:Integer|integers]]:
+:$2, 6, 30, 210, 510 \, 510$
+{{OEIS|A161620}}
+No others are known.
+The corresponding indices of those [[Definition:Primorial|primorials]] are:
+:$2, 3, 5, 7, 17$
+{{OEIS|A215658}}
+The corresponding values of $n$ such that $p\# = \paren {n - 1} n$ are:
+:$2, 3, 6, 15, 715$
+{{OEIS|A215659}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 2 \#
+ | r = 1 \times 2
+}}
+{{eqn | r = 2
+}}
+{{eqn | l = 3 \#
+ | r = 2 \times 3
+}}
+{{eqn | r = 6
+}}
+{{eqn | l = 5 \#
+ | r = 2 \times 3 \times 5
+}}
+{{eqn | r = 5 \times \paren {2 \times 3}
+}}
+{{eqn | r = 5 \times 6
+}}
+{{eqn | r = 30
+}}
+{{eqn | l = 7 \#
+ | r = 2 \times 3 \times 5 \times 7
+}}
+{{eqn | r = \paren {2 \times 7} \times \paren {3 \times 5}
+}}
+{{eqn | r = 14 \times 15
+}}
+{{eqn | r = 210
+}}
+{{eqn | l = 17 \#
+ | r = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17
+}}
+{{eqn | r = \paren {2 \times 3 \times 7 \times 17} \times \paren {5 \times 11 \times 13}
+}}
+{{eqn | r = 714 \times 715
+}}
+{{eqn | r = 510 \, 510
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integers whose Ratio between Sigma and Phi is Square}
+Tags: Sigma Function, Euler Phi Function, Square Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence of integers]] whose [[Definition:Sigma Function|$\sigma$ value]] [[Definition:Integer Division|divided]] by its [[Definition:Euler Phi Function|Euler $\phi$ value]] is a [[Definition:Square Number|square]] begins:
+:$1, 14, 30, 105, 248, 264, 418, 714, 1485, 3080, \ldots$
+{{OEIS|A293391}}
+{{finish|This sequence needs to be added to the individual integer pages.}}
+\end{theorem}
+
+\begin{proof}
+{{refactor|Include this as examples}}
+{{begin-eqn}}
+{{eqn | l = \map \phi {714}
+ | r = 192
+ | c = {{EulerPhiLink|714}}
+}}
+{{eqn | l = \map \sigma {714}
+ | r = 1728
+ | c = {{SigmaLink|714}}
+}}
+{{eqn | l = \map \sigma {714} / \map \phi {714}
+ | r = 1728 / 192
+ | c =
+}}
+{{eqn | r = 9
+ | c =
+}}
+{{eqn | r = 3^2
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|Finish this off}}
+\end{proof}<|endoftext|>
+\section{Implicit Function Theorem/Real Functions}
+Tags: Implicit Function Theorem
+
+\begin{theorem}
+Let $n$ and $k$ be [[Definition:Natural Number|natural numbers]].
+Let $\Omega \subset \R^{n + k}$ be [[Definition:Open Set|open]].
+Let $f: \Omega \to \R^k$ be [[Definition:Continuous Function|continuous]].
+Let the [[Definition:Partial Derivative|partial derivatives]] of $f$ with respect to $\R^k$ be [[Definition:Continuous Function|continuous]].
+Let $\tuple {a, b} \in \Omega$, with $a\in \R^n$ and $b\in \R^k$.
+Let $\map f {a, b} = 0$.
+For $\tuple {x_0, y_0} \in \Omega$, let $D_2 \map f {x_0, y_0}$ denote the [[Definition:Total Derivative|total derivative]] of the function $y \mapsto \map f {x_0, y}$ at $y_0$.
+Let the [[Definition:Linear Mapping|linear map]] $D_2 \map f {a, b}$ be [[Definition:Invertible Linear Mapping|invertible]].
+Then there exist [[Definition:Neighborhood|neighborhoods]] $U \subset \Omega$ of $a$ and $V \subset \R^k$ of $b$ such that there exists a unique [[Definition:Function|function]] $g: U \to V$ such that $\map f {x, \map g x} = 0$ for all $x \in U$.
+Moreover, $g$ is [[Definition:Continuous Mapping|continuous]].
+\end{theorem}<|endoftext|>
+\section{Uniform Contraction Mapping Theorem}
+Tags: Fixed Point Theorems, Implicit Functions, Metric Spaces, Named Theorems
+
+\begin{theorem}
+Let $M$ and $N$ be [[Definition:Metric Space|metric spaces]].
+Let $M$ be [[Definition:Complete Metric Space|complete]].
+Let $f : M \times N \to M$ be a [[Definition:Continuous Mapping|continuous]] [[Definition:Uniform Contraction Mapping|uniform contraction]].
+Then for all $t \in N$ there exists a [[Definition:Unique|unique]] $g \left({t}\right) \in M$ such that $f(g \left({t}\right), t) = g \left({t}\right)$, and the [[Definition:Mapping|mapping]] $g: N \to M$ is [[Definition:Continuous Mapping (Metric Spaces)|continuous]].
+\end{theorem}
+
+\begin{proof}
+For every $t\in N$, the [[Definition:Mapping|mapping]]:
+:$f_t: M \to M : x \mapsto f \left({x, t}\right)$ is a [[Definition:Contraction Mapping|contraction]].
+By the [[Banach Fixed-Point Theorem]], there exists a [[Definition:unique|unique]] $g \left({t}\right) \in M$ such that $f_t \left({g \left({t}\right)}\right) = g \left({t}\right)$.
+We show that $g$ is [[Definition:Continuous Mapping (Metric Spaces)|continuous]].
+Let $K < 1$ be a [[Definition:Uniform Lipschitz Constant|uniform Lipschitz constant]] for $f$.
+Let $s, t \in N$.
+Then
+{{begin-eqn}}
+{{eqn | l = d \left({g \left({s}\right), g \left({t}\right)}\right)
+ | r = d \left({f \left({g \left({s}\right), s}\right), f \left({g \left({t}\right), t}\right)}\right)
+ | c = Definition of $g$
+}}
+{{eqn | o = \le
+ | r = d \left({f \left({g \left({s}\right), s}\right), f \left({g \left({t}\right), s}\right)}\right) + d \left({f \left({g \left({t}\right), s}\right), f \left({g \left({t}\right), t}\right)}\right)
+ | c = {{Defof|Metric}}
+}}
+{{eqn | o = \le
+ | r = K \cdot d \left({g \left({s}\right), g \left({t}\right)}\right) + d \left({f \left({g \left({t}\right), s}\right), f \left({g \left({t}\right), t}\right)}\right)
+ | c = $f$ is a [[Definition:Uniform Contraction Mapping|uniform contraction]]
+}}
+{{end-eqn}}
+and thus:
+:$d \left({g \left({s}\right), g \left({t}\right)}\right) \le \dfrac 1 {1 - K} d \left({f \left({g \left({t}\right), s}\right), f \left({g \left({t}\right), t}\right)}\right)$
+The continuity of $g$ now follows from that of $f$ using the definition of [[Definition:Product Metric|product metric]]
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Mapping at Element is Supremum of Compact Elements implies Mapping at Element is Supremum that Way Below}
+Tags: Way Below Relation
+
+\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 a [[Definition:Mapping|mapping]] such that
+:$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \preceq_1 x \land w}\right.$ is [[Definition:Compact Element|compact]]$\left.{}\right\}$
+Then
+:$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
+\end{theorem}
+
+\begin{proof}
+Let $x \in S$.
+Define $X = \left\{ {f\left({w}\right): w \in S \land w \preceq_1 x \land w}\right.$ is [[Definition:Compact Element|compact]]$\left.{}\right\}$
+Define $Y = \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
+We will prove that
+:$X \subseteq Y$
+Let $b \in X$.
+By definition of $X$:
+:$\exists w \in S: b = f\left({w}\right) \land w \preceq_1 x \land w$ is [[Definition:Compact Element|compact]].
+By definition of [[Definition:Compact Element|compact element]]:
+:$w \ll w$
+By [[Preceding and Way Below implies Way Below]]:
+:$w \ll x$
+Thus by definition of $Y$:
+:$b \in Y$
+{{qed|lemma}}
+We will prove that
+:$f\left({x}\right)$ is [[Definition:Upper Bound of Set|upper bound]] for $Y$.
+Let $b \in Y$.
+By definition of $Y$:
+:$\exists w \in S: b = f\left({w}\right) \land w \ll x$
+By [[Way Below implies Preceding]]:
+:$w \preceq_1 x$
+By [[Mapping at Element is Supremum implies Mapping is Increasing]]:
+:$f$ is an [[Definition:Increasing Mapping|increasing mapping]].
+Thus by definition of [[Definition:Increasing Mapping|increasing mapping]]:
+:$b \preceq_2 f\left({x}\right)$
+{{qed|lemma}}
+We will prove that
+:$\forall b \in T: b$ is [[Definition:Upper Bound of Set|upper bound]] for $Y \implies f\left({x}\right) \preceq_2 b$
+Let $b \in T$ such that
+:$b$ is [[Definition:Upper Bound of Set|upper bound]] for $Y$.
+By [[Upper Bound for Subset]]:
+:$b$ is [[Definition:Upper Bound of Set|upper bound]] for $X$.
+By assumption:
+:$f\left({x}\right) = \sup X$
+Thus by definition of [[Definition:Supremum of Set|supremum]]:
+:$f\left({x}\right) \preceq_2 b$
+{{qed|lemma}}
+Thus by definition of [[Definition:Supremum of Set|supremum]]:
+:$f\left({x}\right) = \sup Y$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Inverse Function Theorem for Real Functions}
+Tags: Implicit Functions
+
+\begin{theorem}
+Let $n\geq1$ and $k\geq1$ be [[Definition:Natural Number|natural numbers]].
+Let $\Omega\subset \R^n$ be [[Definition:Open Set of Real Euclidean Space|open]].
+$f : \Omega \to \R^n$ be a [[Definition:Vector-Valued Function|vector-valued function]] of [[Definition:Differentiability Class|class]] $C^k$.
+Let $a\in\Omega$.
+Let the [[Definition:Differential of Vector-Valued Function|differential]] $Df(a)$ of $f$ at $a$ be [[Definition:Invertible Linear Operator|invertible]].
+Then there exist [[Definition:Open Set of Real Euclidean Space|open sets]] $U\subset\Omega$ and $V\subset\R^n$ such that the [[Definition:Restriction|restriction]] of $f$ to $U$ is a $C^k$-[[Definition:Diffeomorphism|diffeomorphism]] $f:U\to V$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Implicit Functions]]
+s18r12x1y8fezruex4l4o9glbp06lpn
+\end{proof}<|endoftext|>
+\section{Sum of 714 and 715}
+Tags: Prime Numbers, 714, 715
+
+\begin{theorem}
+The [[Definition:Integer Addition|sum]] of $714$ and $715$ is a [[Definition:Digit|$4$-digit]] [[Definition:Integer|integer]] which has $6$ [[Definition:Anagram|anagrams]] which are [[Definition:Prime Number|prime]].
+\end{theorem}
+
+\begin{proof}
+We have that:
+:$714 + 715 = 1429$
+Hence we investigate its [[Definition:Anagram|anagrams]].
+We bother only to check those which do not end in either $2$ or $4$, as those are [[Definition:Even Integer|even]].
+{{begin-eqn}}
+{{eqn | l = 1429
+ | o =
+ | c = is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 1249
+ | o =
+ | c = is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 4129
+ | o =
+ | c = is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 4219
+ | o =
+ | c = is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 2149
+ | r = 7 \times 307
+ | c = and so is not [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 2419
+ | r = 41 \times 59
+ | c = and so is not [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 9241
+ | o =
+ | c = is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 9421
+ | o =
+ | c = is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 2941
+ | r = 17 \times 173
+ | c = and so is not [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 2491
+ | r = 47 \times 53
+ | c = and so is not [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 4291
+ | r = 7 \times 613
+ | c = and so is not [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 4921
+ | r = 7 \times 19 \times 37
+ | c = and so is not [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+Of the above, $6$ are seen to be [[Definition:Prime Number|prime]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{720 is Product of Consecutive Numbers in Two Ways}
+Tags: Factorials, 720
+
+\begin{theorem}
+:$720 = 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8$
+\end{theorem}
+
+\begin{proof}
+Follows from [[Factorial as Product of Two Factorials]]:
+:$10! = 6! \times 7!$
+and so:
+:$\dfrac {10!} {7!} = 10 \times 9 \times 8 = 6 \times 5 \times 4 \times 3 \times 2 \times 1$
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Factorial which is Sum of Two Squares}
+Tags: Factorials, Square Numbers, 720
+
+\begin{theorem}
+The only [[Definition:Factorial|factorial]] which can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]] is:
+{{begin-eqn}}
+{{eqn | l = 6!
+ | r = 12^2 + 24^2
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+We show that for $n \ge 7$, $n!$ cannot be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]].
+By refining the result in [[Interval containing Prime Number of forms 4n - 1, 4n + 1, 6n - 1, 6n + 1]], one can show that:
+:There exists a [[Definition:Prime Number|prime]] of the form $4 k + 3$ strictly between $m$ and $2 m$ whenever $m \ge 4$.
+Let $n \ge 7$. Then $\ceiling {\dfrac n 2} \ge 4$.
+Using the result above, there is a [[Definition:Prime Number|prime]] $p$ of the form $4 k + 3$ such that:
+:$\ceiling {\dfrac n 2} < p < 2 \ceiling {\dfrac n 2}$
+We then have, by multiplying the inequality by $2$:
+:$2 \ceiling {\dfrac n 2} < 2 p < 4 \ceiling {\dfrac n 2}$
+This gives:
+:$p < 2 \ceiling {\dfrac n 2} < 2 p$
+Which implies:
+:$p \le n < 2 p$
+From [[Integer as Sum of Two Squares]]:
+:$n!$ can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]] {{iff}} each of its [[Definition:Prime Factor|prime divisors]] of the form $4 k + 3$ (if any) occur to an [[Definition:Even Power|even power]].
+The inequality above shows that there are no multiples of $p$ which are not greater than $n$ except $p$ itself.
+Hence $p$ occurs to an [[Definition:Odd Power|odd power]], $1$, in $n!$.
+This shows that for $n \ge 7$, $n!$ cannot be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]].
+Checking the rest of the [[Definition:Factorial|factorials]] we see that the only ones satisfying the criteria are:
+{{begin-eqn}}
+{{eqn | l = 0! = 1!
+ | r = 0^2 + 1^2
+}}
+{{eqn | l = 2!
+ | r = 1^2 + 1^2
+}}
+{{eqn | l = 6!
+ | r = 12^2 + 24^2
+}}
+{{end-eqn}}
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Multiplicity of 720 in 720 Factorial}
+Tags: Factorials, 720, De Polignac's Formula
+
+\begin{theorem}
+The [[Definition:Multiplicity of Prime Factor|multiplicity]] of $720$ in $720!$ is $178$.
+That is:
+:$720^{178} \divides 720!$
+but:
+:$720^{179} \nmid 720!$
+where:
+:$720!$ denotes [[Definition:Factorial|$720$ factorial]]
+:$\divides$ denotes [[Definition:Divisor of Integer|divisibility]]
+:$\nmid$ denotes non-[[Definition:Divisor of Integer|divisibility]].
+\end{theorem}
+
+\begin{proof}
+We have that:
+:$720 = 2^4 \times 3^2 \times 5$
+It remains to inspect the [[Definition:Divisor of Integer|divisibility]] of $2$, $3$ and $5$ in $720!$
+Thus:
+=== [[De Polignac's Formula/Examples/2 in 720 Factorial|Multiplicity of $2$ in $720!$]] ===
+{{:De Polignac's Formula/Examples/2 in 720 Factorial}}
+=== [[De Polignac's Formula/Examples/3 in 720 Factorial|Multiplicity of $3$ in $720!$]] ===
+{{:De Polignac's Formula/Examples/3 in 720 Factorial}}
+=== [[De Polignac's Formula/Examples/5 in 720 Factorial|Multiplicity of $5$ in $720!$]] ===
+{{:De Polignac's Formula/Examples/5 in 720 Factorial}}
+We calculate the [[Definition:Multiplicity of Prime Factor|multiplicity]] of the [[Definition:Integer Power|powers]] of $2$ and $3$ in $720!$ thus:
+{{begin-eqn}}
+{{eqn | l = 716
+ | r = 4 \times 179
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \paren {2^4}^{179}
+ | o = \divides
+ | r = 720!
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 356
+ | r = 2 \times 178
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \paren {3^2}^{178}
+ | o = \divides
+ | r = 720!
+ | c =
+}}
+{{end-eqn}}
+Thus it is seen that the smallest [[Definition:Integer Power|power]] of the [[Definition:Prime Power|prime powers]] that are [[Definition:Divisor of Integer|divisors]] of $720$ that [[Definition:Divisor of Integer|divide]] $720!$ is that of $3^2$, which is $178$.
+Hence:
+:$720^{178} \divides 720!$
+but:
+:$720^{179} \nmid 720!$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Continuous iff Mapping at Element is Supremum}
+Tags: Topological Order Theory, Way Below Relation, Continuous Lattices
+
+\begin{theorem}
+Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ 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}}
+:$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
+\end{theorem}
+
+\begin{proof}
+=== Sufficient Condition ===
+Assume that
+:$f$ is [[Definition:Continuous (Topology)|continuous]].
+By [[Continuous iff Directed Suprema Preserving]]:
+:$f$ is [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
+Thus by [[Directed Suprema Preserving Mapping at Element is Supremum]]:
+:$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
+{{qed|lemma}}
+=== Necessary Condition ===
+Assume that
+:$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
+By [[Mapping at Element is Supremum implies Way Below iff There Exists Element that Way Below and Way Below]]:
+:$\forall x \in S, y \in T: y \ll f\left({x}\right) \iff \exists w \in S: w \ll x \land y \ll f\left({w}\right)$
+Thus by [[Continuous iff Way Below iff There Exists Element that Way Below and Way Below]]:
+:$f$ is [[Definition:Continuous (Topology)|continuous]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Synthetic Basis}
+Tags: Topological Bases
+
+\begin{theorem}
+Let $S$ be a [[Definition:Set|set]].
+{{TFAE|def = Synthetic Basis}}
+\end{theorem}
+
+\begin{proof}
+=== 1 implies 2 ===
+Let $U, V \in \mathcal B$.
+Let $x \in U \cap V$.
+[[Definition:By Hypothesis|By hypothesis]]:
+:$\displaystyle \exists \mathcal A \subseteq \mathcal B: U \cap V = \bigcup \mathcal A$
+By definition of [[Definition:Set Union|union]], $\exists W \in\mathcal A : x \in W$.
+By [[Set is Subset of Union/General Result|Set is Subset of Union: General Result]], $W \subset U \cap V$.
+Therefore:
+:$\displaystyle \forall x \in A \cap B: \exists W \in \mathcal A \subseteq \mathcal B: x \in W \subseteq U \cap V$
+{{qed|lemma}}
+=== 2 implies 1 ===
+Let $U, V \in \mathcal B$.
+Define the [[Definition:Set|set]]:
+:$\displaystyle \mathcal A = \left\{{W \in \mathcal B: W \subseteq U \cap V}\right\} \subseteq \mathcal B$
+By [[Union is Smallest Superset/General Result|Union is Smallest Superset: General Result]]:
+:$\displaystyle \bigcup \mathcal A \subseteq U \cap V$
+[[Definition:By Hypothesis|By hypothesis]]:
+:$\displaystyle \forall x \in U \cap V: \exists W \in \mathcal A: x \in W$
+Thus $\displaystyle U \cap V \subseteq \bigcup \mathcal A$
+By definition of [[Definition:Set Equality|set equality]]:
+:$\displaystyle U \cap V = \bigcup \mathcal A$
+{{qed}}
+[[Category:Topological Bases]]
+m15b6r1d0o3c8wzdll9jdc3352t5fbm
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Generated Submodule}
+Tags: Generators of Modules
+
+\begin{theorem}
+Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
+Let $M$ be an [[Definition:Module|$R$-module]].
+Let $S\subset M$ be a [[Definition:Subset|subset]].
+{{TFAE|def = Generated Submodule}}
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Generators of Modules]]
+rqov6fml3sxtmb9wvaaymxl33wl25iu
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Basis of Vector Space}
+Tags: Bases of Vector Spaces
+
+\begin{theorem}
+Let $K$ be a [[Definition:Division Ring|division ring]].
+Let $\struct {G, +_G, \circ}_K$ be an [[Definition:Vector Space|vector space]] over $K$.
+{{TFAE| def = Basis of Vector Space}}
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Bases of Vector Spaces]]
+sor4sfol7raf8l3pr4u27v7w22abozr
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Differentiable Real Function at Point}
+Tags: Differentiable Real Functions
+
+\begin{theorem}
+Let $\openint a b \subset \R$ be an [[Definition:Open Real Interval|open interval]].
+Let $\xi$ be a point in $\openint a b$.
+{{TFAE|def = Differentiable Real Function at Point|view = differentiable real function at a point}}
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|compare [[Equivalence of Definitions of Derivative]]}}
+[[Category:Differentiable Real Functions]]
+34ueqe29g1aqkgizzbncuvs99ypu7tq
+\end{proof}<|endoftext|>
+\section{Continuous iff Mapping at Element is Supremum of Compact Elements}
+Tags: Topological Order Theory, Way Below Relation, Continuous Lattices
+
+\begin{theorem}
+Let $L = \left({S, \preceq_1, \tau_1}\right)$ and $R = \left({T, \preceq_2, \tau_2}\right)$ be [[Definition:Complete Lattice|complete]] [[Definition:Algebraic Ordered Set|algebraic]] [[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}}
+:$\forall x \in S: f \left({x}\right) = \sup \left\{ {f \left({w}\right): w \in S \land w \preceq_1 x \land w}\right.$ is [[Definition:Compact Element|compact]]$\left.{}\right\}$
+\end{theorem}
+
+\begin{proof}
+By [[Algebraic iff Continuous and For Every Way Below Exists Compact Between]]:
+:$L$ and $R$ are [[Definition:Continuous Ordered Set|continuous]].
+=== Sufficient Condition ===
+Assume that@
+:$f$ is [[Definition:Continuous (Topology)|continuous]].
+By [[Continuous iff Mapping at Element is Supremum]]:
+:$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
+Let $x \in S$.
+By definitions of [[Definition:Image of Subset under Mapping|image of set]] and [[Definition:Compact Closure|compact closure]]:
+:$\left\{ {f\left({w}\right): w \in S \land w \preceq_1 x \land w}\right.$ is [[Definition:Compact Element|compact]]$\left.{}\right\} = f\left[{x^{\mathrm{compact} } }\right]$
+By [[Compact Closure is Directed]]:
+:$D := x^{\mathrm{compact} }$ is [[Definition:Directed Subset|directed]].
+By [[Continuous iff Directed Suprema Preserving]]:
+:$f$ [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
+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:Complete Lattice|complete lattice]]:
+:$D$ admits a [[Definition:Supremum of Set|supremum]].
+By definition of [[Definition:Algebraic Ordered Set|algebraic]]:
+:$L$ satisfies [[Definition:Axiom of K-Approximation|axiom of K-approximation]].
+Thus
+{{begin-eqn}}
+{{eqn | l = f\left({x}\right)
+| r = f\left({\sup D}\right)
+| c = [[Definition:Axiom of K-Approximation|axiom of K-approximation]]
+}}
+{{eqn | r = \sup \left({f\left[{D}\right]}\right)
+| c = definition of [[Definition:Mapping Preserves Supremum/Subset|mapping preserves the supremum]]
+}}
+{{eqn | r = \sup \left\{ {f\left({w}\right): w \in S \land w \preceq_1 x \land w \mathrm{\ is\ compact} }\right\}
+}}
+{{end-eqn}}
+{{qed|lemma}}
+=== Necessary Condition ===
+Assume that
+:$\forall x \in S: f \left({x}\right) = \sup \left\{ {f \left({w}\right): w \in S \land w \preceq_1 x \land w}\right.$ is [[Definition:Compact Element|compact]]$\left.{}\right\}$
+By [[Mapping at Element is Supremum of Compact Elements implies Mapping at Element is Supremum that Way Below]]:
+:$\forall x \in S: f \left({x}\right) = \sup \left\{ {f \left({w}\right): w \in S \land w \ll x}\right\}$
+Thus by [[Continuous iff Mapping at Element is Supremum]]:
+:$f$ is [[Definition:Continuous (Topology)|continuous]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Solutions to Approximate Fermat Equation x^3 = y^3 + z^3 Plus or Minus 1}
+Tags: Approximate Fermat Equations
+
+\begin{theorem}
+The [[Definition:Approximate Fermat Equation|approximate Fermat equation]]:
+:$x^3 = y^3 + z^3 \pm 1$
+has the solutions:
+{{begin-eqn}}
+{{eqn | l = 9^3
+ | r = 6^3 + 8^3 + 1
+}}
+{{eqn | l = 103^3
+ | r = 64^3 + 94^3 - 1
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+Performing the arithmetic:
+{{begin-eqn}}
+{{eqn | l = 6^3 + 8^3 + 1
+ | r = 216 + 512 + 1
+}}
+{{eqn | r = 729
+ | c =
+}}
+{{eqn | r = 9^3
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 64^3 + 94^3 - 1
+ | r = 262 \, 144 + 830 \, 584 - 1
+}}
+{{eqn | r = 1 \, 092 \, 727
+ | c =
+}}
+{{eqn | r = 103^3
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cubes which are Sum of Five Cubes}
+Tags: Sums of Cubes
+
+\begin{theorem}
+The following [[Definition:Cube Number|cube numbers]] can be expressed as the [[Definition:Integer Addition|sum]] of $5$ [[Definition:Positive Integer|positive]] [[Definition:Cube Number|cube numbers]]:
+:$9^3, \ldots$
+{{expand|More terms needed. It seems that: $4$ and all numbers $> 8$ can be so expressed only $4, 8, 10, 11, 13$ require repeated cubes}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 9^3
+ | r = 729
+ | c =
+}}
+{{eqn | r = 1 + 27 + 64 + 125 + 512
+ | c =
+}}
+{{eqn | r = 1^3 + 3^3 + 4^3 + 5^3 + 8^3
+ | c =
+}}
+{{end-eqn}}
+{{expand|Add the proof based on $9^3 {{=}} 1^3 + 6^3 + 8^3$ and $6^3 {{=}} 3^3 + 4^3 + 5^3$ from [[Cubes which are Sum of Three Cubes]]}}
+\end{proof}<|endoftext|>
+\section{Period of Reciprocal of 729 is 81}
+Tags: 729, Examples of Reciprocals
+
+\begin{theorem}
+The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $729$ has $\dfrac 1 9$ the maximum [[Definition:Period of Recurrence|period]], that is, $81$:
+:$\dfrac 1 {729} = 0 \cdotp \dot 00137 \, 17421 \, 12482 \, 85322 \, 35939 \, 64334 \, 70507 \, 54458 \, 16186 \, 55692 \, 72976 \, 68038 \, 40877 \, 91495 \, 19890 \, 26063 \, \dot 1$
+The [[Definition:Recurrence of Basis Expansion|recurring part]] can be arranged in groups of $9$ digits each, revealing an interesting pattern:
+{{begin-eqn}}
+{{eqn | l = 001 \, 371 \, 742
+ | o =
+}}
+{{eqn | l = 112 \, 482 \, 853
+ | o =
+}}
+{{eqn | l = 223 \, 593 \, 964
+ | o =
+}}
+{{eqn | l = 334 \, 705 \, 075
+ | o =
+}}
+{{eqn | l = 445 \, 816 \, 186
+ | o =
+}}
+{{eqn | l = 556 \, 927 \, 297
+ | o =
+}}
+{{eqn | l = 668 \, 038 \, 408
+ | o =
+}}
+{{eqn | l = 779 \, 149 \, 519
+ | o =
+}}
+{{eqn | l = 890 \, 260 \, 631
+ | o =
+}}
+{{end-eqn}}
+that is, each row (apart from the last) can be obtained from the previous one by adding $111 \, 111 \, 111$ to it.
+{{OEIS|A021733}}
+\end{theorem}
+
+\begin{proof}
+Performing the calculation using [[Definition:Long Division|long division]]:
+
+ 0.00137174211248285322359396433470507544581618655692729766803840877914951989026063100
+ ---------------------------------------------------------------------------------------
+729)1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000
+ 729 729 3645 2187 5103 5832 3645 5103 5832 3645 4374
+ --- --- ---- ---- ---- ---- ---- ---- ---- ---- ----
+ 2710 910 2350 7030 3700 1180 5050 4870 5680 1450 2260
+ 2187 729 2187 6561 3645 729 4374 4374 5103 729 2187
+ ---- --- ---- ---- ---- ---- ---- ---- ---- ---- ----
+ 5230 1810 1630 4690 5500 4510 6760 4960 5770 7210 730
+ 5103 1458 1458 4374 5103 4374 6561 4374 5103 6561 729
+ ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---
+ 1270 3520 1720 3160 3970 1360 1990 5860 6670 6490 1000
+ 729 2916 1458 2916 3645 729 1458 5832 6561 5832 729
+ ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
+ 5410 6040 2620 2440 3250 6310 5320 2800 1090 6580 ...
+ 5103 5832 2187 2187 2916 5832 5103 2187 729 6561
+ ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
+ 3070 2080 4330 2530 3340 4780 2170 6130 3610 1900
+ 2916 1458 3645 2187 2916 4374 1458 5832 2916 1458
+ ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
+ 1540 6220 6850 3430 4240 4060 7120 2980 6940 4420
+ 1458 5832 6561 2916 3645 3645 6561 2916 6561 4374
+ ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
+ 820 3880 2890 5140 5950 4150 5590 6400 3790 4600
+ 729 3645 2187 5103 5832 3645 5103 5832 3645 4374
+
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Implicit Function Theorem for Differentiable Real Functions}
+Tags: Implicit Functions
+
+\begin{theorem}
+Let $\Omega \subset \R^{n+k}$ be [[Definition:Open Set of Real Euclidean Space|open]].
+Let $f : \Omega \to \R^k$ be [[Definition:Differentiable Vector-Valued Function|differentiable]].
+Let the $i$th [[Definition:Partial Derivative of Real-Valued Function|partial derivatives]] of $f$ be [[Definition:Continuous Function|continuous]] in $\Omega$ for $n+1 \leq i \leq n+k$.
+Let $(a,b) \in \Omega$, with $a\in \R^n$ and $b\in \R^k$.
+Let $f(a,b) = 0$.
+For $(x_0,y_0)\in\Omega$, let $D_2 f(x_0,y_0)$ denote the [[Definition:Differential of Vector-Valued Function|differential]] of the function $y\mapsto f(x_0, y)$ at $y_0$.
+Let the [[Definition:Linear Mapping|linear map]] $D_2 f(a,b)$ be [[Definition:Invertible Linear Mapping|invertible]].
+Then there exist [[Definition:Neighborhood|neighborhoods]] $U\subset\Omega$ of $a$ and $V\subset\R^k$ of $b$ such that there exists a unique [[Definition:Function|function]] $g : U \to V$ such that $f(x, g(x)) = 0$ for all $x\in U$.
+Moreover, $g$ is [[Definition:Differentiable Vector-Valued Function|differentiable]], and its [[Definition:Differential of Vector-Valued Function|differential]] satisfies:
+:$dg (x) = - \left( (D_2f)(x, g(x)) \right)^{-1} \circ (D_1 f)(x, g(x))$ for all $x\in U$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Implicit Functions]]
+4t3eiy6pzrf811n7rr0487hxp0tgrea
+\end{proof}<|endoftext|>
+\section{Implicit Function Theorem for Smooth Real Functions}
+Tags: Implicit Functions
+
+\begin{theorem}
+Let $\Omega \subset \R^{n+k}$ be [[Definition:Open Set of Real Euclidean Space|open]].
+Let $f : \Omega \to \R^k$ be [[Definition:Smooth Vector-Valued Function|smooth]].
+Let $(a,b) \in \Omega$, with $a\in \R^n$ and $b\in \R^k$.
+Let $f(a,b) = 0$.
+For $(x_0,y_0)\in\Omega$, let $D_2 f(x_0,y_0)$ denote the [[Definition:Differential of Vector-Valued Function|differential]] of the function $y\mapsto f(x_0, y)$ at $y_0$.
+Let the [[Definition:Linear Mapping|linear map]] $D_2 f(a,b)$ be [[Definition:Invertible Linear Mapping|invertible]].
+Then there exist [[Definition:Neighborhood|neighborhoods]] $U\subset\Omega$ of $a$ and $V\subset\R^k$ of $b$ such that there exists a unique [[Definition:Function|function]] $g : U \to V$ such that $f(x, g(x)) = 0$ for all $x\in U$.
+Moreover, $g$ is [[Definition:Smooth Vector-Valued Function|smooth]], and its [[Definition:Differential of Vector-Valued Function|differential]] satisfies:
+:$dg (x) = - \left( (D_2f)(x, g(x)) \right)^{-1} \circ (D_1 f)(x, g(x))$ for all $x\in U$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Implicit Functions]]
+op8r7n6voyto8kaqajoonhlny7b7fra
+\end{proof}<|endoftext|>
+\section{Sum of 4 Consecutive Binomial Coefficients forming Square}
+Tags: Binomial Coefficients, Square Numbers
+
+\begin{theorem}
+Consider the [[Definition:Diophantine Equation|Diophantine equation]]:
+:$\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 = m^2$
+where:
+:$\dbinom a b$ denotes a [[Definition:Binomial Coefficient|binomial coefficient]]
+:$n$ is an [[Definition:Integer|integer]]
+:$m$ is a [[Definition:Positive Integer|non-negative integer]].
+Then $n$ has one of the following values:
+:$-1, 0, 2, 7, 15, 74, 767$
+{{OEIS|A047694}}
+The corresponding values of $m$ are:
+:$0, 1, 2, 8, 24, 260, 8672$
+{{OEIS|A047695}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | r = \dbinom {-1} 0 + \dbinom {-1} 1 + \dbinom {-1} 2 + \dbinom {-1} 3
+ | o =
+ | c =
+}}
+{{eqn | r = \left({-1}\right)^0 \dbinom 0 0 + \left({-1}\right)^1 \dbinom 1 1 + \left({-1}\right)^2 \dbinom 2 2 + \left({-1}\right)^3 \dbinom 3 3
+ | c = [[Negated Upper Index of Binomial Coefficient/Corollary 1|Negated Upper Index of Binomial Coefficient: Corollary 1]]
+}}
+{{eqn | r = 1 - 1 + 1 - 1
+ | c = [[Binomial Coefficient with Self]]
+}}
+{{eqn | r = 0
+ | c =
+}}
+{{eqn | r = 0^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | r = \dbinom 0 0 + \dbinom 0 1 + \dbinom 0 2 + \dbinom 0 3
+ | o =
+ | c =
+}}
+{{eqn | r = 1 + 0 + 0 + 0
+ | c = [[Binomial Coefficient with Zero]]
+}}
+{{eqn | r = 1
+ | c =
+}}
+{{eqn | r = 1^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | r = \dbinom 2 0 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3
+ | o =
+ | c =
+}}
+{{eqn | r = 1 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3
+ | c = [[Binomial Coefficient with Zero]]
+}}
+{{eqn | r = 1 + 2 + \dbinom 2 2 + \dbinom 2 3
+ | c = [[Binomial Coefficient with One]]
+}}
+{{eqn | r = 1 + 2 + 1 + \dbinom 2 3
+ | c = [[Binomial Coefficient with Self]]
+}}
+{{eqn | r = 1 + 2 + 1 + 0
+ | c = {{Defof|Binomial Coefficient}}
+}}
+{{eqn | r = 4
+ | c =
+}}
+{{eqn | r = 2^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | r = \dbinom 7 0 + \dbinom 7 1 + \dbinom 7 2 + \dbinom 7 3
+ | o =
+ | c =
+}}
+{{eqn | r = \dfrac {7!} {7! \, 0!} + \dfrac {7!} {6! \, 1!} + \dfrac {7!} {5! \, 2!} + \dfrac {7!} {4! \, 3!}
+ | c = {{Defof|Binomial Coefficient}}
+}}
+{{eqn | r = \dfrac {7!} {7! \times 1} + \dfrac 7 1 + \dfrac {7 \times 6} {2 \times 1} + \dfrac {7 \times 6 \times 5} {3 \times 2 \times 1}
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = 1 + 7 + \dfrac {42} 2 + \dfrac {210} 6
+ | c =
+}}
+{{eqn | r = 1 + 7 + 21 + 35
+ | c =
+}}
+{{eqn | r = 64
+ | c =
+}}
+{{eqn | r = 8^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | r = \dbinom {15} 0 + \dbinom {15} 1 + \dbinom {15} 2 + \dbinom {15} 3
+ | o =
+ | c =
+}}
+{{eqn | r = \dfrac {15!} {15! \, 0!} + \dfrac {15!} {14! \, 1!} + \dfrac {15!} {13! \, 2!} + \dfrac {15!} {12! \, 3!}
+ | c = {{Defof|Binomial Coefficient}}
+}}
+{{eqn | r = \dfrac {15!} {15! \times 1} + \dfrac {15} 1 + \dfrac {15 \times 14} {2 \times 1} + \dfrac {15 \times 14 \times 13} {3 \times 2 \times 1}
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = 1 + 15 + \dfrac {210} 2 + \dfrac {2730} 6
+ | c =
+}}
+{{eqn | r = 1 + 15 + 105 + 455
+ | c =
+}}
+{{eqn | r = 576
+ | c =
+}}
+{{eqn | r = 24^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | r = \dbinom {74} 0 + \dbinom {74} 1 + \dbinom {74} 2 + \dbinom {74} 3
+ | o =
+ | c =
+}}
+{{eqn | r = \dfrac {74!} {74! \, 0!} + \dfrac {74!} {73! \, 1!} + \dfrac {74!} {72! \, 2!} + \dfrac {74!} {71! \, 3!}
+ | c = {{Defof|Binomial Coefficient}}
+}}
+{{eqn | r = \dfrac {74!} {74! \times 1} + \dfrac {74} 1 + \dfrac {74 \times 73} {2 \times 1} + \dfrac {74 \times 73 \times 72} {3 \times 2 \times 1}
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = 1 + 74 + \dfrac {5402} 2 + \dfrac {388 \, 944} 6
+ | c =
+}}
+{{eqn | r = 1 + 74 + 2701 + 64 \, 824
+ | c =
+}}
+{{eqn | r = 67 \, 600
+ | c =
+}}
+{{eqn | r = 260^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | r = \dbinom {767} 0 + \dbinom {767} 1 + \dbinom {767} 2 + \dbinom {767} 3
+ | o =
+ | c =
+}}
+{{eqn | r = \dfrac {767!} {767! \, 0!} + \dfrac {767!} {766! \, 1!} + \dfrac {767!} {765! \, 2!} + \dfrac {767!} {764! \, 3!}
+ | c = {{Defof|Binomial Coefficient}}
+}}
+{{eqn | r = \dfrac {767!} {767! \times 1} + \dfrac {767} 1 + \dfrac {767 \times 766} {2 \times 1} + \dfrac {767 \times 766 \times 765} {3 \times 2 \times 1}
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = 1 + 767 + \dfrac {587 \, 522} 2 + \dfrac {449 \, 454 \, 330} 6
+ | c =
+}}
+{{eqn | r = 1 + 767 + 293 \, 761 + 74 \, 909 \, 055
+ | c =
+}}
+{{eqn | r = 75 \, 203 \, 584
+ | c =
+}}
+{{eqn | r = 8672^2
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to show there are no more.}}
+\end{proof}<|endoftext|>
+\section{Implicit Function Theorem for Lipschitz Contractions}
+Tags: Implicit Functions
+
+\begin{theorem}
+Let $M$ and $N$ be [[Definition:Metric Space|metric spaces]].
+Let $M$ be [[Definition:Complete Metric Space|complete]].
+Let $f : M \times N \to M$ be a [[Definition:Lipschitz Continuous|Lipschitz continuous]] [[Definition:Uniform Contraction Mapping|uniform contraction]].
+Then for all $t\in N$ there exists a [[Definition:Unique|unique]] $g(t) \in M$ such that $f(g(t), t) = g(t)$, and the [[Definition:Mapping|mapping]] $g : N \to M$ is [[Definition:Lipschitz Continuous|Lipschitz continuous]].
+\end{theorem}
+
+\begin{proof}
+For every $t\in N$, the [[Definition:Mapping|mapping]]:
+:$f_t : M \to M : x \mapsto f(x,t)$ is a [[Definition:Contraction Mapping|contraction]].
+By the [[Banach Fixed-Point Theorem]], there exists a [[Definition:unique|unique]] $g(t) \in M$ such that $f_t(g(t)) = g(t)$.
+We show that $g$ is [[Definition:Lipschitz Continuous|Lipschitz continuous]].
+Let $K<1$ be a [[Definition:Uniform Lipschitz Constant|uniform Lipschitz constant]] for $f$.
+Let $L$ be a [[Definition:Lipschitz Constant|Lipschitz constant]] for $f$.
+Let $s,t\in N$.
+Then
+{{begin-eqn}}
+{{eqn | l = d(g(s), g(t))
+ | r = d(f(g(s), s), f(g(t), t))
+ | c = Definition of $g$
+}}
+{{eqn | o = \leq
+ | r = d(f(g(s), s), f(g(t), s)) + d(f(g(t), s), f(g(t), t))
+ | c = {{defof|Metric}}
+}}
+{{eqn | o = \leq
+ | r = K \cdot d(g(s), g(t)) + d(f(g(t), s), f(g(t), t))
+ | c = $f$ is a [[Definition:Uniform Contraction Mapping|uniform contraction]]
+}}
+{{end-eqn}}
+and thus:
+{{begin-eqn}}
+{{eqn | l = d(g(s), g(t))
+ | o = \leq
+ | r = \dfrac1{1-K}d(f(g(t), s), f(g(t), t))
+}}
+{{eqn | o = \leq
+ | r = \dfrac L{1-K} d(s,t)
+ | c = $f$ is [[Definition:Lipschitz Continuous|lipschitz continuous]]
+}}
+{{end-eqn}}
+Thus $g$ is [[Definition:Lipschitz Continuous|lipschitz continuous]].
+{{qed}}
+[[Category:Implicit Functions]]
+66rdm27e18ivrvk98jlfgzalfq9xt22
+\end{proof}<|endoftext|>
+\section{Implicit Function Theorem for Lipschitz Contraction at Point}
+Tags: Implicit Functions
+
+\begin{theorem}
+Let $M$ and $N$ be [[Definition:Metric Space|metric spaces]].
+Let $M$ be [[Definition:Complete Metric Space|complete]].
+Let $f : M \times N \to M$ be a [[Definition:Uniform Contraction Mapping|uniform contraction]].
+Then for all $t\in N$ there exists a [[Definition:Unique|unique]] $g(t) \in M$ such that $f(g(t), t) = g(t)$, and if $f$ is [[Definition:Lipschitz Continuous at Point|Lipschitz continuous]] at a point $(g(t),t)$, then $g$ is [[Definition:Lipschitz Continuous at Point|Lipschitz continuous]] at $t$.
+\end{theorem}
+
+\begin{proof}
+For every $t\in N$, the [[Definition:Mapping|mapping]]:
+:$f_t : M \to M : x \mapsto f(x,t)$ is a [[Definition:Contraction Mapping|contraction]].
+By the [[Banach Fixed-Point Theorem]], there exists a [[Definition:unique|unique]] $g(t) \in M$ such that $f_t(g(t)) = g(t)$.
+Let $f$ be [[Definition:Lipschitz Continuous at Point|Lipschitz continuous]] at $(g(t),t)$.
+We show that $g$ is [[Definition:Lipschitz Continuous at Point|Lipschitz continuous]] at $t$.
+Let $K<1$ be a [[Definition:Uniform Lipschitz Constant|uniform Lipschitz constant]] for $f$.
+Let $L$ be a [[Definition:Lipschitz Constant|Lipschitz constant]] for $f$ at $a$.
+Let $s\in N$.
+Then
+{{begin-eqn}}
+{{eqn | l = d(g(s), g(t))
+ | r = d(f(g(s), s), f(g(t), t))
+ | c = Definition of $g$
+}}
+{{eqn | o = \leq
+ | r = d(f(g(s), s), f(g(t), s)) + d(f(g(t), s), f(g(t), t))
+ | c = {{defof|Metric}}
+}}
+{{eqn | o = \leq
+ | r = K \cdot d(g(s), g(t)) + d(f(g(t), s), f(g(t), t))
+ | c = $f$ is a [[Definition:Uniform Contraction Mapping|uniform contraction]]
+}}
+{{end-eqn}}
+and thus:
+{{begin-eqn}}
+{{eqn | l = d(g(s), g(t))
+ | o = \leq
+ | r = \dfrac1{1-K}d(f(g(t), s), f(g(t), t))
+}}
+{{eqn | o = \leq
+ | r = \dfrac L{1-K} d(s,t)
+ | c = $f$ is [[Definition:Lipschitz Continuous at Point|lipschitz continuous]] at $(g(t),t)$
+}}
+{{end-eqn}}
+Thus $g$ is [[Definition:Lipschitz Continuous|lipschitz continuous]] at $t$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Local Normal Form for Immersions}
+Tags: Immersions, Implicit Functions
+
+\begin{theorem}
+Let $\Omega\subset\R^k$ be [[Definition:Open Set of Real Euclidean Space|open]].
+Let $f: \Omega \to \R^n$ be an [[Definition:Immersion|immersion]].
+Let $p \in \Omega$.
+Then:
+:$k \le n$
+and there exists a [[Definition:Local Diffeomorphism|local diffeomorphism]] $\phi$ around $\map f p$ such that:
+:$\phi \circ \map f x = \tuple {x, 0}$
+for all $x$ in a [[Definition:Neighborhood (Topology)|neighborhood]] of $p$.
+\end{theorem}<|endoftext|>
+\section{Local Normal Form for Submersions}
+Tags: Submersions, Implicit Functions
+
+\begin{theorem}
+Let $\Omega\subset\R^n$ be [[Definition:Open Set of Real Euclidean Space|open]].
+Let $f : \Omega \to \R^k$ be an [[Definition:Submersion|submersion]].
+Let $p\in\Omega$.
+Then $n\geq k$, and there exists a [[Definition:Local Diffeomorphism|local diffeomorphism]] $\phi$ around $f(p)$ such that
+:$\phi\circ f (x, y) = x$ for all $(x, y)$ in a [[Definition:Neighborhood (Topology)|neighborhood]] of $p$.
+\end{theorem}<|endoftext|>
+\section{Smallest Square Inscribed in Two Pythagorean Triangles}
+Tags: Pythagorean Triangles
+
+\begin{theorem}
+The smallest [[Definition:Square (Geometry)|square]] with [[Definition:Integer|integer]] [[Definition:Side of Polygon|sides]] that can be [[Definition:Inscribe/Polygon within Polygon|inscribed]] within two different [[Definition:Pythagorean Triangle|Pythagorean triangles]] so that one [[Definition:Side of Polygon|side]] of the [[Definition:Square (Geometry)|square]] lies on the [[Definition:Hypotenuse|hypotenuse]] has [[Definition:Side of Polygon|side]] [[Definition:Length of Line|length]] $780$.
+The two [[Definition:Pythagorean Triangle|Pythagorean triangles]] in question have [[Definition:Side of Polygon|side]] [[Definition:Length of Line|lengths]] $\tuple {1443, 1924, 2405}$ and $\tuple {1145, 2748, 2977}$.
+\end{theorem}<|endoftext|>
+\section{Sequence of Numbers Divisible by Sequence of Primes}
+Tags: Prime Numbers, Divisors, Recreational Mathematics
+
+\begin{theorem}
+The [[Definition:Positive Integer|integers]] in this [[Definition:Integer Sequence|sequence]]:
+:$788, 789, 790, 791, 792, 793$
+are [[Definition:Divisor of Integer|divisible]] by:
+:$2, 3, 5, 7, 11, 13$
+respectively.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 788
+ | r = 2 \times 394
+}}
+{{eqn | l = 789
+ | r = 3 \times 263
+}}
+{{eqn | l = 790
+ | r = 5 \times 158
+}}
+{{eqn | l = 791
+ | r = 7 \times 113
+}}
+{{eqn | l = 792
+ | r = 11 \times 72
+}}
+{{eqn | l = 793
+ | r = 13 \times 61
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers whose Square is Palindromic with Even Number of Digits}
+Tags: Square Numbers, Palindromic Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] whose [[Definition:Square (Algebra)|square]] is a [[Definition:Palindromic Number|palindromic number]] with an [[Definition:Even Integer|even number]] of [[Definition:Digit|digits]] begins:
+:$836, 798 \, 644, 64 \, 030 \, 648, 83 \, 163 \, 115 \, 486, 6 \, 360 \, 832 \, 925 \, 898, \ldots$
+{{OEIS|A016113}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 836^2
+ | r = 698 \, 896
+ | c = $6$ [[Definition:Digit|digits]]
+}}
+{{eqn | l = 798 \, 644^2
+ | r = 637 \, 832 \, 238 \, 736
+ | c = $12$ [[Definition:Digit|digits]]
+}}
+{{eqn | l = 64 \, 030 \, 648^2
+ | r = 4 \, 099 \, 923 \, 883 \, 299 \, 904
+ | c = $16$ [[Definition:Digit|digits]]
+}}
+{{eqn | l = 83 \, 163 \, 115 \, 486^2
+ | r = 6 \, 916 \, 103 \, 777 \, 337 \, 773 \, 016 \, 196
+ | c = $22$ [[Definition:Digit|digits]]
+}}
+{{eqn | l = 6 \, 360 \, 832 \, 925 \, 898^2
+ | r = 40 \, 460 \, 195 \, 511 \, 188 \, 111 \, 559 \, 106 \, 404
+ | c = $26$ [[Definition:Digit|digits]]
+}}
+{{end-eqn}}
+\end{proof}<|endoftext|>
+\section{Sum of Sequence of Factorials}
+Tags: Factorials, Sums of Sequences
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] $S = \sequence {s_n}$ defined as:
+:$\displaystyle s_n = \sum_{k \mathop = 1}^n k!$
+begins:
+:$1, 3, 9, 33, 153, 873, 5913, 46 \, 233, 409 \, 113, 4 \, 037 \, 913, \ldots$
+{{OEIS|A007489}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = s_1
+ | r = 1!
+ | c =
+}}
+{{eqn | r = 1
+ | c = {{Defof|Factorial}}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = s_2
+ | r = s_1 + 2!
+ | c =
+}}
+{{eqn | r = 1 + 2
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = 3
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = s_3
+ | r = s_2 + 3!
+ | c =
+}}
+{{eqn | r = 3 + 6
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = 9
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = s_4
+ | r = s_3 + 4!
+ | c =
+}}
+{{eqn | r = 9 + 24
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = 33
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = s_5
+ | r = s_4 + 5!
+ | c =
+}}
+{{eqn | r = 33 + 120
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = 153
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = s_6
+ | r = s_5 + 6!
+ | c =
+}}
+{{eqn | r = 153 + 720
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = 873
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = s_7
+ | r = s_6 + 7!
+ | c =
+}}
+{{eqn | r = 873 + 5040
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = 5913
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = s_8
+ | r = s_7 + 8!
+ | c =
+}}
+{{eqn | r = 5913 + 40 \, 320
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = 46 \, 223
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = s_9
+ | r = s_8 + 9!
+ | c =
+}}
+{{eqn | r = 46 \, 223 + 362 \, 880
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = 409 \, 113
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = s_{10}
+ | r = s_9 + 10!
+ | c =
+}}
+{{eqn | r = 409 \, 113 + 3 \, 628 \, 800
+ | c = {{Defof|Factorial}}
+}}
+{{eqn | r = 4 \, 037 \, 913
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Factorials]]
+[[Category:Sums of Sequences]]
+korxjeya3vqwt1ew8ur173dgderib20
+\end{proof}<|endoftext|>
+\section{Number of Magic Squares of Order 4}
+Tags: Magic Squares, 880
+
+\begin{theorem}
+The number of different [[Definition:Magic Square|magic squares]] of [[Definition:Order of Magic Square|order $4$]], up to rotation and reflection, is $880$.
+\end{theorem}<|endoftext|>
+\section{Primitive Semiperfect Numbers which are not Primitive Abundant}
+Tags: Primitive Semiperfect Numbers, Primitive Abundant Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Primitive Semiperfect Number|primitive semiperfect numbers]] which are not also [[Definition:Primitive Abundant Number|primitive abundant]] starts:
+:$6, 28, 350, 490, 496, 770, 910, 1190, \ldots$
+These are [[Definition:Semiperfect Number|semiperfect numbers]] which are either:
+: [[Definition:Perfect Number|perfect]]
+or:
+: whose only [[Definition:Abundant Number|abundant]] [[Definition:Aliquot Part|aliquot parts]] are [[Definition:Weird Number|weird]].
+\end{theorem}
+
+\begin{proof}
+A [[Definition:Primitive Semiperfect Number|primitive semiperfect number]] is a [[Definition:Semiperfect Number|semiperfect number]] which has no [[Definition:Aliquot Part|aliquot parts]] which are themselves [[Definition:Semiperfect Number|semiperfect]].
+Thus by definition a [[Definition:Primitive Semiperfect Number|primitive semiperfect number]] is either [[Definition:Perfect Number|perfect]] or [[Definition:Abundant Number|abundant]].
+A [[Definition:Primitive Abundant Number|primitive abundant number]] is an [[Definition:Abundant Number|abundant number]] whose [[Definition:Aliquot Part|aliquot parts]] are all [[Definition:Deficient Number|deficient]].
+Thus the [[Definition:Perfect Number|perfect numbers]]:
+:$6, 28, 496, \ldots$
+are not [[Definition:Primitive Abundant Number|primitive abundant]].
+However, by [[Divisor of Perfect Number is Deficient]], the [[Definition:Perfect Number|perfect numbers]] are all by definition [[Definition:Primitive Semiperfect Number|primitive semiperfect]].
+Hence the presence of the [[Definition:Perfect Number|perfect numbers]] in this [[Definition:Integer Sequence|sequence]].
+Next, consider the [[Definition:Integer Sequence|sequence]] of [[Definition:Weird Number|weird numbers]]:
+:$70, 836, \ldots$
+These are numbers which are [[Definition:Abundant Number|abundant]] but not [[Definition:Semiperfect Number|semiperfect]].
+Thus an [[Definition:Abundant Number|abundant number]] whose [[Definition:Aliquot Part|aliquot parts]] are all [[Definition:Deficient Number|deficient]] except for one or more [[Definition:Weird Number|weird numbers]] is a [[Definition:Primitive Semiperfect Number|primitive semiperfect number]] which is not [[Definition:Primitive Abundant Number|primitive abundant]].
+However, such a number is not [[Definition:Primitive Abundant Number|primitive abundant]] because it has an [[Definition:Aliquot Part|aliquot part]] which is [[Definition:Abundant Number|abundant]], though [[Definition:Weird Number|weird]].
+So:
+:while $70$ is [[Definition:Primitive Abundant Number|primitive abundant]] it is not [[Definition:Primitive Semiperfect Number|primitive semiperfect number]]
+:while $350$, $490$, $710$, $910$, $1190$, and so on, are [[Definition:Primitive Semiperfect Number|primitive semiperfect]], they are not [[Definition:Primitive Abundant Number|primitive abundant]].
+Hence the result.
+{{qed}}
+[[Category:Primitive Semiperfect Numbers]]
+[[Category:Primitive Abundant Numbers]]
+8aalkwm0o7fcvh38xcqvh31x3muet2s
+\end{proof}<|endoftext|>
+\section{Sequence of Odd Abundant Numbers}
+Tags: Abundant Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Odd Integer|odd]] [[Definition:Abundant Number|abundant numbers]] begins:
+:$945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, \ldots$
+{{OEIS|A005231}}
+\end{theorem}<|endoftext|>
+\section{Set of 5 Triplets whose Sums and Products are Equal}
+Tags: Recreational Mathematics, 981, 1,425,600
+
+\begin{theorem}
+The following [[Definition:Set|set]] of $5$ [[Definition:Ordered Triple|triplets]] of [[Definition:Positive Integer|integers]] have the property that:
+:the [[Definition:Integer Addition|sum]] of the [[Definition:Positive Integer|integers]] in each [[Definition:Ordered Triple|triplet]] are equal
+and:
+:the [[Definition:Integer Multiplication|product]] of the [[Definition:Positive Integer|integers]] in each [[Definition:Ordered Triple|triplet]] are equal:
+:$\tuple {6, 480, 495}$, $\tuple {11, 160, 810}$, $\tuple {12, 144, 825}$, $\tuple {20, 81, 880}$, $\tuple {33, 48, 900}$
+The [[Definition:Integer Addition|sum]] is $981$, and the [[Definition:Integer Multiplication|product]] is $1 \, 425 \, 600$.
+This is the only known such [[Definition:Set|set]] of $5$ [[Definition:Ordered Triple|triplets]] of [[Definition:Positive Integer|integers]] with this property.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 6 + 480 + 495
+ | r = 981
+}}
+{{eqn | l = 11 + 160 + 810
+ | r = 981
+}}
+{{eqn | l = 12 + 144 + 825
+ | r = 981
+}}
+{{eqn | l = 20 + 81 + 880
+ | r = 981
+}}
+{{eqn | l = 33 + 48 + 900
+ | r = 981
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6 \times 480 \times 495
+ | r = \paren {2 \times 3} \times \paren {2^5 \times 3 \times 5} \times \paren {3^2 \times 5 \times 11}
+}}
+{{eqn | r = 2^6 \times 3^4 \times 5^2 \times 11
+}}
+{{eqn | r = 1 \, 425 \, 600
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 11 \times 160 \times 810
+ | r = 11 \times \paren {2^5 \times 5} \times \paren {2 \times 3^4 \times 5}
+}}
+{{eqn | r = 2^6 \times 3^4 \times 5^2 \times 11
+}}
+{{eqn | r = 1 \, 425 \, 600
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 12 \times 144 \times 825
+ | r = \paren {2^2 \times 3} \times \paren {2^4 \times 3^2} \times \paren {3 \times 5^2 \times 11}
+}}
+{{eqn | r = 2^6 \times 3^4 \times 5^2 \times 11
+}}
+{{eqn | r = 1 \, 425 \, 600
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 20 \times 81 \times 880
+ | r = \paren {2^2 \times 5} \times 3^4 \times \paren {2^4 \times 5 \times 11}
+}}
+{{eqn | r = 2^6 \times 3^4 \times 5^2 \times 11
+}}
+{{eqn | r = 1 \, 425, 600
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 33 \times 48 \times 900
+ | r = \paren {3 \times 11} \times \paren {2^4 \times 3} \times \paren {2^2 \times 3^2 \times 5^2}
+}}
+{{eqn | r = 2^6 \times 3^4 \times 5^2 \times 11
+}}
+{{eqn | r = 1 \, 425 \, 600
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum of Pandigital Triplet of 3-Digit Primes}
+Tags: Prime Numbers, 999
+
+\begin{theorem}
+The smallest [[Definition:Integer|integer]] which is the [[Definition:Integer Addition|sum]] of a [[Definition:Set|set]] of $3$ [[Definition:Digit|three-digit]] [[Definition:Prime Number|primes]] using all $9$ [[Definition:Digit|digits]] from $1$ to $9$ once each is $999$:
+:$149 + 263 + 587 = 999$
+\end{theorem}
+
+\begin{proof}
+All [[Definition:Digit|three-digit]] [[Definition:Prime Number|primes]] end in $1, 3, 7, 9$.
+Suppose $1$ is used as the units digit of a [[Definition:Prime Number|prime]].
+Since the digit $1$ cannot be used again, the sum of the [[Definition:Prime Number|primes]] is at least:
+:$221 + 333 + 447 = 1001$
+so $1$ cannot be used as a units digit .
+The units digits of the [[Definition:Prime Number|primes]] are $3, 7, 9$.
+To minimise the sum, the hundreds digits must be $1, 2, 4$.
+This leaves $5, 6, 8$ be the tens digits.
+The [[Definition:Prime Number|primes]] satisfying these conditions are:
+:$157, 163, 167$
+:$257, 263, 269, 283$
+:$457, 463, 467, 487$
+Only $269$ contain $9$, so we must choose it.
+Only $157$ does not contain $6$, so must choose it next.
+But then all [[Definition:Prime Number|primes]] beginning with $4$ have some digit coinciding with the above [[Definition:Prime Number|primes]].
+Hence the next minimal sum of the [[Definition:Prime Number|primes]] (if they exist) is:
+:$10^2 \paren {1 + 2 + 5} + 10 \paren {4 + 6 + 8} + \paren {3 + 7 + 9} = 999$
+and we have shown that these [[Definition:Prime Number|primes]] do exist.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Divisibility Test for 7, 11 and 13}
+Tags: Divisibility Tests, 7, 11, 13, 1001
+
+\begin{theorem}
+Mark off the [[Definition:Integer|integer]] $N$ being tested into groups of $3$ [[Definition:Digit|digits]].
+Because of the standard way of presenting [[Definition:Integer|integers]], this may already be done, for example:
+:$N = 22 \, 846 \, 293 \, 462 \, 733 \, 356$
+Number the groups of $3$ from the right:
+:$N = \underbrace{22}_6 \, \underbrace{846}_5 \, \underbrace{293}_4 \, \underbrace{462}_3 \, \underbrace{733}_2 \, \underbrace{356}_1$
+Considering each group a [[Definition:Digit|$3$-digit]] [[Definition:Integer|integer]], [[Definition:Integer Addition|add]] the [[Definition:Even Integer|even]] numbered groups together, and [[Definition:Integer Subtraction|subtract]] the [[Definition:Odd Integer|odd]] numbered groups:
+:$22 - 846 + 293 - 462 + 733 - 356 = -616$
+where the [[Definition:Sign of Number|sign]] is irrelevant.
+If the result is [[Definition:Divisor of Integer|divisible]] by $7$, $11$ or $13$, then so is $N$.
+In this case:
+:$616 = 2^3 \times 7 \times 11$
+and so $N$ is divisible by $7$ and $11$ but not $13$.
+\end{theorem}
+
+\begin{proof}
+Let $N$ be expressed as:
+:$N = \displaystyle \sum_{k \mathop = 0}^n a_k 1000^k = a_0 + a_1 1000 + a_2 1000^2 + \cdots + a_n 1000^n$
+where $n$ is the number of groups of $3$ [[Definition:Digit|digits]].
+We have that:
+:$1000 \equiv -1 \pmod {1001}$
+Hence from [[Congruence of Powers]]:
+:$1000^r \equiv \paren {-1}^r \pmod {1001}$
+Thus:
+:$N \equiv a_0 + \paren {-1} a_1 + \paren {-1}^2 a_2 + \cdots + \paren {-1}^n a_n \pmod {1001}$
+from the definition of [[Definition:Modulo Addition|Modulo Addition]].
+Then we note that:
+:$1001 = 7 \times 11 \times 13$
+and the result follows.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Solutions to Diophantine Equation x (x + 1) = y (y + 5) (y + 10) (y + 15)}
+Tags: Diophantine Equations
+
+\begin{theorem}
+The [[Definition:Diophantine Equation|Diophantine equation]]
+: $n = x \left({x + 1}\right) = y \left({y + 5}\right) \left({y + 10}\right) \left({y + 15}\right)$
+has exactly $2$ solutions:
+{{begin-eqn}}
+{{eqn | l = 1056
+ | r = 32 \times 33 = 1 \times 6 \times 11 \times 16
+}}
+{{eqn | l = 43 \, 056
+ | r = 207 \times 208 = 8 \times 13 \times 18 \times 23
+}}
+{{end-eqn}}
+\end{theorem}<|endoftext|>
+\section{Numbers Reversed when Multiplying by 9}
+Tags: Reversals, 1089
+
+\begin{theorem}
+Numbers of the form $\sqbrk {10 (9) 89}_{10}$ are [[Definition:Reversal|reversed]] when they are [[Definition:Integer Multiplication|multiplied]] by $9$:
+{{begin-eqn}}
+{{eqn | l = 1089 \times 9
+ | r = 9801
+}}
+{{eqn | l = 10 \, 989 \times 9
+ | r = 98 \, 901
+}}
+{{eqn | l = 109 \, 989 \times 9
+ | r = 989 \, 901
+}}
+{{end-eqn}}
+and so on.
+\end{theorem}
+
+\begin{proof}
+Let k represent the number of $9$s in the middle of the number.
+For $k > 0$ We can rewrite the number as follows:
+{{begin-eqn}}
+{{eqn | l = \sqbrk {10 (9) 89}_{10}
+ | r = 10 \times 10^{k + 2 } + 900 \sum_{i \mathop = 0}^{k - 1} 10^i + 89
+ | c = {{Defof|Geometric Series}}
+}}
+{{end-eqn}}
+Taking numbers of this form and multiplying by $9$ produces:
+{{begin-eqn}}
+{{eqn | l = 9 \times \paren {10 \times 10^{k + 2 } + 900 \sum_{i \mathop = 0}^{k - 1} 10^i + 89 }
+ | r = 90 \times 10^{k + 2 } + 8100 \sum_{i \mathop = 0}^{k - 1} 10^i + 801
+}}
+{{end-eqn}}
+The first part is composed of $k + 4$ [[Definition:Digit|digits]]. The first [[Definition:Digit|digit]] will be $9$ followed by $k + 3$ [[Definition:Digit|digits]] of $0$
+{{begin-eqn}}
+{{eqn | l = 90 \times 10^{k + 2 }
+ | r = 9 \times 10^{k + 3 }
+}}
+{{end-eqn}}
+The sum in the middle is composed of $k + 3$ [[Definition:Digit|digits]]. The first [[Definition:Digit|digit]] will be $8$ followed by $k - 1$ [[Definition:Digit|digits]] of $9$ and then the remaining three [[Definition:Digit|digits]] at the end are $100$
+{{begin-eqn}}
+{{eqn | l = 8100 \sum_{i \mathop = 0}^{k - 1} 10^i
+ | r = 899 \cdots 99100
+}}
+{{end-eqn}}
+Summing the three pieces, the final answer will have $k + 4$ [[Definition:Digit|digits]].
+The first [[Definition:Digit|digit]] is $9$
+followed by $8$ which is the first [[Definition:Digit|digit]] of the middle part
+followed by $k$ [[Definition:Digit|digits]] of $9$ where the last $9$ is the sum of the $1$ from the middle part and the $8$ of the last part
+and then ending in $01$:
+{{begin-eqn}}
+{{eqn | l = \sqbrk {10 (9) 89}_{10} \times 9
+ | r = \sqbrk {98 (9) 01}_{10}
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Reciprocal of 1089}
+Tags: 1089, Examples of Reciprocals
+
+\begin{theorem}
+:$\dfrac 1 {1089} = 0 \cdotp \dot 00091 \, 82736 \, 45546 \, 37281 \, 9 \dot 1$
+\end{theorem}
+
+\begin{proof}
+Performing the calculation using [[Definition:Long Division|long division]]:
+
+ 0.00091827364554637281910009...
+ --------------------------------------------
+1089)1.00000000000000000000000000...
+ 9801 5445 1089
+ ---- ---- ----
+ 1990 5950 9910
+ 1089 5445 9801
+ ---- ---- ----
+ 9010 5050 1090
+ 8712 4356 1089
+ ---- ---- ----
+ 2980 6940 10000
+ 2178 6534 9801
+ ---- ---- ----
+ 8020 4060 ....
+ 7623 3267
+ ---- ----
+ 3970 7930
+ 3267 7623
+ ---- ----
+ 7030 3070
+ 6534 2178
+ ---- ----
+ 4960 8920
+ 4356 8712
+ ---- ----
+ 6040 2080
+ 5445 1089
+
{{qed}}
+\end{proof}<|endoftext|>
+\section{Square which is Difference between Square and Square of Reversal}
+Tags: Square Numbers, Reversals
+
+\begin{theorem}
+$33^2 = 65^2 - 56^2$
+This is the only [[Definition:Square Number|square]] of a $2$-[[Definition:Digit|digit]] number which has this property.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 33^2
+ | r = 1089
+ | c =
+}}
+{{eqn | l = 65^2 - 56^2
+ | r = 4225 - 3136
+ | c =
+}}
+{{eqn | r = 1089
+ | c =
+}}
+{{end-eqn}}
+Let $\sqbrk {xy}$ be a $2$-[[Definition:Digit|digit]] number such that $x \le y$ and $\sqbrk {xy}^2 - \sqbrk {yx}^2$ is a [[Definition:Square Number|square]] of a $2$-[[Definition:Digit|digit]] number.
+The case $x = y$ gives the solution $\sqbrk {xy}^2 - \sqbrk {yx}^2 = 0$, which is not a [[Definition:Square Number|square]] of a $2$-[[Definition:Digit|digit]] number.
+For $x \ne y$:
+{{begin-eqn}}
+{{eqn | l = \sqbrk {xy}^2 - \sqbrk {yx}^2
+ | r = \paren {10 x + y}^2 - \paren {10 y + x}^2
+}}
+{{eqn | r = 100 x^2 + 20 x y + y^2 - 100 y^2 - 20 y x - x^2
+}}
+{{eqn | r = 99 \paren {x^2 - y^2}
+}}
+{{eqn | r = 3^2 \times 11 \paren {x - y} \paren {x + y}
+}}
+{{eqn | ll = \leadsto
+ | l = \paren {x - y} \paren {x + y}
+ | r = 11 n^2
+ | c = for some integer $n$
+}}
+{{end-eqn}}
+By [[Euclid's Lemma for Prime Divisors]], one of $x - y, x + y$ must be divisible by $11$.
+Hence from [[Absolute Value of Integer is not less than Divisors]], either $x - y$ or $x + y$ must be greater than or equal to $11$.
+Since $x - y < x < 9$, $x - y$ cannot be a multiple of $11$.
+From $1 = 1 + 0 \le x + y < 9 + 9 = 18$, we have that $x + y = 11$.
+This implies that $x - y$ is a [[Definition:Square Number|square number]].
+$x + y = 11$ gives $\tuple {x,y} = \tuple {6,5}, \tuple {7,4}, \tuple {8,3}, \tuple {9,2}$ as the possible solutions.
+Among these solutions, only $\tuple {6,5}$ has a difference of a [[Definition:Square Number|square number]]: $1$.
+Therefore the only [[Definition:Square Number|square]] of a $2$-[[Definition:Digit|digit]] number expressible as a difference between a [[Definition:Square Number|square]] and the [[Definition:Square Number|square]] of its [[Definition:Reversal|reversal]] is:
+:$65^2 - 56^2 = 33^2$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers for which Sixth Power plus 1091 is Composite}
+Tags: Sixth Powers
+
+\begin{theorem}
+The number $1091$ has the property that:
+:$x^6 + 1091$
+is [[Definition:Composite Number|composite]] for all [[Definition:Integer|integer]] values of $x$ from $1$ to $3905$.
+\end{theorem}
+
+\begin{proof}
+We check the result and show that it cannot be improved further by showing:
+:$3906$ is the smallest $x$ such that $x^6 + 1091$ is [[Definition:Prime Number|prime]].
+Suppose $x^6 + 1091$ is [[Definition:Prime Number|prime]].
+Then:
+:$x$ is a multiple of $42$
+:$x$ ends in $0$, $4$ or $6$ in decimal notation
+:$x \not \equiv \pm 1, \pm 3, \pm 4 \pmod {13}$
+:$x \not \equiv \pm 4, \pm 6, \pm 9 \pmod {19}$
+The proof is split into $6$ parts:
+=== $x$ is a multiple of $2$ ===
+Suppose not. Then $x$ is [[Definition:Odd Integer|odd]], and so is $x^6$.
+Hence $x^6 + 1091$ is [[Definition:Even Integer|even]], and thus [[Definition:Composite Number|composite]].
+Thus we must require $x$ to be [[Definition:Even Integer|even]].
+{{qed|lemma}}
+=== $x$ is a multiple of $3$ ===
+Suppose not. Write $x = 3 k \pm 1$.
+Hence:
+{{begin-eqn}}
+{{eqn | l = x^6 + 1091
+ | r = \paren {3 k \pm 1}^6 + 1091
+}}
+{{eqn | o = \equiv
+ | r = \paren {\pm 1}^6 - 1
+ | rr = \pmod 3
+ | c = [[Congruence of Powers]]
+}}
+{{eqn | o = \equiv
+ | r = 0
+ | rr = \pmod 3
+}}
+{{end-eqn}}
+showing that $x^6 + 1091$ is [[Definition:Composite Number|composite]] ([[Definition:Divisor of Integer|divisible]] by $3$).
+Thus we must require $x$ to be a multiple of $3$.
+{{qed|lemma}}
+=== $x$ is a multiple of $7$ ===
+Suppose not. Then $x \perp 7$.
+Then:
+{{begin-eqn}}
+{{eqn | l = n^6 + 1091
+ | o = \equiv
+ | r = 1 + 1091
+ | rr = \pmod 7
+ | c = [[Fermat's Little Theorem]]
+}}
+{{eqn | o = \equiv
+ | r = 0
+ | rr = \pmod 7
+}}
+{{end-eqn}}
+showing that $x^6 + 1091$ is [[Definition:Composite Number|composite]] ([[Definition:Divisor of Integer|divisible]] by $7$).
+Thus we must require $x$ to be a multiple of $7$.
+{{qed|lemma}}
+=== $x$ ends in $0$, $4$ or $6$ ===
+From the above we require $x$ to be [[Definition:Even Integer|even]].
+Suppose $x$ ends in $2$ or $8$.
+Then $x^6$ ends in $4$.
+Thus $x^6 + 1091$ ends in $5$, which by [[Divisibility by 5]] is [[Definition:Divisor of Integer|divisible]] by $5$.
+Hence we must require $x$ to end in $0$, $4$ or $6$.
+{{qed|lemma}}
+=== $x \not \equiv \pm 1, \pm 3, \pm 4 \pmod {13}$ ===
+Here is a table of $x^6 \pmod {13}$:
+:$\begin{array}{|c|c|c|c|c|c|c|c|}
+\hline x \bmod {13} & 0 & \pm 1 & \pm 2 & \pm 3 & \pm 4 & \pm 5 & \pm 6 \\
+\hline x^6 \bmod {13} & 0 & 1 & -1 & 1 & 1 & -1 & -1 \\
+\hline
+\end{array}$
+For $x \equiv \pm 1, \pm 3, \pm 4 \pmod {13}$:
+{{begin-eqn}}
+{{eqn | l = x^6 + 1091
+ | o = \equiv
+ | r = 1 - 1
+ | rr = \pmod {13}
+}}
+{{eqn | o = \equiv
+ | r = 0
+ | rr = \pmod {13}
+}}
+{{end-eqn}}
+showing that $x^6 + 1091$ is [[Definition:Composite Number|composite]] ([[Definition:Divisor of Integer|divisible]] by $13$).
+Thus we must require $x \not \equiv \pm 1, \pm 3, \pm 4 \pmod {13}$.
+{{qed|lemma}}
+=== $x \not \equiv \pm 4, \pm 6, \pm 9 \pmod {19}$ ===
+Here is a table of $x^6 \pmod {19}$:
+:$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
+\hline x \bmod {19} & 0 & \pm 1 & \pm 2 & \pm 3 & \pm 4 & \pm 5 & \pm 6 & \pm 7 & \pm 8 & \pm 9 \\
+\hline x^6 \bmod {19} & 0 & 1 & 7 & 7 & 11 & 6 & 11 & 1 & 1 & 11\\
+\hline
+\end{array}$
+For $x \equiv \pm 4, \pm 6, \pm 9 \pmod {19}$:
+{{begin-eqn}}
+{{eqn | l = x^6 + 1091
+ | o = \equiv
+ | r = 11 + 8
+ | rr = \pmod {19}
+}}
+{{eqn | o = \equiv
+ | r = 0
+ | rr = \pmod {19}
+}}
+{{end-eqn}}
+showing that $x^6 + 1091$ is [[Definition:Composite Number|composite]] ([[Definition:Divisor of Integer|divisible]] by $19$).
+Thus we must require $x \not \equiv \pm 4, \pm 6, \pm 9 \pmod {19}$.
+{{qed|lemma}}
+Here is a list of $x^6 + 1091$, where $42 \divides x$ and does not end in $2$ or $8$:
+{{begin-eqn}}
+{{eqn | l = 84^6 + 1091
+ | r = 23 \times 15273827509
+}}
+{{eqn | l = 126
+ | o = \equiv
+ | r = -4 \pmod {13}
+}}
+{{eqn | l = 210^6 + 1091
+ | r = 109 \times 786845146799
+}}
+{{eqn | l = 294
+ | o = \equiv
+ | r = +9 \pmod {19}
+}}
+{{eqn | l = 336
+ | o = \equiv
+ | r = -6 \pmod {19}
+}}
+{{eqn | l = 420
+ | o = \equiv
+ | r = +4 \pmod {13}
+}}
+{{eqn | l = 504
+ | o = \equiv
+ | r = -3 \pmod {13}
+}}
+{{eqn | l = 546^6 + 1091
+ | r = 2129 \times 12444578592403
+}}
+{{eqn | l = 630^6 + 1091
+ | r = 347 \times 41957 \times 4294468229
+}}
+{{eqn | o = etc.}}
+{{end-eqn}}
+{{finish|1/8 of the way there. I'll fill it in later}}
+The above results are obtained using an integer factorization calculator.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Wieferich's Criterion}
+Tags: Fermat's Last Theorem, Wieferich Primes
+
+\begin{theorem}
+Suppose [[Definition:Fermat's Equation|Fermat's equation]]:
+:$x^p + y^p = z^p$
+has a solution in which $p$ is an [[Definition:Odd Prime|odd prime]] that does not [[Definition:Divisor of Integer|divide]] any of $x$, $y$ or $z$.
+Then $2^{p - 1} - 1$ is [[Definition:Divisor of Integer|divisible]] by $p^2$.
+\end{theorem}<|endoftext|>
+\section{1105 as Sum of Two Squares}
+Tags: Sums of Squares, 1105
+
+\begin{theorem}
+$1105$ can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]] in more ways than any smaller [[Definition:Integer|integer]]:
+{{begin-eqn}}
+{{eqn | l = 1105
+ | m = 1089 + 16
+ | mo= =
+ | r = 33^2 + 4^2
+ | c =
+}}
+{{eqn | m = 1024 + 81
+ | mo= =
+ | r = 32^2 + 9^2
+ | c =
+}}
+{{eqn | m = 961 + 144
+ | mo= =
+ | r = 31^2 + 12^2
+ | c =
+}}
+{{eqn | m = 625 + 529
+ | mo= =
+ | r = 24^2 + 23^2
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Demonstrate there are no smaller with as many ways of doing this}}
+\end{proof}<|endoftext|>
+\section{Integer as Difference between Two Squares}
+Tags: Square Numbers
+
+\begin{theorem}
+Let $n$ be a [[Definition:Positive Integer|positive integer]].
+Then $n$ can be expressed as:
+:$n = a^2 - b^2$
+{{iff}} $n$ has at least two [[Definition:Distinct|distinct]] [[Definition:Divisor of Integer|divisors]] of the same [[Definition:Parity|parity]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = n
+ | r = a^2 - b^2
+ | c =
+}}
+{{eqn | r = \left({a + b}\right) \left({a - b}\right)
+ | c = [[Difference of Two Squares]]
+}}
+{{end-eqn}}
+Thus $n = p q$ where:
+{{begin-eqn}}
+{{eqn | n = 1
+ | l = p
+ | r = \left({a + b}\right)
+ | c =
+}}
+{{eqn | n = 2
+ | l = q
+ | r = \left({a - b}\right)
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = p + q
+ | r = 2 a
+ | c = $(1) + (2)$
+}}
+{{eqn | l = p - q
+ | r = 2 b
+ | c = $(1) - (2)$
+}}
+{{eqn | ll= \leadsto
+ | l = a
+ | r = \dfrac {p + q} 2
+ | c =
+}}
+{{eqn | l = b
+ | r = \dfrac {p - q} 2
+ | c =
+}}
+{{end-eqn}}
+Thus for $a$ and $b$ to be [[Definition:Integer|integers]], both $a$ and $b$ must be:
+:[[Definition:Distinct|distinct]], otherwise $p = q$ and so $b = 0$
+:either both [[Definition:Even Integer|even]] or both [[Definition:Odd Integer|odd]], otherwise both $\dfrac {p + q} 2$ and $\dfrac {p - q} 2$ will be [[Definition:Odd Integer|odd]].
+Hence the result.
+{{qed}}
+[[Category:Square Numbers]]
+mxz53irrff1a78q6brxyd4g53y7yiax
+\end{proof}<|endoftext|>
+\section{Difference between Two Squares equal to Repunit/Corollary 1}
+Tags: Difference between Two Squares equal to Repunit
+
+\begin{theorem}
+{{begin-eqn}}
+{{eqn | l = 6^2 - 5^2
+ | r = 11
+ | c =
+}}
+{{eqn | l = 56^2 - 45^2
+ | r = 1111
+ | c =
+}}
+{{eqn | l = 556^2 - 445^2
+ | r = 111 \, 111
+ | c =
+}}
+{{eqn | o = :
+ | c =
+}}
+{{end-eqn}}
+and in general for [[Definition:Integer|integer]] $n$:
+:$R_{2 n} = {\underbrace {55 \ldots 56}_{n - 1 \ 5 \text{'s} } }^2 - {\underbrace {44 \ldots 45}_{n - 1 \ 4 \text{'s} } }^2$
+that is:
+:$\displaystyle \sum_{k \mathop = 0}^{2 n - 1} 10^k = \paren {\sum_{k \mathop = 1}^{n - 1} 5 \times 10^k + 6}^2 - \paren {\sum_{k \mathop = 1}^{n - 1} 4 \times 10^k + 5}^2$
+\end{theorem}
+
+\begin{proof}
+From [[Difference between Two Squares equal to Repunit]], $R_{2 n} = x^2 - y^2$ exactly when $R_{2 n} = a b$ where $x = \dfrac {a + b} 2$ and $y = \dfrac {a - b} 2$.
+By the [[Basis Representation Theorem]]
+
+{{begin-eqn}}
+{{eqn | l = R_{2n}
+ | r = \sum_{k \mathop = 0}^{2 n - 1} 10^k
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 0}^{n - 1} 10^k + \sum_{k \mathop = n}^{2 n - 1} 10^k
+ | c = splitting the [[Definition:Summation|summation]] into two
+}}
+{{eqn | r = \sum_{k \mathop = 0}^{n - 1} 10^k + 10^n \sum_{k \mathop = 0}^{n - 1} 10^k
+ | c = factoring $10^n$ out of the second part
+}}
+{{eqn | r = \paren {10^n + 1} \sum_{k \mathop = 0}^{n - 1} 10^k
+ | c =
+}}
+{{eqn | r = \underbrace {100 \ldots 01}_{n - 1 \ 0 \text{'s} } \times \underbrace {111 \ldots 1}_{n \ 1 \text{'s} }
+ | c =
+}}
+{{end-eqn}}
+
+Thus, let:
+:$a = \paren {10^n + 1}$
+:$b = \displaystyle \sum_{k \mathop = 0}^{n - 1}$
+So:
+{{begin-eqn}}
+{{eqn | l = a + b
+ | r = \sum_{k \mathop = 0}^n 10^k + 1
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^n 10^k + 2
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \dfrac {a + b} 2
+ | r = \sum_{k \mathop = 1}^n \frac {10^k} 2 + 1
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^n 5 \times 10^{k - 1} + 1
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 0}^{n - 1} 5 \times 10^k + 1
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^{n - 1} 5 \times 10^k + 6
+ | c =
+}}
+{{eqn | r = \underbrace {55 \ldots 50}_{n - 1 \ 5 \text{'s} } + 6
+ | c =
+}}
+{{eqn | r = \underbrace {55 \ldots 56}_{n - 1 \ 5 \text{'s} }
+ | c =
+}}
+{{end-eqn}}
+Similarly:
+{{begin-eqn}}
+{{eqn | l = a - b
+ | r = \paren {10^n + 1} - \sum_{k \mathop = 0}^{n - 1} 10^k
+ | c =
+}}
+{{eqn | r = \paren {\sum_{k \mathop = 0}^{n - 1} 9 \times 10^k + 1} + 1 - \sum_{k \mathop = 0}^{n - 1} 10^k
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 0}^{n - 1} 8 \times 10^k + 2
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \dfrac {a - b} 2
+ | r = \sum_{k \mathop = 0}^{n - 1} 4 \times 10^k + 1
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^{n - 1} 4 \times 10^k + 5
+ | c =
+}}
+{{eqn | r = \underbrace {44 \ldots 45}_{n - 1 \ 4 \text{'s} }
+ | c =
+}}
+{{end-eqn}}
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Difference between Two Squares equal to Repunit/Corollary 2}
+Tags: Difference between Two Squares equal to Repunit
+
+\begin{theorem}
+{{begin-eqn}}
+{{eqn | l = 6^2 - 5^2
+ | r = 11
+ | c =
+}}
+{{eqn | l = 56^2 - 45^2
+ | r = 1111
+ | c =
+}}
+{{eqn | l = 5056^2 - 5045^2
+ | r = 111 \, 111
+ | c =
+}}
+{{eqn | o = :
+ | c =
+}}
+{{end-eqn}}
+and in general for [[Definition:Integer|integer]] $n$:
+:$R_{2 n} = {\underbrace{5050 \ldots 56}_{n - 1 \ 5 \text{'s} } }^2 - {\underbrace{5050 \ldots 45}_{n - 1 \ 5 \text{'s} } }^2$
+that is:
+:$\displaystyle \sum_{k \mathop = 0}^{2 n - 1} 10^k = \left({\sum_{k \mathop = 1}^{n - 1} 5 \times 10^{2 k - 1} + 6}\right)^2 - \left({\sum_{k \mathop = 1}^{n - 1} 5 \times 10^{2 k - 1} - 5}\right)^2$
+\end{theorem}
+
+\begin{proof}
+From [[Difference between Two Squares equal to Repunit]], $R_{2n} = x^2 - y^2$ exactly when $R_{2n} = a b$ where $x = \dfrac {a + b} 2$ and $y = \dfrac {a - b} 2$.
+By the [[Basis Representation Theorem]]
+
+{{begin-eqn}}
+{{eqn | l = R_{2n}
+ | r = \sum_{0 \mathop \le k \mathop < 2 n} 10^k
+ | c =
+}}
+{{eqn | r = \sum_{\substack {0 \mathop \le k \mathop < 2 n \\ k \text { even} } } 10^k + \sum_{\substack {0 \mathop \le k \mathop < 2 n \\ k \text { odd} } } 10^k
+ | c =
+}}
+{{eqn | r = \sum_{\substack {0 \mathop \le k \mathop < 2 n \\ k \text { even} } } 10^k + 10 \times \sum_{\substack {0 \mathop \le k \mathop < 2 n \\ k \text { even} } } 10^k
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 0}^{n - 1} 10^{2 k} + 10 \times \sum_{k \mathop = 0}^n 10^{2 k}
+ | c = change of indices
+}}
+{{eqn | r = \sum_{k \mathop = 0}^{n - 1} 11 \times 10^{2 k}
+ | c =
+}}
+{{eqn | r = 11 \sum_{k \mathop = 0}^{n - 1} 10^{2 k}
+ | c =
+}}
+{{eqn | r = 11 \times \underbrace {10101 \ldots 01}_{n \ 1 \text{'s} }
+ | c =
+}}
+{{end-eqn}}
+Thus, let:
+:$a = \displaystyle \sum_{k \mathop = 0}^{n - 1} \times 10^{2 k}$
+:$b = 11$
+So:
+{{begin-eqn}}
+{{eqn | l = a + b
+ | r = \sum_{k \mathop = 0}^{n - 1} 10^{2 k} + 11
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^{n - 1} 10^{2 k} + 12
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \dfrac {a + b} 2
+ | r = \sum_{k \mathop = 1}^{n - 1} \frac {10^{2 k} } 2 + 6
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^{n - 1} 5 \times 10^{2 k - 1} + 6
+ | c =
+}}
+{{eqn | r = \underbrace {5050 \ldots 50}_{n - 1 \ 5 \text{'s} } + 6
+ | c =
+}}
+{{eqn | r = \underbrace{5050 \ldots 56}_{n - 1 \ 5 \text{'s} }
+ | c =
+}}
+{{end-eqn}}
+Similarly:
+{{begin-eqn}}
+{{eqn | l = a - b
+ | r = \sum_{k \mathop = 0}^{n - 1} \times 10^{2 k} - 11
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^{n - 1} 10^{2 k} - 10
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \dfrac {a - b} 2
+ | r = \sum_{k \mathop = 1}^{n - 1} \frac {10^{2 k} } 2 - 5
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^{n - 1} 5 \times 10^{2 k - 1} - 5
+ | c =
+}}
+{{eqn | r = \underbrace {5050 \ldots 50}_{n - 1 \ 5 \text{'s} } - 5
+ | c =
+}}
+{{eqn | r = \underbrace{5050 \ldots 45}_{n - 1 \ 5 \text{'s} }
+ | c =
+}}
+{{end-eqn}}
+Hence the result.
+{{qed}}
+[[Category:Difference between Two Squares equal to Repunit]]
+6a93mucf5ecaddxmcog2ma01dprm4w5
+\end{proof}<|endoftext|>
+\section{Numbers whose Squares are Consecutive Odd or Even Integers Juxtaposed}
+Tags: Square Numbers, Recreational Mathematics
+
+\begin{theorem}
+[[Definition:Integer|Integers]] whose [[Definition:Square (Algebra)|squares]] consists of $2$ consecutive [[Definition:Odd Integer|odd]] or [[Definition:Even Integer|even integers]] juxtaposed include:
+:$1127^2 = 01 \, 270 \, 129$
+:$8874^2 = 78 \, 747 \, 876$
+Such [[Definition:Integer|integers]] come in [[Definition:Ordered Pair|pairs]] which [[Definition:Integer Addition|add]] to $1$ more than a [[Definition:Integer Power|power]] of $10$:
+:$1127 + 8874 = 10 \, 001$
+{{expand|A great deal more to be done here}}
+\end{theorem}<|endoftext|>
+\section{Sixth Power as Sum of 7 Sixth Powers}
+Tags: Sixth Powers, 1141
+
+\begin{theorem}
+The smallest known [[Definition:Integer|integer]] whose [[Definition:Sixth Power|$6$th power]] can be expressed as the [[Definition:Integer Addition|sum]] of $7$ smaller [[Definition:Sixth Power|$6$th powers]] is $1141$:
+:$1141^6 = 74^6 + 234^6 + 402^6 + 474^6 + 702^6 + 894^6 + 1077^6$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | r = 74^6 + 234^6 + 402^6 + 474^6 + 702^6 + 894^6 + 1077^6
+ | o =
+ | c =
+}}
+{{eqn | r = 164 \, 206 \, 490 \, 176
+ | c =
+}}
+{{eqn | o =
+ | ro= +
+ | r = 164 \, 170 \, 508 \, 913 \, 216
+ | c =
+}}
+{{eqn | o =
+ | ro= +
+ | r = 4 \, 220 \, 426 \, 278 \, 476 \, 864
+ | c =
+}}
+{{eqn | o =
+ | ro= +
+ | r = 11 \, 341 \, 488 \, 324 \, 787 \, 776
+ | c =
+}}
+{{eqn | o =
+ | ro= +
+ | r = 119 \, 680 \, 300 \, 997 \, 734 \, 464
+ | c =
+}}
+{{eqn | o =
+ | ro= +
+ | r = 510 \, 534 \, 520 \, 424 \, 456 \, 256
+ | c =
+}}
+{{eqn | o =
+ | ro= +
+ | r = 1 \, 560 \, 609 \, 404 \, 742 \, 322 \, 089
+ | c =
+}}
+{{eqn | r = 2 \, 206 \, 550 \, 475 \, 483 \, 180 \, 841
+ | c =
+}}
+{{eqn | r = 1141^6
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers Not Expressible as Sum of no more than 5 Squares of Composite Numbers}
+Tags: Number Theory
+
+\begin{theorem}
+There are $256$ [[Definition:Positive Integer|integers]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of no more than $5$ [[Definition:Square (Algebra)|squares]] of [[Definition:Composite Number|composite numbers]]:
+:$1, 2, 3, \ldots, 1167$
+{{finish}}
+\end{theorem}<|endoftext|>
+\section{Square Numbers which are Sum of Sequence of Odd Cubes}
+Tags: Square Numbers, Sums of Sequences, Cube Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Square Number|square numbers]] which can be expressed as the [[Definition:Integer Addition|sum]] of a [[Definition:Integer Sequence|sequence]] of [[Definition:Odd Number|odd]] [[Definition:Cube Number|cubes]] from $1$ begins:
+:$1, 1225, 1 \, 413 \, 721, 1 \, 631 \, 432 \, 881, \dotsc$
+{{OEIS|A046177}}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Square Root|square roots]] of this [[Definition:Integer Sequence|sequence]] is:
+:$1, 35, 1189, 40 \, 391, \dotsc$
+{{OEIS|A046176}}
+\end{theorem}
+
+\begin{proof}
+We have that:
+{{begin-eqn}}
+{{eqn | l = 1225
+ | r = 35^2
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^5 \paren {2 k - 1}^3 = 1^3 + 3^3 + 5^3 + 7^3 + 9^3
+ | c =
+}}
+{{eqn | l = 1 \, 413 \, 721
+ | r = 1189^2
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^{29} \paren {2 k - 1}^3 = 1^3 + 3^3 + 5^3 + \dotsb + 55^3 + 57^3
+ | c =
+}}
+{{end-eqn}}
+From [[Sum of Sequence of Odd Cubes]] we have:
+:$\displaystyle \sum_{j \mathop = 1}^n \paren {2 j - 1}^3 = 1^3 + 3^3 + 5^3 + \dotsb + \paren {2 n − 1}^3 = n^2 \paren {2 n^2 − 1}$
+Thus we need to find all $n$ such that $2 n^2 − 1$ is [[Definition:Square Number|square]].
+{{finish|see if there's a proof that can be offered up}}
+\end{proof}<|endoftext|>
+\section{Product of Injective Spaces is Injective}
+Tags: Topology
+
+\begin{theorem}
+Let $I$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]].
+Let $\left({\left({S_i, \tau_i}\right)}\right)_{i \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Injective Space|injective]] [[Definition:Topological Space|topological spaces]].
+Then $\displaystyle \prod_{i \mathop \in I} \left({S_i, \tau_i}\right)$ is [[Definition:Injective Space|injective space]].
+\end{theorem}<|endoftext|>
+\section{Retract of Injective Space is Injective}
+Tags: Topology
+
+\begin{theorem}
+Let $T = \left({S, \tau}\right)$ be an [[Definition:Injective Space|injective topological space]].
+Let $R = \left({Z, \tau'}\right)$ be a [[Definition:Retract (Topology)|retract]] of $T$.
+Then $R$ is [[Definition:Injective Space|injective]].
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Retract (Topology)|retract]]:
+:there exists a [[Definition:Continuous (Topology)|continuous]] [[Definition:Retraction (Topology)|retraction]] $r: S \to Z$ of $T$.
+Let $\mathcal Y = \left({Y, \sigma}\right)$ be a [[Definition:Topological Space|topological space]].
+Let $f: Y \to Z$ be a [[Definition:Continuous (Topology)|continuous]] [[Definition:Mapping|mapping]].
+Let $\mathcal X = \left({X, \sigma'}\right)$ such that
+:$\mathcal Y$ is [[Definition:Topological Subspace|topological subspace]] of $\mathcal X$.
+By [[Inclusion Mapping is Continuous]]:
+:$i_Z$ is [[Definition:Continuous (Topology)|continuous]] $\left({R \to T}\right)$
+where $i_Z$ denotes the [[Definition:Inclusion Mapping|inclusion mapping]] from $Z$ in $S$.
+By [[Composite of Continuous Mappings is Continuous]]:
+:$i_Z \circ f: Y \to S$ is [[Definition:Continuous (Topology)|continuous]].
+By definition of [[Definition:Injective Space|injective space]]:
+:there exists a [[Definition:Continuous (Topology)|continuous]] [[Definition:Mapping|mapping]] $g:X \to S: g \restriction Y = i_Z \circ f$
+Define $h := r \circ g$
+By [[Composite of Continuous Mappings is Continuous]]:
+:$h$ is [[Definition:Continuous (Topology)|continuous]].
+We will prove that
+:$h \restriction Y = f$
+By definition of [[Definition:Topological Subspace|topological subspace]]:
+:$Y \subseteq X$ and $Z \subseteq S$
+Thus by definitions of [[Definition:Composition of Mappings|composition of mappings]] and [[Definition:Restriction of Mapping|restriction of mapping]]:
+:$h \restriction Y: Y \to Z$ and $f: Y \to Z$
+Let $y \in Y$.
+By [[Restriction of Composition is Composition of Restriction]]:
+:$h \restriction Y = r \circ \left({g \restriction Y}\right)$
+By definition of [[Definition:Mapping|mapping]]:
+:$f\left({y}\right) \in Z$
+Thus
+{{begin-eqn}}
+{{eqn | l = \left({h \restriction Y}\right)\left({y}\right)
+| r = r\left({\left({i_Z \circ f}\right)\left({y}\right)}\right)
+| c = definition of [[Definition:Composition of Mappings|composition of mappings]]
+}}
+{{eqn | r = r\left({i_Z\left({f\left({y}\right)}\right)}\right)
+| c = definition of [[Definition:Composition of Mappings|composition of mappings]]
+}}
+{{eqn | r = r\left({f\left({y}\right)}\right)
+| c = definition of [[Definition:Inclusion Mapping|inclusion mapping]]
+}}
+{{eqn | r = f\left({y}\right)
+| c = definition of [[Definition:Retraction (Topology)|retraction]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers equal to Sum of Squares of two Parts}
+Tags: Sums of Squares, Recreational Mathematics
+
+\begin{theorem}
+[[Definition:Integer|Integers]] that can be split into two parts whose [[Definition:Square (Algebra)|squares]] add up to it include:
+:$1233 = 12^2 + 33^2$
+:$8833 = 88^2 + 33^2$
+{{expand|Need to establish the parameters of this}}
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Need to establish exactly what is to be proved}}
+\end{proof}<|endoftext|>
+\section{Triples of Consecutive Sphenic Numbers}
+Tags: Sphenic Numbers
+
+\begin{theorem}
+The [[Definition:Sequence|sequence]] of [[Definition:Ordered Triple|triplets]] of consecutive [[Definition:Sphenic Number|sphenic numbers]] starts:
+:$\tuple {1309, 1310, 1311}, \tuple {1885, 1886, 1887}, \tuple {2013, 2014, 2015}, \ldots$
+{{OEIS|A066509|order = first}}
+{{OEIS|A248202|order = middle}}
+\end{theorem}
+
+\begin{proof}
+Note that there cannot be [[Definition:Ordered Quadruple|quadruplets]] of such numbers, since one of the [[Definition:Ordered Quadruple|quadruplets]] must be [[Definition:Divisor of Integer|divisible]] by $4$, making it non-[[Definition:Sphenic Number|sphenic]].
+We have:
+{{begin-eqn}}
+{{eqn | l = 1309
+ | r = 7 \times 11 \times 17
+}}
+{{eqn | l = 1310
+ | r = 2 \times 5 \times 131
+}}
+{{eqn | l = 1311
+ | r = 3 \times 19 \times 23
+}}
+{{eqn | l = 1885
+ | r = 5 \times 13 \times 29
+}}
+{{eqn | l = 1886
+ | r = 2 \times 23 \times 41
+}}
+{{eqn | l = 1887
+ | r = 3 \times 17 \times 37
+}}
+{{eqn | l = 2013
+ | r = 3 \times 11 \times 61
+}}
+{{eqn | l = 2014
+ | r = 2 \times 19 \times 53
+}}
+{{eqn | l = 2015
+ | r = 5 \times 13 \times 31
+}}
+{{end-eqn}}
+hence each number above is [[Definition:Sphenic Number|sphenic]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cube Number as Sum of Three Consecutive Odd Squares}
+Tags: Cube Numbers, Sums of Squares, 1331
+
+\begin{theorem}
+:$1331 = 11^3 = 19^2 + 21^2 + 23^2$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 19^2 + 21^2 + 23^2
+ | r = 361 + 441 + 529
+ | c =
+}}
+{{eqn | r = 1331
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Triplet of Consecutive Integers Divisible by Cube}
+Tags: Cube Numbers
+
+\begin{theorem}
+The smallest [[Definition:Sequence|sequence]] of [[Definition:Ordered Triple|triplets]] of consecutive [[Definition:Integer|integers]] each of which is [[Definition:Divisor of Integer|divisible]] by a [[Definition:Cube Number|cube]] greater than $1$ is:
+:$\tuple {1375, 1376, 1377}$
+\end{theorem}
+
+\begin{proof}
+We will show that:
+{{begin-eqn}}
+{{eqn | l = 1375
+ | r = 11 \times 5^3
+ | c =
+}}
+{{eqn | l = 1376
+ | r = 172 \times 2^3
+ | c =
+}}
+{{eqn | l = 1377
+ | r = 51 \times 3^3
+ | c =
+}}
+{{end-eqn}}
+is the smallest such [[Definition:Ordered Triple|triplet]].
+Each number in such [[Definition:Ordered Triple|triplets]] of consecutive [[Definition:Integer|integers]] is [[Definition:Divisor of Integer|divisible]] by a [[Definition:Cube Number|cube]] of some [[Definition:Prime Number|prime number]].
+Only $2, 3, 5, 7, 11$ are less than $\sqrt [3] {1377}$.
+Since the numbers involved are small, we can check the result by brute force.
+For general results one is encouraged to use the [[Chinese Remainder Theorem]].
+=== Case $1$: a number is divisible by $11^3$ ===
+The only multiple of $11^3$ less than $1377$ is $1331$, and:
+{{begin-eqn}}
+{{eqn | l = 1330
+ | r = 2 \times 5 \times 7 \times 19
+}}
+{{eqn | l = 1332
+ | r = 2^2 \times 3^2 \times 37
+}}
+{{end-eqn}}
+Since neither $1330$ nor $1332$ are [[Definition:Divisor of Integer|divisible]] by a [[Definition:Cube Number|cube]] of some [[Definition:Prime Number|prime number]], $1331$ is not in a [[Definition:Ordered Triple|triplet]].
+{{qed|lemma}}
+=== Case $2$: a number is divisible by $7^3$ ===
+The only multiples of $7^3$ less than $1377$ are $343, 686, 1029, 1372$, and:
+{{begin-eqn}}
+{{eqn | l = 342
+ | r = 2 \times 3^2 \times 19
+}}
+{{eqn | l = 344
+ | r = 2^3 \times 43
+}}
+{{eqn | l = 345
+ | r = 3 \times 5 \times 23
+}}
+{{eqn | l = 685
+ | r = 5 \times 137
+}}
+{{eqn | l = 687
+ | r = 3 \times 229
+}}
+{{eqn | l = 1028
+ | r = 2^2 \times 257
+}}
+{{eqn | l = 1030
+ | r = 2 \times 5 \times 103
+}}
+{{eqn | l = 1371
+ | r = 3 \times 457
+}}
+{{eqn | l = 1373
+ | o = \text {is}
+ | r = \text {prime}
+}}
+{{end-eqn}}
+Hence none of these numbers is in a [[Definition:Ordered Triple|triplet]].
+{{qed|lemma}}
+=== Case $3$: the numbers are divisible by $2^3, 3^3, 5^3$ respectively ===
+Let $n = k \times 5^3$.
+We show that $k$ cannot be [[Definition:Divisor of Integer|divisible]] by $3$ or $4$.
+Suppose $3 \divides k$.
+Then none of $n \pm 1, n \pm 2$ are [[Definition:Divisor of Integer|divisible]] by $3$, and consequently $3^3$.
+Suppose $4 \divides k$.
+Then none of $n \pm 1, n \pm 2$ are [[Definition:Divisor of Integer|divisible]] by $4$, and consequently $2^3$.
+The only multiples of $5^3$ less than $1377$ are $125, 250, 375, 500, 625, 750, 875, 1000, 1125, 1250, 1375$, and we eliminate $375, 500, 750, 1000, 1125$ due to the reasons above.
+Now:
+{{begin-eqn}}
+{{eqn | l = 124
+ | r = 2^2 \times 31
+}}
+{{eqn | l = 126
+ | r = 2 \times 3^2 \times 7
+}}
+{{eqn | l = 249
+ | r = 3 \times 83
+}}
+{{eqn | l = 251
+ | o = \text {is}
+ | r = \text {prime}
+}}
+{{eqn | l = 623
+ | r = 7 \times 89
+}}
+{{eqn | l = 624
+ | r = 2^4 \times 3 \times 13
+}}
+{{eqn | l = 626
+ | r = 2 \times 313
+}}
+{{eqn | l = 874
+ | r = 2 \times 19 \times 23
+}}
+{{eqn | l = 876
+ | r = 2^2 \times 3 \times 73
+}}
+{{eqn | l = 1030
+ | r = 2 \times 5 \times 103
+}}
+{{eqn | l = 1249
+ | o = \text {is}
+ | r = \text {prime}
+}}
+{{eqn | l = 1251
+ | r = 3^2 \times 139
+}}
+{{end-eqn}}
+Hence none of these numbers is in a [[Definition:Ordered Triple|triplet]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Quadruplet of Consecutive Integers Divisible by Cube}
+Tags: Cube Numbers
+
+\begin{theorem}
+The smallest [[Definition:Sequence|sequence]] of [[Definition:Ordered Quadruple|quadruplets]] of consecutive [[Definition:Integer|integers]] each of which is [[Definition:Divisor of Integer|divisible]] by a [[Definition:Cube Number|cube]] greater than $1$ is:
+:$\tuple {22 \, 624, 22 \, 625, 22 \, 626, 22 \, 627}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 22 \, 624
+ | r = 2828 \times 2^3
+ | c =
+}}
+{{eqn | l = 22 \, 625
+ | r = 181 \times 5^3
+ | c =
+}}
+{{eqn | l = 22 \, 626
+ | r = 838 \times 3^3
+ | c =
+}}
+{{eqn | l = 22 \, 627
+ | r = 17 \times 11^3
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown there is no smaller such quadruplet.}}
+\end{proof}<|endoftext|>
+\section{Riemann Zeta Function as a Multiple Integral}
+Tags: Riemann Zeta Function, Analytic Number Theory
+
+\begin{theorem}
+For $n \in \Z_{> 0}$, the [[Definition:Riemann Zeta Function|Riemann zeta function]] is given by:
+:$\displaystyle \map \zeta n = \int_{\closedint 0 1^n} \frac 1 {1 - \prod_{i \mathop = 1}^n x_i} \prod_{i \mathop = 1}^n \rd x_i$
+where $\closedint 0 1^n$ denotes the [[Definition:Cartesian Space|Cartesian $n$th power]] of the [[Definition:Closed Real Interval|closed real interval]] $\closedint 0 1$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int_{\closedint 0 1^n} \frac 1 {1 - \prod_{i \mathop = 1}^n x_i} \prod_{i \mathop = 1}^n \rd x_i
+ | r = \int_{\closedint 0 1^n} \sum_{j \mathop = 1}^\infty \paren {\prod_{i \mathop = 1}^n x_i}^{j - 1} \prod_{i \mathop = 1}^n \rd x_i
+ | c = [[Sum of Infinite Geometric Sequence]]
+}}
+{{eqn | r = \sum_{j \mathop = 1}^\infty \prod_{i \mathop = 1}^n \int_0^1 x^{j - 1}_i \rd x_i
+ | c = [[Fubini's Theorem]]
+}}
+{{eqn | r = \sum_{j \mathop = 1}^\infty \frac 1 {j^n}
+}}
+{{eqn | r = \map \zeta n
+ | c = {{Defof|Riemann Zeta Function}}
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Riemann Zeta Function]]
+[[Category:Analytic Number Theory]]
+jwrs1ween35w33fz057fglkz65ihfz8
+\end{proof}<|endoftext|>
+\section{Closed Form for Hexagonal Pyramidal Numbers}
+Tags: Closed Forms, Pyramidal Numbers
+
+\begin{theorem}
+The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Hexagonal Pyramidal Number|hexagonal pyramidal number]] is:
+:$S_n = \dfrac {n \paren {n + 1} \paren {4 n - 1} } 6$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = S_n
+ | r = \sum_{k \mathop = 1}^n H_k
+ | c = {{Defof|Hexagonal Pyramidal Number}}
+}}
+{{eqn | r = \sum_{k \mathop = 1}^n k \paren {2 k - 1}
+ | c = [[Closed Form for Hexagonal Numbers]]
+}}
+{{eqn | r = 3 \sum_{k \mathop = 1}^n 2 k^2 - \sum_{k \mathop = 1}^n k
+ | c =
+}}
+{{eqn | r = \frac {2 n \paren {n + 1} \paren {2 n + 1} } 6 - \sum_{k \mathop = 1}^n k
+ | c = [[Sum of Sequence of Squares]]
+}}
+{{eqn | r = \frac {2 n \paren {n + 1} \paren {2 n + 1} } 6 - \dfrac {n \paren {n + 1} } 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{eqn | r = \frac {2 n \paren {n + 1} \paren {2 n + 1} - 3 n \paren {n + 1} } 6
+ | c =
+}}
+{{eqn | r = \frac {n \paren {n + 1} \paren {4 n + 2 - 3} } 6
+ | c =
+}}
+{{eqn | r = \frac {n \paren {n + 1} \paren {4 n - 1} } 6
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Closed Forms]]
+[[Category:Pyramidal Numbers]]
+1b9e5x8004owfzmnt6dfbmrdwpbcibr
+\end{proof}<|endoftext|>
+\section{Tetrahedral and Triangular Numbers}
+Tags: Triangular Numbers, Pyramidal Numbers
+
+\begin{theorem}
+The only [[Definition:Positive Integer|positive integers]] which are simultaneously [[Definition:Tetrahedral Number|tetrahedral]] and [[Definition:Triangular Number|triangular]] are:
+:$1, 10, 120, 1540, 7140$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1
+ | r = \dfrac {1 \paren {1 + 1} \paren {1 + 2} } 6
+ | c = [[Closed Form for Tetrahedral Numbers]]
+}}
+{{eqn | r = \dfrac {1 \times \paren {1 + 1} } 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 10
+ | r = \dfrac {3 \paren {3 + 1} \paren {3 + 2} } 6
+ | c = [[Closed Form for Tetrahedral Numbers]]
+}}
+{{eqn | r = \dfrac {4 \times \paren {4 + 1} } 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 120
+ | r = \dfrac {8 \paren {8 + 1} \paren {8 + 2} } 6
+ | c = [[Closed Form for Tetrahedral Numbers]]
+}}
+{{eqn | r = \dfrac {15 \times \paren {15 + 1} } 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 1540
+ | r = \dfrac {20 \paren {20 + 1} \paren {20 + 2} } 6
+ | c = [[Closed Form for Tetrahedral Numbers]]
+}}
+{{eqn | r = \dfrac {55 \times \paren {55 + 1} } 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 7140
+ | r = \dfrac {34 \paren {34 + 1} \paren {34 + 2} } 6
+ | c = [[Closed Form for Tetrahedral Numbers]]
+}}
+{{eqn | r = \dfrac {119 \times \paren {119 + 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{Restriction of Composition is Composition of Restriction}
+Tags: Mapping Theory
+
+\begin{theorem}
+Let $X, Y, Z$ be [[Definition:Set|sets]].
+Let $f: X \to Y$ and $g: Y \to Z$ be [[Definition:Mapping|mappings]].
+Let $S \subseteq X$.
+Then:
+: $\left({g \circ f}\right) \restriction S = g \circ \left({f \restriction S}\right)$
+\end{theorem}
+
+\begin{proof}
+By definitions of [[Definition:Composition of Mappings|composition of mappings]] and [[Definition:Restriction of Mapping|restriction of mapping]]:
+:$\left({g \circ f}\right) \restriction S: S \to Z$ and $g \circ \left({f \restriction S}\right):S \to Z$
+Let $s \in S$.
+By definition of [[Definition:Restriction of Mapping|restriction of mapping]]:
+:$\left({\left({g \circ f}\right) \restriction S}\right)\left({s}\right) = \left({g \circ f}\right)\left({s}\right)$
+Thus
+{{begin-eqn}}
+{{eqn | l = \left({g \circ \left({f \restriction S}\right)}\right)\left({s}\right)
+ | r = g\left({\left({f \restriction S}\right)\left({s}\right)}\right)
+ | c = {{Defof|Composition of Mappings}}
+}}
+{{eqn | r = g\left({f\left({s}\right)}\right)
+ | c = {{Defof|Restriction of Mapping}}
+}}
+{{eqn | r = \left({\left({g \circ f}\right) \restriction S}\right)\left({s}\right)
+ | c = {{Defof|Composition of Mappings}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Odd Numbers not Sum of Prime and Power}
+Tags: Powers, Prime Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Odd Integer|odd numbers]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of a [[Definition:Perfect Power|perfect power]] and a [[Definition:Prime Number|prime number]] begins:
+:$1, 5, 1549, 1 \, 771 \, 561, \ldots$
+{{OEIS|A119747}}
+It is not known if there are any more terms.
+\end{theorem}
+
+\begin{proof}
+The cases $1$ and $5$ are trivial.
+Now we show that $1549 - a^b$ is never [[Definition:Prime Number|prime]] for $a \ge 1$ and $b \ge 2$.
+It suffices to show the result for [[Definition:Prime Number|prime]] values of $b$.
+We first prove:
+{{begin-eqn}}
+{{eqn | n = 1
+ | l = 2
+ | o = \divides
+ | r = a
+}}
+{{eqn | n = 2
+ | l = 3
+ | o = \divides
+ | r = a
+ | rr = \text {if } b = 2
+}}
+{{eqn | n = 3
+ | l = a^b
+ | o = \not \equiv
+ | r = 4
+ | rr = \pmod {10}
+}}
+{{end-eqn}}
+For $(1)$:
+Suppose $a$ is [[Definition:Odd Integer|odd]].
+Then so is $a^b$.
+Therefore $1549 - a^b$ is [[Definition:Even Integer|even]].
+Hence $1549 - a^b$ is [[Definition:Prime Number|prime]] only if it is equal to $2$.
+But $1547$ is not a [[Definition:Perfect Power|perfect power]].
+{{qed|lemma}}
+For $(2)$:
+Suppose $b = 2$ and $a$ is not [[Definition:Divisor of Integer|divisible]] by $3$.
+By [[Square Modulo 3/Corollary 3|Corollary to Square Modulo $3$]]:
+:$3 \divides \paren {a^2 - 1}$
+Hence:
+:$3 \divides \paren {1548 - a^2 + 1} = 1549 - a^b$
+So $1549 - a^b$ is [[Definition:Prime Number|prime]] only if it is equal to $3$.
+But $1547$ is not a [[Definition:Square Number|perfect square]].
+{{qed|lemma}}
+For $(3)$:
+Suppose $a^b \equiv 4 \pmod {10}$.
+Then $1549 - a^b \equiv 5 \pmod {10}$
+So $1549 - a^b$ is [[Definition:Prime Number|prime]] only if it is equal to $5$.
+But $1544$ is not a [[Definition:Perfect Power|perfect power]].
+{{qed|lemma}}
+Hence we only need to check the cases:
+:For $b = 2: \: 6^2, 12^2, 18^2, 24^2, 30^2, 36^2$ since $42^2 > 1549$
+:For $b \ne 2: \: 2^3, 2^5, 2^7, 4^3, 4^5, 6^3, 8^3, 10^3$ since $2^{11}, 4^7, 6^5 > 1549$
+of which $12^2 \equiv 18^2 \equiv 4^3 \equiv 4^5 \equiv 4 \pmod {10}$, so they are rejected.
+We have:
+{{begin-eqn}}
+{{eqn | l = 1549 - 6^2
+ | r = 1513
+}}
+{{eqn | r = 17 \times 89
+}}
+{{eqn | l = 1549 - 24^2
+ | r = 973
+}}
+{{eqn | r = 7 \times 139
+}}
+{{eqn | l = 1549 - 30^2
+ | r = 649
+}}
+{{eqn | r = 11 \times 59
+}}
+{{eqn | l = 1549 - 36^2
+ | r = 253
+}}
+{{eqn | r = 11 \times 23
+}}
+{{eqn | l = 1549 - 2^3
+ | r = 1541
+}}
+{{eqn | r = 23 \times 67
+}}
+{{eqn | l = 1549 - 2^5
+ | r = 1517
+}}
+{{eqn | r = 37 \times 41
+}}
+{{eqn | l = 1549 - 2^7
+ | r = 1421
+}}
+{{eqn | r = 7^2 \times 29
+}}
+{{eqn | l = 1549 - 6^3
+ | r = 1333
+}}
+{{eqn | r = 31 \times 43
+}}
+{{eqn | l = 1549 - 8^3
+ | r = 1037
+}}
+{{eqn | r = 17 \times 61
+}}
+{{eqn | l = 1549 - 10^3
+ | r = 549
+}}
+{{eqn | r = 3^2 \times 61
+}}
+{{end-eqn}}
+and none of the above are [[Definition:Prime Number|prime]].
+{{ProofWanted|For $1771561 {{=}} 11^6$}}
+{{ProofWanted|In addition to the above, don't we also need to demonstrate that for $5 < n < 1549$, an odd number *does* have such an expression?}}
+\end{proof}<|endoftext|>
+\section{One-Digit Number is Harshad}
+Tags: Harshad Numbers
+
+\begin{theorem}
+Let $n$ be a [[Definition:Digit|$1$-digit]] [[Definition:Positive Integer|positive integer]].
+Then $n$ is a [[Definition:Harshad Number|harshad number]].
+\end{theorem}
+
+\begin{proof}
+By definition, a [[Definition:Harshad Number|harshad number]] is [[Definition:Divisor of Integer|divisible]] by the [[Definition:Integer Addition|sum]] of its [[Definition:Digit|digits]] [[Definition:Decimal Notation|base $10$]].
+Let $n$ be a [[Definition:Digit|$1$-digit]] [[Definition:Positive Integer|positive integer]].
+The [[Definition:Integer Addition|sum]] of the [[Definition:Digit|digits]] of $n$ is trivially $n$.
+The result follows from [[Integer Divides Itself]].
+{{qed}}
+[[Category:Harshad Numbers]]
+kxykvvanwxcz0ibl71hfbpqgniu0v7p
+\end{proof}<|endoftext|>
+\section{Smallest Fermat Pseudoprime to Bases 2, 3 and 5}
+Tags: Fermat Pseudoprimes, 1729
+
+\begin{theorem}
+The smallest [[Definition:Fermat Pseudoprime|Fermat pseudoprime]] to bases $2$, $3$ and $5$ is $1729$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|We have the list of [[Definition:Poulet Number|Poulet numbers]] and [[Definition:Fermat Pseudoprime/Base 3|Fermat pseudoprimes base $3$]], but not of base $5$. Once we get that list, we can find the numbers on the list for both.}}
+\end{proof}<|endoftext|>
+\section{Numbers that Factorise into Sum of Digits and Reversal}
+Tags: Fermat Pseudoprimes, 1729
+
+\begin{theorem}
+The following [[Definition:Positive Integer|positive integers]] can each be expressed as the [[Definition:Integer Multiplication|product]] of the [[Definition:Integer Addition|sum]] of its [[Definition:Digit|digits]] and the [[Definition:Reversal|reversal]] of the [[Definition:Integer Addition|sum]] of its [[Definition:Digit|digits]]:
+:$1, 81, 1458, 1729$
+{{OEIS|A110921}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1
+ | r = 1 \times 1
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 81
+ | r = 9 \times 9
+}}
+{{eqn | r = 9 \times \paren {8 + 1}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 1458
+ | r = 81 \times 18
+}}
+{{eqn | r = 81 \times \paren {1 + 4 + 5 + 8}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 1729
+ | r = 91 \times 19
+}}
+{{eqn | r = 91 \times \paren {1 + 7 + 2 + 9}
+}}
+{{end-eqn}}
+Let $n$ be a [[Definition:Positive Integer|positive integer]].
+Let $S$ the [[Definition:Integer Addition|sum]] of its [[Definition:Digit|digits]] and $S'$ be the [[Definition:Reversal|reversal]] of the [[Definition:Integer Addition|sum]] of its [[Definition:Digit|digits]].
+We wish to determine [[Definition:Integer|integers]] $n$ that satisfy:
+:$n = S S'$
+From this we have:
+{{begin-eqn}}
+{{eqn | l = n
+ | o = \equiv
+ | r = S S'
+ | rr = \pmod 9
+ | c = [[Equal Numbers are Congruent]]
+}}
+{{eqn | ll = \leadsto
+ | l = S
+ | o = \equiv
+ | r = S^2
+ | rr = \pmod 9
+ | c = [[Congruence of Sum of Digits to Base Less 1]]
+}}
+{{eqn | ll = \leadsto
+ | l = S
+ | o = \equiv
+ | r = 0 \text { or } 1
+ | rr = \pmod 9
+ | c =
+}}
+{{end-eqn}}
+Suppose $n$ is a $d$-[[Definition:Digit|digit]] [[Definition:Integer|integer]].
+Suppose $d \le 4$. Then $S \le 9 d \le 36$.
+The values of $S S'$ are:
+{{begin-eqn}}
+{{eqn | l = 1 \times 1
+ | r = 1
+ | c = and we have $1 = 1$
+}}
+{{eqn | l = 9 \times 9
+ | r = 81
+ | c = and we have $8 + 1 = 9$
+}}
+{{eqn | l = 10 \times 01
+ | r = 10
+ | c = and we have $1 + 0 \ne 10$
+}}
+{{eqn | l = 18 \times 81
+ | r = 1458
+ | c = and we have $1 + 4 + 5 + 8 = 18$
+}}
+{{eqn | l = 19 \times 91
+ | r = 1729
+ | c = and we have $1 + 7 + 2 + 9 = 19$
+}}
+{{eqn | l = 27 \times 72
+ | r = 1944
+ | c = and we have $1 + 9 + 4 + 4 \ne 27$
+}}
+{{eqn | l = 28 \times 82
+ | r = 2296
+ | c = and we have $2 + 2 + 9 + 6 \ne 28$
+}}
+{{eqn | l = 36 \times 63
+ | r = 2268
+ | c = and we have $2 + 2 + 6 + 8 \ne 36$
+}}
+{{end-eqn}}
+Among these values, only $1, 81, 1458, 1729$ have the desired property.
+We claim that there are no [[Definition:Integer|integers]] with more than $4$ [[Definition:Digit|digits]] with this property.
+{{AimForCont}} $d \ge 5$.
+Suppose $d$ is a $k$-[[Definition:Digit|digit]] [[Definition:Integer|integer]].
+We have $k = 1 + \floor {\log d}$.
+We show that $2 k + 2 \le d - 1$:
+:$d = 5,6$ are single-[[Definition:Digit|digit]] [[Definition:Integer|integers]], so $k = 1$.
+:$d - 1 \ge 4 = 2 k + 2$
+:For $d \ge 7$, consider the function $\map f d = d - 2 \log d - 5$.
+:Then $\map f 7 > 7 - 2 - 5 = 0$.
+:We also have $\map {f'} d = 1 - \dfrac 2 {d \ln 10} > 1 - \dfrac 1 d$,
+:so $\map {f'} d > 0$ for all $d \ge 7$.
+:By [[Real Function with Strictly Positive Derivative is Strictly Increasing]], $f$ is [[Definition:Strictly Increasing Real Function|strictly increasing]] for all $d \ge 7$.
+:Then:
+{{begin-eqn}}
+{{eqn | l = d - 2 \log d - 5
+ | o = >
+ | r = \map f 7
+ | c = $f$ is [[Definition:Strictly Increasing Real Function|strictly increasing]] for all $d \ge 7$.
+}}
+{{eqn | o = >
+ | r = 0
+ | c =
+}}
+{{eqn | ll = \leadsto
+ | l = d - 1
+ | o = >
+ | r = 2 \log d + 4
+}}
+{{eqn | o = \ge
+ | r = 2 \floor {\log d} + 4
+ | c = {{Defof|Floor Function}}
+}}
+{{eqn | r = 2 k + 2
+ | c = $k = 1 + \floor {\log d}$
+}}
+{{end-eqn}}
+:So we have $2 k + 2 \le d - 1$ for all $d \ge 5$.
+$9 d$ has not more than $k + 1$ [[Definition:Digit|digits]].
+Since $S \le 9 d$, $S$ cannot have more [[Definition:Digit|digits]] than $9 d$.
+We also have that $S'$ cannot have more [[Definition:Digit|digits]] than $S$.
+Therefore we have $S, S' < 10^{k + 1}$.
+Then $n = S S' < 10^{2 k + 2} \le 10^{d - 1} \le n$, which is a [[Proof by Contradiction|contradiction]].
+The result follows by [[Proof by Contradiction]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{1782 is 3 Times Sum of all 2-Digit Numbers from its Digits}
+Tags: Recreational Mathematics, 1782
+
+\begin{theorem}
+$1782$ equals $3$ [[Definition:Integer Multiplication|multiplied by]] the [[Definition:Integer Addition|sum]] of all the [[Definition:Digit|$2$-digit]] [[Definition:Integer|integers]] that can be formed from its [[Definition:Digit|digits]].
+\end{theorem}
+
+\begin{proof}
+The number of [[Definition:Digit|$2$-digit]] [[Definition:Integer|integers]] that can be formed from the [[Definition:Digit|digits]] of $1782$ equals the number of [[Definition:Permutation (Ordered Selection)|$2$-permutations]] of $\set {1, 7, 8, 2}$.
+That is:
+:$\set {17, 18, 12, 71, 78, 72, 81, 87, 82, 21, 27, 28}$
+Hence:
+:$17 + 18 + 12 + 71 + 78 + 72 + 81 + 87 + 82 + 21 + 27 + 28 = 594 = \dfrac {1782} 3$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Triple of Consecutive Happy Numbers}
+Tags: Happy Numbers
+
+\begin{theorem}
+The smallest [[Definition:Ordered Triple|triple]] of consecutive [[Definition:Integer|integers]] all of which are [[Definition:Happy Number|happy]] is:
+:$\left({1880, 1881, 1882}\right)$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | o =
+ | r = 1880
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 1^2 + 8^2 + 8^2 + 0^2
+ | r = 1 + 64 + 64 + 0
+ | c =
+}}
+{{eqn | r = 129
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 1^2 + 2^2 + 9^2
+ | r = 1 + 4 + 81
+ | c =
+}}
+{{eqn | r = 86
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 8^2 + 6^2
+ | r = 64 + 36
+ | c =
+}}
+{{eqn | r = 100
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 1^2 + 0^2 + 0^2
+ | r = 1 + 0 + 0
+ | c =
+}}
+{{eqn | r = 1
+ | c = and so $1880$ is [[Definition:Happy Number|happy]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | o =
+ | r = 1881
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 1^2 + 8^2 + 8^2 + 1^2
+ | r = 1 + 64 + 64 + 1
+ | c =
+}}
+{{eqn | r = 130
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 1^2 + 3^2 + 0^2
+ | r = 1 + 9 + 0
+ | c =
+}}
+{{eqn | r = 10
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 1^2 + 0^2
+ | r = 1 + 0
+ | c =
+}}
+{{eqn | r = 1
+ | c = and so $1881$ is [[Definition:Happy Number|happy]]
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | o =
+ | r = 1882
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 1^2 + 8^2 + 8^2 + 2^2
+ | r = 1 + 64 + 64 + 4
+ | c =
+}}
+{{eqn | r = 133
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 1^2 + 3^2 + 3^2
+ | r = 1 + 9 + 9
+ | c =
+}}
+{{eqn | r = 19
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 1^2 + 9^2
+ | r = 1 + 81
+ | c =
+}}
+{{eqn | r = 82
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 8^2 + 2^2
+ | r = 64 + 4
+ | c =
+}}
+{{eqn | r = 68
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 6^2 + 8^2
+ | r = 36 + 64
+ | c =
+}}
+{{eqn | r = 100
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 1^2 + 0^2 + 0^2
+ | r = 1 + 0 + 0
+ | c =
+}}
+{{eqn | r = 1
+ | c = and so $1882$ is [[Definition:Happy Number|happy]]
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that this is the smallest such triple}}
+\end{proof}<|endoftext|>
+\section{Numbers whose Digits are Unchanged when Subtracting Reversal}
+Tags: Reversals, Anagrams
+
+\begin{theorem}
+The following [[Definition:Integer Sequence|sequence]] consists of the [[Definition:Integer|integers]] which have the property that [[Definition:Integer Subtraction|subtraction]] of their [[Definition:Reversal|reversals]] results in [[Definition:Anagram|anagrams]] of them:
+:$954, 1980, 2961, 3870, 5823, 7641, 9108, 19980, 29880, 29961, 32760, \ldots$
+{{OEIS|A121969}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 954 - 459
+ | r = 495
+}}
+{{eqn | l = 1980 - 0891
+ | r = 1089
+}}
+{{eqn | l = 2961 - 1692
+ | r = 1269
+}}
+{{eqn | l = 3870 - 0783
+ | r = 3087
+}}
+{{eqn | l = 5823 - 3285
+ | r = 2538
+}}
+{{eqn | l = 7641 - 1467
+ | r = 6174
+}}
+{{eqn | l = 9108 - 8019
+ | r = 1089
+}}
+{{eqn | l = 19 \, 980 - 08 \, 991
+ | r = 10 \, 989
+}}
+{{eqn | l = 29 \, 880 - 08 \, 892
+ | r = 20 \, 988
+}}
+{{eqn | l = 29 \, 961 - 16 \, 992
+ | r = 12 \, 969
+}}
+{{eqn | l = 32 \, 760 - 06 \, 732
+ | r = 26 \, 037
+}}
+{{end-eqn}}
+{{ProofWanted|that there are no more in between these}}
+\end{proof}<|endoftext|>
+\section{Sequence of Composite Mersenne Numbers}
+Tags: Mersenne Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Mersenne Number|Mersenne numbers]] which are [[Definition:Composite Number|composite]] begins:
+:$2047, 8 \, 388 \, 607, 536 \, 870 \, 911, 137 \, 438 \, 953 \, 471, 2 \, 199 \, 023 \, 255 \, 551,\ldots$
+{{OEIS|A065341}}
+The [[Definition:Integer Sequence|sequence]] of corresponding indices $p$ such that $2^p - 1$ is [[Definition:Composite Number|composite]] begins:
+:$11, 23, 29, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, \ldots$
+{{OEIS|A054723}}
+The [[Definition:Integer Sequence|sequence]] of corresponding [[Definition:Integer|integers]] $n$ such that the $n$th [[Definition:Prime Number|prime number]] $p \left({n}\right)$ is such that $2^{p \left({n}\right)} - 1$ is [[Definition:Composite Number|composite]] begins:
+:$5, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, \ldots$
+{{OEIS|A135980}}
+\end{theorem}
+
+\begin{proof}
+Established by inspecting the [[Definition:Mersenne Number/Sequence|sequence of Mersenne numbers]]:
+:$3, 7, 31, 127, 2047, 8191, 131 \, 071, 524 \, 287, 8 \, 388 \, 607, 536 \, 870 \, 911, 2 \, 147 \, 483 \, 647, \ldots$
+and removing from it the [[Mersenne Prime/Current Status|sequence of Mersenne primes]]:
+:$3, 7, 31, 127, 8191, 131 \, 071, 524 \, 287, 2 \, 147 \, 483 \, 647, \ldots$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers Reversed when Multiplying by 4}
+Tags: Reversals, 2178
+
+\begin{theorem}
+Numbers of the form $\sqbrk {21 \paren 9 78}_{10}$ are [[Definition:Reversal|reversed]] when they are [[Definition:Integer Multiplication|multiplied]] by $4$:
+{{begin-eqn}}
+{{eqn | l = 2178 \times 4
+ | r = 8712
+}}
+{{eqn | l = 21 \, 978 \times 4
+ | r = 87 \, 912
+}}
+{{eqn | l = 219 \, 978 \times 4
+ | r = 879 \, 912
+}}
+{{end-eqn}}
+and so on.
+\end{theorem}
+
+\begin{proof}
+Let k represent the number of $9$s in the middle of the number.
+For $k > 0$ We can rewrite the number as follows:
+{{begin-eqn}}
+{{eqn | l = \sqbrk {21 (9) 78}_{10}
+ | r = 21 \times 10^{k + 2 } + 900 \sum_{i \mathop = 0}^{k - 1} 10^i + 78
+ | c = {{Defof|Geometric Series}}
+}}
+{{end-eqn}}
+Taking numbers of this form and multiplying by $4$ produces:
+{{begin-eqn}}
+{{eqn | l = 4 \times \paren {21 \times 10^{k + 2 } + 900 \sum_{i \mathop = 0}^{k - 1} 10^i + 78 }
+ | r = 84 \times 10^{k + 2 } + 3600 \sum_{i \mathop = 0}^{k - 1} 10^i + 312
+}}
+{{end-eqn}}
+The first part is composed of $k + 4$ [[Definition:Digit|digits]]. The first two [[Definition:Digit|digits]] will be $84$ followed by $k +2$ [[Definition:Digit|digits]] of $0$
+{{begin-eqn}}
+{{eqn | l = 84 \times 10^{k + 2 }
+ | r = 84 \times 10^{k + 2 }
+}}
+{{end-eqn}}
+The sum in the middle is composed of $k + 3$ [[Definition:Digit|digits]]. The first [[Definition:Digit|digit]] will be $3$ followed by $k - 1$ [[Definition:Digit|digits]] of $9$ and then the remaining three [[Definition:Digit|digits]] at the end are $600$
+{{begin-eqn}}
+{{eqn | l = 3600 \sum_{i \mathop = 0}^{k - 1} 10^i
+ | r = 399 \cdots 99600
+}}
+{{end-eqn}}
+Summing the three pieces, the final answer will have $k + 4$ [[Definition:Digit|digits]].
+The first [[Definition:Digit|digit]] is $8$
+followed by $7$ which is the sum of the $4$ from the first part and the $3$ of the middle part
+followed by $k$ [[Definition:Digit|digits]] of $9$ where the last $9$ is the sum of the $6$ from the middle part and the $3$ of the last part
+and then ending in $12$:
+{{begin-eqn}}
+{{eqn | l = \sqbrk {21 (9) 78}_{10} \times 4
+ | r = \sqbrk {87 (9) 12}_{10}
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{17 Consecutive Integers each with Common Factor with Product of other 16}
+Tags: Recreational Mathematics
+
+\begin{theorem}
+The $17$ consecutive [[Definition:Integer|integers]] from $2184$ to $2200$ have the property that each one is not [[Definition:Coprime Integers|coprime]] with the [[Definition:Integer Multiplication|product]] of the other $16$.
+\end{theorem}
+
+\begin{proof}
+We obtain the [[Definition:Prime Decomposition|prime decomposition]] of all $17$ of these [[Definition:Integer|integers]]:
+{{begin-eqn}}
+{{eqn | l = 2184
+ | r = 2^3 \times 3 \times 7 \times 13
+}}
+{{eqn | l = 2185
+ | r = 5 \times 19 \times 23
+}}
+{{eqn | l = 2186
+ | r = 2 \times 1093
+}}
+{{eqn | l = 2187
+ | r = 3^7
+}}
+{{eqn | l = 2188
+ | r = 2^2 \times 547
+}}
+{{eqn | l = 2189
+ | r = 11 \times 199
+}}
+{{eqn | l = 2190
+ | r = 2 \times 3 \times 5 \times 73
+}}
+{{eqn | l = 2191
+ | r = 7 \times 313
+}}
+{{eqn | l = 2192
+ | r = 2^4 \times 137
+}}
+{{eqn | l = 2193
+ | r = 3 \times 17 \times 43
+}}
+{{eqn | l = 2194
+ | r = 2 \times 1097
+}}
+{{eqn | l = 2195
+ | r = 5 \times 439
+}}
+{{eqn | l = 2196
+ | r = 2^2 \times 3^2 \times 61
+}}
+{{eqn | l = 2197
+ | r = 13^3
+}}
+{{eqn | l = 2198
+ | r = 2 \times 7 \times 157
+}}
+{{eqn | l = 2199
+ | r = 3 \times 733
+}}
+{{eqn | l = 2200
+ | r = 2^3 \times 5^2 \times 11
+}}
+{{end-eqn}}
+It can be seen by inspection that each of the [[Definition:Integer|integers]] in this [[Definition:Integer Sequence|sequence]] shares at least one [[Definition:Prime Factor|prime factor]] with at least one other.
+It is then worth noting that:
+{{begin-eqn}}
+{{eqn | l = 2183
+ | r = 37 \times 59
+}}
+{{eqn | l = 2201
+ | r = 31 \times 71
+}}
+{{end-eqn}}
+and it can be seen that the [[Definition:Integer Sequence|sequence]] can be extended neither upwards nor downwards.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Relational Structure admits Lower Topology}
+Tags: Topological Order Theory
+
+\begin{theorem}
+Let $R = \left({S, \preceq}\right)$ be a [[Definition:Relational Structure|relational structure]].
+Then there exists a [[Definition:Relational Structure with Topology|relational structure]] with [[Definition:Lower Topology|lower topology]] $T = \left({S, \preceq, \tau}\right)$ such that $T$ is a [[Definition:Topological Space|topological space]].
+\end{theorem}
+
+\begin{proof}
+Define $B := \left\{ {\complement_S\left({x^\succeq}\right): x \in S}\right\}$
+where $x^\succeq$ denotes the [[Definition:Upper Closure of Element|upper closure]] of $x$.
+By definition of [[Definition:Topology Generated by Synthetic Sub-Basis|generated topology]]:
+:$\tau\left({B}\right)$ is a [[Definition:Topology|topology]] on $S$
+where $B$ is a [[Definition:Synthetic Sub-Basis|sub-basis]] of $\tau \left({B}\right)$.
+Thus by definition of [[Definition:Lower Topology|lower topology]]:
+:$T := \left({S, \preceq, \tau\left({B}\right)}\right)$ has a [[Definition:Lower Topology|lower topology]].
+Thus by definition:
+:$T$ is a [[Definition:Topological Space|topological space]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Lower and Upper Bounds for Sequences/Warning}
+Tags: Limits of Sequences
+
+\begin{theorem}
+Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
+Let $x_n \to l$ as $n \to \infty$.
+Then it is '''not''' the case that:
+:$(1): \quad \forall n \in \N: x_n > a \implies l > a$
+:$(2): \quad \forall n \in \N: x_n < b \implies l < b$
+\end{theorem}
+
+\begin{proof}
+Take the examples:
+:$(1): \quad \sequence {x_n} = \dfrac 1 n$
+:$(2): \quad \sequence {y_n} = -\dfrac 1 n$
+Then :
+:$\forall n \in \N_{>0}: \dfrac 1 n > 0, -\dfrac 1 n < 0$
+From [[Sequence of Reciprocals is Null Sequence]], we have
+:$x_n \to 0$
+:$y_n \to 0$
+as $n \to \infty$.
+However, it is clearly '''false''' that $0 > 0$ and $0 < 0$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Squares of 23...3}
+Tags: Recreational Mathematics
+
+\begin{theorem}
+The following pattern holds:
+{{begin-eqn}}
+{{eqn | l = 3^2
+ | r = 9
+}}
+{{eqn | l = 23^2
+ | r = 529
+}}
+{{eqn | l = 233^2
+ | r = 54 \, 289
+}}
+{{eqn | l = 2333^2
+ | r = 5 \, 442 \, 889
+}}
+{{eqn | l = 23333^2
+ | r = 544 \, 428 \, 889
+}}
+{{end-eqn}}
+and so on.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Simple but tedious.}}
+\end{proof}<|endoftext|>
+\section{Smallest Fourth Power as Sum and Difference of Fourth Powers}
+Tags: Fourth Powers, 2401
+
+\begin{theorem}
+The smallest [[Definition:Fourth Power|$4$th power]] that can be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Fourth Power|$4$th powers]] [[Definition:Integer Subtraction|minus]] a $3$rd is:
+:$2401 = 7^4 = 227^4 + 157^4 - 239^4$
+with all numbers less than $10^4$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | o =
+ | r = 227^4 + 157^4 - 239^4
+ | c =
+}}
+{{eqn | r = 2 \, 655 \, 237 \, 841 + 607 \, 573 \, 201 - 3 \, 262 \, 808 \, 641
+ | c =
+}}
+{{eqn | r = 2401
+ | c =
+}}
+{{eqn | r = 7^4
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that this is the smallest such.}}
+\end{proof}<|endoftext|>
+\section{Zsigmondy's Theorem for Sums}
+Tags: Number Theory
+
+\begin{theorem}
+Let $a > b > 0$ be [[Definition:Coprime Integers|coprime]] [[Definition:Positive Integer|positive integers]].
+Let $n \ge 1$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
+Then there is a [[Definition:Prime Number|prime number]] $p$ such that
+:$p$ [[Definition:Divisor of Integer|divides]] $a^n + b^n$
+:$p$ does not [[Definition:Divisor of Integer|divide]] $a^k + b^k$ for all $k < n$
+with the following exception:
+:$n = 3$, $a = 2$, $b = 1$
+\end{theorem}
+
+\begin{proof}
+By [[Zsigmondy's Theorem]], there exists a [[Definition:Prime Divisor|prime divisor]] $p$ of $a^{2 n} - b^{2 n}$ which does not [[Definition:Divisor of Integer|divide]] $a^k - b^k$ for all $k < 2 n$ unless:
+:$n = 1$ and $a + b$ is a [[Definition:Integer Power|power]] of $2$
+:$n = 3$, $a = 2$, $b = 1$
+In particular, $p$ does not [[Definition:Divisor of Integer|divide]] $a^{2 k} - b^{2 k} = \paren {a^k - b^k} \paren {a^k + b^k}$ for $k < n$.
+It remains to check the case $n = 1$ and $a + b$ a [[Definition:Integer Power|power]] of $2$.
+We have to show that $a^2 + b^2$ has an [[Definition:Odd Integer|odd]] [[Definition:Prime Divisor|prime divisor]].
+Since $a$ and $b$ are [[Definition:Coprime Integers|coprime]], both $a$ and $b$ are [[Definition:Odd Integer|odd]].
+By [[Square Modulo 4]], $a^2 + b^2 \equiv 2 \pmod 4$.
+Because $a > b > 0$, $a^2 + b^2 > 2$.
+But $4 \divides 2^k$ for $k > 1$.
+Thus $a^2 + b^2$ is not a [[Definition:Integer Power|power]] of $2$.
+Hence $a^2 + b^2$ has an [[Definition:Odd Integer|odd]] [[Definition:Prime Divisor|prime divisor]].
+{{qed}}
+{{Namedfor|Karl Zsigmondy|cat = Zsigmondy}}
+[[Category:Number Theory]]
+3spcigpduvdts8wuzbojjnubqlu3z3v
+\end{proof}<|endoftext|>
+\section{Cyclotomic Polynomial of Index times Prime Power}
+Tags: Cyclotomic Polynomials
+
+\begin{theorem}
+Let $n, k \ge 1$ be [[Definition:Natural Number|natural numbers]].
+Let $p$ be a [[Definition:Prime Number|prime number]].
+Let $\Phi_n$ denote the $n$th [[Definition:Cyclotomic Polynomial|cyclotomic polynomial]].
+Then $\map {\Phi_{p^k n}} x = \begin{cases}
+\map {\Phi_n} {x^{p^k}} & \text{if } p \divides n\\
+\dfrac {\map {\Phi_n} {x^{p^k}}} {\map {\Phi_n} {x^{p^{k - 1}}}} & \text{if } p \nmid n \end{cases}$
+\end{theorem}
+
+\begin{proof}
+Suppose $p \divides n$.
+Then for all $m \in \Z$:
+{{begin-eqn}}
+{{eqn | l = m \perp n
+ | o = \implies
+ | r = m \perp n \land m \perp p
+ | c = [[Law of Identity]]; [[Divisor of One of Coprime Numbers is Coprime to Other]]
+}}
+{{eqn | o = \implies
+ | r = m \perp p^k n
+ | c = [[Integer Coprime to all Factors is Coprime to Whole]]
+}}
+{{eqn | o = \implies
+ | r = m \perp n
+ | c = [[Divisor of One of Coprime Numbers is Coprime to Other]]
+}}
+{{eqn | ll = \leadsto
+ | l = m \perp p^k n
+ | o = \iff
+ | r = m \perp n
+ | c = {{Defof|Biconditional}}
+}}
+{{end-eqn}}
+Hence:
+{{begin-eqn}}
+{{eqn | l = \map {\Phi_{p^k n} } x
+ | r = \prod_{\zeta} \paren {x - \zeta}
+ | c = where the product runs over all [[Definition:Primitive Complex Root of Unity|primitive complex $p^k n$th roots of unity]]
+}}
+{{eqn | r = \prod_{\substack {1 \mathop \le m \mathop \le p^k n \\ \gcd \set {m, p^k n} = 1} } \paren {x - \map \exp {\frac {2 \pi i m} {p^k n} } }
+ | c = [[Condition for Complex Root of Unity to be Primitive]]
+}}
+{{eqn | r = \prod_{\substack {1 \mathop \le m \mathop \le p^k n \\ \gcd \set {m, n} = 1} } \paren {x - \map \exp {\frac {2 \pi i m} {p^k n} } }
+ | c = as $m \perp p^k n \iff m \perp n$
+}}
+{{eqn | r = \prod_{q \mathop = 0}^{p^k - 1} \prod_{\substack {1 \mathop \le r \mathop \le n \\ \gcd \set {q n + r, n} = 1} } \paren {x - \map \exp {\frac {2 \pi i \paren {q n + r} } {p^k n} } }
+ | c = Writing $m = q n + r$ by [[Division Theorem]]
+}}
+{{eqn | r = \prod_{\substack {1 \mathop \le r \mathop \le n \\ \gcd \set {r, n} = 1} } \prod_{q \mathop = 0}^{p^k - 1} \paren {x - \map \exp {\frac {2 \pi i} {p^k} }^q \map \exp {\frac {2 \pi i r} {p^k n} } }
+ | c = rearranging; [[GCD with Remainder]]
+}}
+{{eqn | r = \prod_{\substack {1 \mathop \le r \mathop \le n \\ \gcd \set {r, n} = 1} } \paren {x^{p^k} - \map \exp {\frac {2 \pi i r} n} }
+ | c = [[Factorisation of z^n-a|Factorisation of $z^n - a$]]
+}}
+{{eqn | r = \prod_{\zeta} \paren {x^{p^k} - \zeta}
+ | c = where the product runs over all [[Definition:Primitive Complex Root of Unity|primitive complex $n$th roots of unity]]; [[Condition for Complex Root of Unity to be Primitive]]
+}}
+{{eqn | r = \map {\Phi_n} {x^{p^k} }
+ | c = {{Defof|Cyclotomic Polynomial}}
+}}
+{{end-eqn}}
+{{qed|lemma}}
+Now suppose $p \nmid n$.
+We still have $p \divides p n$.
+Write $p^k = p^{k - 1} p n$.
+Notice that the result we proved above holds trivially for $k = 0$:
+:$\map {\Phi_{p^0 n} } x = \map {\Phi_n } x = \map {\Phi_n } {x^1} = \map {\Phi_n } {x^{p^0}}$
+Hence from the above:
+:$\map {\Phi_{p^k n} } x = \map {\Phi_{p n}} {x^{p^{k - 1}}}$
+We need the following result:
+:the sets $\set {m \in \Z: m \perp p n}$ and $\set {p r: r \perp n}$ are [[Definition:Disjoint Sets|disjoint]] and has [[Definition:Set Union|union]] $\set {m \in \Z: m \perp n}$
+First to show that they are indeed [[Definition:Disjoint Sets|disjoint]]:
+Suppose $x \in \set {p r: r \perp n}$.
+Then $p \divides x$.
+Since $p \divides p n$:
+:$x \not \perp p n$
+and thus:
+:$x \notin \set {m \in \Z: m \perp p n}$
+Hence the sets are [[Definition:Disjoint Sets|disjoint]].
+Now we show that their [[Definition:Set Union|union]] is indeed $\set {m \in \Z: m \perp n}$.
+By [[Divisor of One of Coprime Numbers is Coprime to Other]]:
+:$\forall m \in \Z: m \perp p n \implies \paren {m \perp p \land m \perp n}$
+This gives:
+:$\set {m \in \Z: m \perp p n} \subseteq \set {m \in \Z: m \perp n}$
+Let $x \in \set {p r: r \perp n}$.
+We are given that $p \perp n$.
+By [[Integer Coprime to all Factors is Coprime to Whole]]:
+:$x \perp n$
+Hence $x \in \set {m \in \Z: m \perp n}$.
+This gives:
+:$\set {p r: r \perp n} \subseteq \set {m \in \Z: m \perp n}$
+By [[Union of Subsets is Subset]]:
+:$\set {m \in \Z: m \perp p n} \cup \set {p r: r \perp n} \subseteq \set {m \in \Z: m \perp n}$
+For the other direction, we let $x \notin \set {m \in \Z: m \perp p n} \cup \set {p r: r \perp n}$.
+Then by [[De Morgan's Laws (Set Theory)/Set Complement]]:
+:$x \in \set {m \in \Z: m \not \perp p n} \cap \set {p r: r \not \perp n}$.
+By definition of [[Definition:Set Intersection|intersection]]:
+:$x \in \set {p r: r \not \perp n}$
+Thus:
+:$\exists d \in \Z: d > 1: d \divides r \divides x \land d \divides n$
+Therefore $x \not \perp n$.
+This gives:
+:$x \notin \set {m \in \Z: m \perp n}$
+Hence:
+:$\set {m \in \Z: m \perp n} \subseteq \set {m \in \Z: m \perp p n} \cup \set {p r: r \perp n}$
+and we have our result by definition of [[Definition:Set Equality|set equality]].
+Therefore:
+{{begin-eqn}}
+{{eqn | l = \map {\Phi_{p n} } {x^{p^{k - 1} } }
+ | r = \prod_{\zeta} \paren {x^{p^{k - 1} } - \zeta}
+ | c = where the product runs over all [[Definition:Primitive Complex Root of Unity|primitive complex $p n$th roots of unity]]
+}}
+{{eqn | r = \prod_{\substack {1 \mathop \le m \mathop \le p n \\ m \mathop \perp p n} } \paren {x^{p^{k - 1} } - \map \exp {\frac {2 \pi i m} {p n} } }
+ | c = [[Condition for Complex Root of Unity to be Primitive]]
+}}
+{{eqn | r = \prod_{\substack {1 \mathop \le m \mathop \le p n \\ m \mathop \perp n} } \paren {x^{p^{k - 1} } - \map \exp {\frac {2 \pi i m} {p n} } } / \prod_{\substack {1 \mathop \le p r \mathop \le p n \\ \gcd \set {r, n} = 1} } \paren {x^{p^{k - 1} } - \map \exp {\frac {2 \pi i m} {p n} } }
+ | c = from above
+}}
+{{eqn | r = \prod_{\substack {1 \mathop \le m \mathop \le p n \\ m \mathop \perp n} } \paren {x^{p^{k - 1} } - \map \exp {\frac {2 \pi i m} {p n} } } / \prod_{\zeta} \paren {x^{p^k} - \zeta}
+ | c = where the product runs over all [[Definition:Primitive Complex Root of Unity|primitive complex $n$th roots of unity]]; [[Condition for Complex Root of Unity to be Primitive]]
+}}
+{{eqn | r = \prod_{\substack {1 \mathop \le m \mathop \le p n \\ m \mathop \perp n} } \paren {x^{p^{k - 1} } - \map \exp {\frac {2 \pi i m} {p n} } } / \map {\Phi_n} {x^{p^{k - 1} } }
+ | c = {{Defof|Cyclotomic Polynomial}}
+}}
+{{eqn | r = \prod_{q \mathop = 0}^{p - 1} \prod_{\substack {1 \mathop \le r \mathop \le n \\ \gcd \set {q n + r, n} = 1} } \paren {x^{p^{k - 1} } - \map \exp {\frac {2 \pi i \paren {q n + r} } {p n} } } / \map {\Phi_n} {x^{p^{k - 1} } }
+ | c = Writing $m = q n + r$ by [[Division Theorem]]
+}}
+{{eqn | r = \prod_{\substack {1 \mathop \le r \mathop \le n \\ \gcd \set {r, n} = 1} } \prod_{q \mathop = 0}^{p - 1} \paren {x^{p^{k - 1} } - \map \exp {\frac {2 \pi i} p}^q \map \exp {\frac {2 \pi i r} {p n} } } / \map {\Phi_n} {x^{p^{k - 1} } }
+ | c = rearranging; [[GCD with Remainder]]
+}}
+{{eqn | r = \prod_{\substack {1 \mathop \le r \mathop \le n \\ \gcd \set {r, n} = 1} } \paren {x^{p^k} - \map \exp {\frac {2 \pi i r} n} } / \map {\Phi_n} {x^{p^{k - 1} } }
+ | c = [[Factorisation of z^n-a|Factorisation of $z^n - a$]]
+}}
+{{eqn | r = \prod_{\zeta} \paren {x^{p^k} - \zeta} / \map {\Phi_n} {x^{p^{k - 1} } }
+ | c = where the product runs over all [[Definition:Primitive Complex Root of Unity|primitive complex $n$th roots of unity]]; [[Condition for Complex Root of Unity to be Primitive]]
+}}
+{{eqn | r = \map {\Phi_n} {x^{p^k} } / \map {\Phi_n} {x^{p^{k - 1} } }
+ | c = {{Defof|Cyclotomic Polynomial}}
+}}
+{{end-eqn}}
+as required.
+{{qed}}
+[[Category:Cyclotomic Polynomials]]
+bfpzg3nvi0k44pocm217165uhcntqfo
+\end{proof}<|endoftext|>
+\section{Multiplicative Order of Roots of Cyclotomic Polynomial Modulo Prime}
+Tags: Cyclotomic Polynomials
+
+\begin{theorem}
+Let $n\geq1$ be a [[Definition:Natural Number|natural number]].
+Let $p$ be a [[Definition:Prime Number|prime number]]
+Let $n=p^\alpha q$ where $\alpha = \nu_p(n)$ is the [[Definition:P-adic Valuation|valuation]] of $p$ in $n$.
+Let $a\in\Z$ with $\Phi_n(a)\equiv0\pmod p$.
+Then the [[Definition:Multiplicative Order|order]] of $a$ modulo $p$ is $q$:
+:$\operatorname{ord}_p(a) = q$.
+\end{theorem}
+
+\begin{proof}
+By [[Product of Cyclotomic Polynomials]], $p\mid \Phi_n(a) \mid a^n-1$.
+Thus $a$ is [[Definition:Coprime Integers|coprime]] to $p$.
+By [[Fermat's Little Theorem]], $1\equiv a^n\equiv a^q\pmod p$.
+Thus $\operatorname{ord}_p(a) \leq q$.
+Suppose $\operatorname{ord}_p(a) = k < q$.
+By [[Product of Cyclotomic Polynomials]], $p\mid \Phi_d(a)$ for some $d\mid k$.
+Then $a$ is a [[Definition:Double Root of Polynomial|double root]] of $x^q-1$ modulo $p$.
+By [[Double Root of Polynomial is Root of Derivative]], $q\equiv0\pmod p$.
+This is a contradiction, thus $k=q$.
+{{qed}}
+[[Category:Cyclotomic Polynomials]]
+888kvf5vm9j2n8ofx0xc0w4sdd4dxze
+\end{proof}<|endoftext|>
+\section{Lower Topology is Unique}
+Tags: Topological Order Theory
+
+\begin{theorem}
+Let $T_1 = \left({S, \preceq, \tau_1}\right)$ and $T_2 = \left({S, \preceq, \tau_2}\right)$ be [[Definition:Relational Structure with Topology|relational structures]] with [[Definition:Lower Topology|lower topologies]].
+Then:
+: $\tau_1 = \tau_2$
+\end{theorem}
+
+\begin{proof}
+Define:
+: $B := \left\{ {\complement_S \left({x^\succeq}\right): x \in S}\right\}$
+where $x^\succeq$ denotes the [[Definition:Upper Closure of Element|upper closure]] of $x$.
+Thus:
+{{begin-eqn}}
+{{eqn | l = \tau_1
+ | r = \tau \left({B}\right)
+ | c = {{Defof|Topology Generated by Synthetic Sub-Basis}}
+}}
+{{eqn | r = \tau_2
+ | c = {{Defof|Topology Generated by Synthetic Sub-Basis}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Homogeneous Cyclotomic Polynomial is Symmetric}
+Tags: Cyclotomic Polynomials
+
+\begin{theorem}
+Let $n>1$ be a [[Definition:Natural Number|natural number]].
+Let $\Phi_n(x,y)$ be the $n$th [[Definition:Homogeneous Cyclotomic Polynomial|homogeneous cyclotomic polynomial]].
+Then $\Phi_n(x,y) = \Phi_n(y,x)$, that is, $\Phi_n(x,y)$ is [[Definition:Symmetric Polynomial|symmetric]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Cyclotomic Polynomials]]
+1klkb4mge55s34ezlu50w1to0q0ktsa
+\end{proof}<|endoftext|>
+\section{Cyclotomic Polynomial of Index Power of Two}
+Tags: Cyclotomic Polynomials
+
+\begin{theorem}
+Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
+Then the $2^n$th [[Definition:Cyclotomic Polynomial|cyclotomic polynomial]] is:
+:$\map {\Phi_{2^n} } x = x^{2^{n - 1} } + 1$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \map {\Phi_{2^n} } x
+ | r = \prod_{\zeta} \paren {x - \zeta}
+ | c = where the product runs over all [[Definition:Primitive Complex Root of Unity|primitive complex $2^n$th roots of unity]]
+}}
+{{eqn | r = \prod_{\substack {1 \mathop \le k \mathop \le 2^n \\ \gcd \set {k, 2^n} = 1} } \paren {x - \map \exp {\frac {2 \pi i k} {2^n} } }
+ | c = [[Condition for Complex Root of Unity to be Primitive]]
+}}
+{{eqn | r = \prod_{\substack {1 \mathop \le k \mathop \le 2^n \\ k \text { odd} } } \paren {x - \map \exp {\frac {k i \pi} {2^{n - 1} } } }
+ | c =
+}}
+{{eqn | r = \prod_{k \mathop = 0}^{2^{n - 1} - 1} \paren {x - \map \exp {\frac {\paren {2 k + 1} i \pi} {2^{n - 1} } } }
+ | c = by writing each [[Definition:Odd Integer|odd number]] as $2 k + 1$
+}}
+{{eqn | r = x^{2^{n - 1} } + 1
+ | c = [[Factorisation of z^n+1|Factorisation of $z^n + 1$]]
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Cyclotomic Polynomials]]
+5f6f3syoexhclaxk8bbaw0tlv0k0h4y
+\end{proof}<|endoftext|>
+\section{Trivial Estimate for Cyclotomic Polynomials}
+Tags: Cyclotomic Polynomials
+
+\begin{theorem}
+Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
+Let $\Phi_n$ be the $n$th [[Definition:Cyclotomic Polynomial|cyclotomic polynomial]].
+Let $\phi$ be the [[Definition:Euler Totient Function|Euler totient function]].
+Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
+Then:
+:$\size {\size z - 1}^{\map \phi n} \le \size {\map {\Phi_n} z} \le \paren {\size z + 1}^{\map \phi n}$
+where:
+:the first inequality becomes an equality only if:
+:::$n = 1$ and $z \in \R_{\ge 0}$
+::or:
+:::$n = 2$ and $z \in \R_{\le 0}$
+:the second inequality becomes an equality only if:
+:::$n = 1$ and $z \in \R_{\le 0}$
+::or:
+:::$n = 2$ and $z \in \R_{\ge 0}$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Cyclotomic Polynomials]]
+8irnibmc93ig6ztlrhmeb6bb0y6sfpt
+\end{proof}<|endoftext|>
+\section{2520 equals Sum of 4 Divisors in 6 Ways}
+Tags: Recreational Mathematics, 2520
+
+\begin{theorem}
+The number $2520$ can be expressed as the [[Definition:Integer Addition|sum]] of $4$ of its [[Definition:Divisor of Integer|divisors]] in $6$ different ways:
+{{begin-eqn}}
+{{eqn | l = 2520
+ | r = 1260 + 630 + 504 + 126
+}}
+{{eqn | r = 1260 + 630 + 421 + 210
+}}
+{{eqn | r = 1260 + 840 + 360 + 60
+}}
+{{eqn | r = 1260 + 840 + 315 + 105
+}}
+{{eqn | r = 1260 + 840 + 280 + 140
+}}
+{{eqn | r = 1260 + 840 + 252 + 168
+}}
+{{end-eqn}}
+This is the maximum possible number of ways it is possible to express an [[Definition:Integer|integer]] as the sum of $4$ of its [[Definition:Divisor of Integer|divisors]].
+\end{theorem}
+
+\begin{proof}
+We apply [[1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways]]:
+{{:1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways}}
+Find the maximum [[Definition:Integer Power|powers]] of the [[Definition:Prime Number|primes]] in each equation, and choose the largest that appears:
+{{begin-eqn}}
+{{eqn | l = 42
+ | r = 2^1 \times 3^1 \times 7^1
+}}
+{{eqn | l = 24
+ | r = 2^3 \times 3^1
+}}
+{{eqn | l = 18
+ | r = 2^1 \times 3^2
+}}
+{{eqn | l = 30
+ | r = 2^1 \times 3^1 \times 5^1
+}}
+{{eqn | l = 20
+ | r = 2^2 \times 5^1
+}}
+{{eqn | l = 12
+ | r = 2^2 \times 3^1
+}}
+{{end-eqn}}
+Therefore the smallest number would be:
+:$2^3 \times 3^2 \times 5^1 \times 7^1 = 2520$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Complete List of Special Highly Composite Numbers}
+Tags: Special Highly Composite Numbers
+
+\begin{theorem}
+There are exactly $6$ [[Definition:Special Highly Composite Number|special highly composite numbers]]:
+:$1, 2, 6, 12, 60, 2520$
+{{OEIS|A106037}}
+\end{theorem}
+
+\begin{proof}
+We have the following:
+: [[Special Highly Composite Number/Examples/1|$1$ is a Special Highly Composite Number]]
+: [[Special Highly Composite Number/Examples/2|$2$ is a Special Highly Composite Number]]
+: [[Special Highly Composite Number/Examples/6|$6$ is a Special Highly Composite Number]]
+: [[Special Highly Composite Number/Examples/12|$12$ is a Special Highly Composite Number]]
+: [[Special Highly Composite Number/Examples/60|$60$ is a Special Highly Composite Number]]
+: [[Special Highly Composite Number/Examples/2520|$2520$ is a Special Highly Composite Number]]
+By inspection of the [[Definition:Sequence of Highly Composite Number|sequence of highly composite numbers]]:
+:$1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, \ldots$
+it can be seen that there are no more [[Definition:Special Highly Composite Number|special highly composite numbers]] less than $2520$.
+Let $n > 18$.
+From [[Ratio between Consecutive Highly Composite Numbers Greater than 2520 is Less than 2]], the $n$th [[Definition:Highly Composite Number|highly composite number]] does not [[Definition:Divisor of Integer|divide]] the $n+1$th.
+Hence the $n$th [[Definition:Highly Composite Number|highly composite number]] is not a [[Definition:Special Highly Composite Number|special highly composite number]].
+The result follows.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Lifting The Exponent Lemma for Sums for p=2}
+Tags: Lifting The Exponent Lemma
+
+\begin{theorem}
+Let $x, y \in \Z$ be [[Definition:Integer|integers]] with $x + y \ne 0$.
+Let $n \ge 1$ be an [[Definition:Odd Integer|odd]] [[Definition:Natural Number|natural number]].
+Let:
+:$2 \divides x + y$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
+Then:
+:$\map {\nu_2} {x^n + y^n} = \map {\nu_2} {x + y}$
+where $\nu_2$ denotes [[Definition:P-adic Valuation|$2$-adic valuation]].
+
+\end{theorem}
+
+\begin{proof}
+This follows from the [[Lifting The Exponent Lemma for p=2]] with $y$ replaced by $-y$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Prime Decomposition of Highly Composite Number}
+Tags: Highly Composite Numbers
+
+\begin{theorem}
+Let $n$ be a [[Definition:Highly Composite Number|highly composite number]].
+Let the [[Definition:Prime Decomposition|prime decomposition]] of $n$ be expressed as:
+:$n = \displaystyle \prod_{k \mathop \in \N} {p_k}^{r_k}$
+where $p_k$ denotes the [[Definition:Prime Number|$k$th prime]].
+Then the [[Definition:Integer Sequence|sequence]] $\left\langle{r_k}\right\rangle$ is [[Definition:Decreasing Sequence|decreasing]].
+That is:
+:$\forall k \in \N: r_k \ge r_{k + 1}$
+\end{theorem}
+
+\begin{proof}
+Let $n = \displaystyle \prod_{k \mathop \in \N} {p_k}^{r_k}$ be [[Definition:Highly Composite Number|highly composite]].
+By definition of [[Definition:Tau Function|$\tau$ function]]:
+:$\tau \left({n}\right) = \displaystyle \prod_{k \mathop \in \N} \left({r_k + 1}\right)$
+{{AimForCont}} $r_{l + 1} > r_l$ for some $l \in \N$.
+Consider $m \in \Z$ whose [[Definition:Prime Decomposition|prime decomposition]] of $n$ is expressed as:
+:$m = \displaystyle \prod_{k \mathop \in \N} {p_k}^{s_k}$
+where:
+:$\forall j < l: s_j = r_j$
+:$\forall j > l + 1: s_j = r_j$
+:$s_l = r_{l + 1}$
+:$s_{l + 1} = r_l$
+We have that:
+{{begin-eqn}}
+{{eqn | l = \tau \left({m}\right)
+ | r = \prod_{k \mathop \in \N} \left({s_k + 1}\right)
+ | c = {{Defof|Tau Function}}
+}}
+{{eqn | r = \prod_{k \mathop \in \N} \left({r_k + 1}\right)
+ | c = as all the $s$s are the same as all the $r$s, but in a different order
+}}
+{{eqn | r = \tau \left({n}\right)
+ | c = {{Defof|Tau Function}}
+}}
+{{end-eqn}}
+Thus:
+: $\tau \left({m}\right) = \tau \left({n}\right)$
+Now we have that:
+:$r_l < r_{l + 1}$
+and so:
+:$s_l > s_{l + 1}$
+and so:
+{{begin-eqn}}
+{{eqn | l = \frac { {p_l}^{r_l} {p_{l + 1} }^{r_{l + 1} } } { {p_l}^{s_l} {p_{l + 1} }^{s_{l + 1} } }
+ | r = \frac { {p_l}^{r_l} {p_{l + 1} }^{r_{l + 1} } } { {p_l}^{r_{l + 1} } {p_{l + 1} }^{r_l} }
+ | c =
+}}
+{{eqn | r = \frac { {p_l}^c} { {p_{l + 1}^c} }
+ | c = where $c = r_{l + 1} - r_l$
+}}
+{{end-eqn}}
+Then:
+:$p_{l + 1} > p_l$
+and so:
+:${p_{l + 1} }^c > {p_l}^c$
+from which it follows that:
+:$m < n$
+while:
+:$\tau \left({m}\right) = \tau \left({n}\right)$
+But $n$ is [[Definition:Highly Composite Number|highly composite]].
+This means that if $\tau \left({m}\right) = \tau \left({n}\right)$, then $n \le m$.
+This is [[Definition:Contradiction|contradicted]] by $n > m$.
+Thus by [[Proof by Contradiction]]:
+:$\forall k \in \N: r_k \ge r_{k + 1}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Complement of Upper Closure of Element is Open in Lower Topology}
+Tags: Topological Order Theory
+
+\begin{theorem}
+Let $T = \left({S, \preceq, \tau}\right)$ be a [[Definition:Relational Structure with Topology|relational structure]] with [[Definition:Lower Topology|lower topology]].
+Let $x \in S$.
+Then $\complement_S\left({x^\succeq}\right)$ is [[Definition:Open Set (Topology)|open]] and $x^\succeq$ is [[Definition:Closed Set (Topology)|closed]].
+\end{theorem}
+
+\begin{proof}
+Define $B := \left\{ {\complement_S\left({y^\succeq}\right): y \in S}\right\}$
+By definition of [[Definition:Lower Topology|lower topology]]:
+:$B$ is [[Definition:Analytic Sub-Basis|sub-basis]] of $T$.
+By definition of [[Definition:Analytic Sub-Basis|sub-basis]]:
+:$B \subseteq \tau$
+By definition of $B$:
+:$\complement_S\left({x^\succeq}\right) \in B$
+Thus by definition of [[Definition:Subset|subset]]:
+:$\complement_S\left({x^\succeq}\right) \in \tau$
+Thus by definition:
+:$x^\succeq$ is [[Definition:Closed Set (Topology)|closed]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Open Subset is Lower in Lower Topology}
+Tags: Topological Order Theory
+
+\begin{theorem}
+Let $T = \left({S, \preceq, \tau}\right)$ be a [[Definition:Transitive Relation|transitive]] [[Definition:Relational Structure with Topology|relational structure]] with [[Definition:Lower Topology|lower topology]].
+Let $A \subseteq S$ such that
+:$A$ is [[Definition:Open Set (Topology)|open]].
+Then $A$ is [[Definition:Lower Set|lower]].
+\end{theorem}
+
+\begin{proof}
+Define $B = \left\{ {\complement_S\left({x^\succeq}\right): x \in S}\right\}$
+By definition of [[Definition:Lower Topology|lower topology]]:
+:$B$ is [[Definition:Analytic Sub-Basis|sub-basis]] of $T$.
+By definitions of [[Definition:Analytic Sub-Basis|sub-basis]] and [[Definition:Analytic Basis|basis]]:
+:$\displaystyle \mathcal B = \left\{ {\bigcap \mathcal F: \mathcal F \subseteq B, \, \mathcal F \text{ is finite}}\right\}$ is a [[Definition:Analytic Basis|basis]].
+By definition of [[Definition:Analytic Basis|basis]]:
+:$\displaystyle \tau \subseteq \left\{ {\bigcup X: X \subseteq \mathcal B}\right\}$
+Let $x \in A$, $y \in S$ such that
+:$y \preceq x$
+By definition of [[Definition:Open Set (Topology)|open set]]:
+:$A \in \tau$
+By definition of [[Definition:Subset|subset]]:
+:$\exists Y \subseteq \mathcal B: A = \bigcup Y$
+By definition of [[Definition:Set Union/General Definition|union]]:
+:$\exists Z \in Y: x \in Z$
+By definition of [[Definition:Subset|subset]]:
+:$\exists X \subseteq B: Z = \bigcap X \land X$ is [[Definition:Finite Set|finite]].
+we will prove that
+:$\forall Q \in X: y \in Q$
+Let $Q \in X$.
+By definition of [[Definition:Subset|subset]]:
+:$Q \in B$
+By definition of $B$:
+:$\exists z \in S: Q = \complement_S\left({z^\succeq}\right)$
+By definition of [[Definition:Intersection of Set of Sets|intersection]]:
+:$x \in Q$
+By [[Upper Closure is Upper Set]]:
+:$z^\succeq$ is an [[Definition:Upper Set|upper set]].
+By [[Complement of Upper Set is Lower Set]]:
+:$Q$ is a [[Definition:Lower Set|lower set]].
+Thus by definition of [[Definition:Lower Set|lower set]]:
+:$y \in Q$
+{{qed|lemma}}
+By definition of [[Definition:Intersection of Set of Sets|intersection]]:
+:$y \in Z$
+Thus by definition of [[Definition:Set Union/General Definition|union]]:
+:$y \in A$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Ratio between Consecutive Highly Composite Numbers Greater than 2520 is Less than 2}
+Tags: Highly Composite Numbers
+
+\begin{theorem}
+The [[Definition:Ratio|ratio]] between $2$ consecutive [[Definition:Highly Composite Number|highly composite numbers]] both greater than $2520$ is less than $2$.
+\end{theorem}
+
+\begin{proof}
+{{AimForCont}} $n$ and $m$ are consecutive [[Definition:Highly Composite Number|highly composite numbers]] such that:
+: $2520 < n < m$
+: $m / n \ge 2$
+By definition of [[Definition:Highly Composite Number|highly composite]]:
+:$\tau \left({m}\right) > \tau \left({n}\right)$
+and, [[Definition:By Hypothesis|by hypothesis]], $m$ is the smallest such [[Definition:Integer|integer]].
+We have that:
+:$\tau \left({2 n}\right) > \tau \left({n}\right)$
+so it follows that $m \le 2 n$, otherwise $m$ would ''not'' be the smallest such [[Definition:Integer|integer]].
+So from $m / n \ge 2$ and $m \le 2 n$, it follows that $m = 2 n$.
+We have that [[Special Highly Composite Number/Examples/2520|$2520$ is a special highly composite number]].
+The [[Definition:Prime Decomposition|prime decomposition]] of $2520$ s given by:
+:$2520 = 2^3 \times 3^2 \times 5 \times 7$
+We have that $n$ is a [[Definition:Highly Composite Number|highly composite number]] such that $n > 2520$.
+As $2520$ is a [[Definition:Special Highly Composite Number|special highly composite number]], $2520$ is a [[Definition:Divisor of Integer|divisor]] of $n$.
+Thus $n$ can be expressed as:
+:$n = 2^a \times 3^b \times 5^c \times 7^d \times 11^e \times 13^f \times r$
+where:
+:$a \ge 3$
+:$b \ge 2$
+:$c \ge 1$
+:$d \ge 1$
+:$r$ is a possibly [[Definition:Vacuous Product|vacuous]] [[Definition:Integer Multiplication|product]] of [[Definition:Prime Number|prime numbers]] strictly greater than $13$.
+We have that $n$ and $m = 2 n$ are consecutive [[Definition:Highly Composite Number|highly composite numbers]].
+Hence it follows that:
+:$\tau \left({3 n / 2}\right) \le \tau \left({n}\right)$
+and:
+:$\tau \left({4 n / 3}\right) \le \tau \left({n}\right)$
+otherwise $3 n / 2$ or $4 n / 3$ would be [[Definition:Highly Composite Number|highly composite numbers]] between $n$ and $2 n$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \tau \left({3 n / 2}\right)
+ | o = \le
+ | r = \tau \left({n}\right)
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \tau \left({2^{a - 1} \times 3^{b + 1} \times 5^c \times 7^d \times 11^e \times 13^f \times r}\right)
+ | o = \le
+ | r = \tau \left({2^a \times 3^b \times 5^c \times 7^d \times 11^e \times 13^f \times r}\right)
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \tau \left({2^{a - 1} \times 3^{b + 1} }\right) \times \tau \left({5^c \times 7^d \times 11^e \times 13^f \times r}\right)
+ | o = \le
+ | r = \tau \left({2^a \times 3^b}\right) \times \tau \left({5^c \times 7^d \times 11^e \times 13^f \times r}\right)
+ | c = [[Tau Function is Multiplicative]]
+}}
+{{eqn | ll= \leadsto
+ | l = \tau \left({2^{a - 1} \times 3^{b + 1} }\right)
+ | o = \le
+ | r = \tau \left({2^a \times 3^b}\right)
+ | c = simplifying
+}}
+{{eqn | ll= \leadsto
+ | l = \left({\left({a - 1}\right) + 1}\right) \left({\left({b + 1}\right) + 1}\right)
+ | o = \le
+ | r = \left({a + 1}\right) \left({b + 1}\right)
+ | c = {{Defof|Tau Function}}
+}}
+{{eqn | ll= \leadsto
+ | l = a \left({b + 2}\right)
+ | o = \le
+ | r = \left({a + 1}\right) \left({b + 1}\right)
+ | c = simplifying
+}}
+{{end-eqn}}
+and:
+{{begin-eqn}}
+{{eqn | l = \tau \left({4 n / 3}\right)
+ | o = \le
+ | r = \tau \left({n}\right)
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \tau \left({2^{a + 2} \times 3^{b - 1} \times 5^c \times 7^d \times 11^e \times 13^f \times r}\right)
+ | o = \le
+ | r = \tau \left({2^a \times 3^b \times 5^c \times 7^d \times 11^e \times 13^f \times r}\right)
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \tau \left({2^{a + 2} \times 3^{b - 1} }\right) \times \tau \left({5^c \times 7^d \times 11^e \times 13^f \times r}\right)
+ | o = \le
+ | r = \tau \left({2^a \times 3^b}\right) \times \tau \left({5^c \times 7^d \times 11^e \times 13^f \times r}\right)
+ | c = [[Tau Function is Multiplicative]]
+}}
+{{eqn | ll= \leadsto
+ | l = \tau \left({2^{a + 2} \times 3^{b - 1} }\right)
+ | o = \le
+ | r = \tau \left({2^a \times 3^b}\right)
+ | c = simplifying
+}}
+{{eqn | ll= \leadsto
+ | l = \left({\left({a + 2}\right) + 1}\right) \left({\left({b - 1}\right) + 1}\right)
+ | o = \le
+ | r = \left({a + 1}\right) \left({b + 1}\right)
+ | c = {{Defof|Tau Function}}
+}}
+{{eqn | ll= \leadsto
+ | l = \left({a + 3}\right) b
+ | o = \le
+ | r = \left({a + 1}\right) \left({b + 1}\right)
+ | c = simplifying
+}}
+{{end-eqn}}
+This leads to:
+{{begin-eqn}}
+{{eqn | l = a
+ | o = \le
+ | r = b + 1
+ | c =
+}}
+{{eqn | l = 2 b - 1
+ | o = \le
+ | r = a
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 2 b - 1
+ | o = \le
+ | r = b + 1
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = b
+ | o = \le
+ | r = 2
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = a
+ | o = \le
+ | r = 3
+ | c =
+}}
+{{end-eqn}}
+It has already been established that:
+:$a \ge 3$
+:$b \ge 2$
+{{begin-eqn}}
+{{eqn | l = a
+ | o = \ge
+ | r = 3
+}}
+{{eqn | l = b
+ | o = \le
+ | r = 2
+}}
+{{end-eqn}}
+so it is now possible to state:
+{{begin-eqn}}
+{{eqn | l = a
+ | r = 3
+}}
+{{eqn | l = b
+ | r = 2
+}}
+{{end-eqn}}
+Suppose:
+:$(1): \quad f \ge 1$
+Then:
+{{begin-eqn}}
+{{eqn | l = 2^5 \times 3^3 \times 5^c \times 7^d \times 11^e \times 13^{f - 1} \times r
+ | o = <
+ | r = 2^3 \times 3^2 \times 5^c \times 7^d \times 11^e \times 13^f \times r
+ | c = because $12 = 2^2 \times 3 < 13$
+}}
+{{eqn | ll= \leadsto
+ | l = \tau \left({2^5 \times 3^3 \times 5^c \times 7^d \times 11^e \times 13^{f - 1} \times r}\right)
+ | o = <
+ | r = \tau \left({2^3 \times 3^2 \times 5^c \times 7^d \times 11^e \times 13^f \times r}\right)
+ | c = as $2^3 \times 3^2 \times 5^c \times 7^d \times 11^e \times 13^f \times r$ is [[Definition:Highly Composite Number|highly composite]]
+}}
+{{eqn | ll= \leadsto
+ | l = \tau \left({2^5 \times 3^3 \times 13^{f - 1} }\right) \times \tau \left({5^c \times 7^d \times 11^e \times r}\right)
+ | o = <
+ | r = \tau \left({2^3 \times 3^2 \times 13^f}\right) \times \tau \left({5^c \times 7^d \times 11^e \times r}\right)
+ | c = [[Tau Function is Multiplicative]]
+}}
+{{eqn | ll= \leadsto
+ | l = \tau \left({2^5 \times 3^3 \times 13^{f - 1} }\right)
+ | o = <
+ | r = \tau \left({2^3 \times 3^2 \times 13^f}\right)
+ | c = simplifying
+}}
+{{eqn | ll= \leadsto
+ | l = \left({5 + 1}\right) \left({3 + 1}\right) \left({\left({f - 1}\right) + 1}\right)
+ | o = <
+ | r = \left({3 + 1}\right) \left({2 + 1}\right) \left({f + 1}\right)
+ | c = {{Defof|Tau Function}}
+}}
+{{eqn | ll= \leadsto
+ | l = 24 f
+ | o = <
+ | r = 12 \left({f + 1}\right)
+ | c = {{Defof|Tau Function}}
+}}
+{{eqn | ll= \leadsto
+ | l = f
+ | o = <
+ | r = 1
+ | c = which is a [[Definition:Contradiction|contradiction]] of $(1)$
+}}
+{{end-eqn}}
+So $f = 0$ and so by [[Prime Decomposition of Highly Composite Number]] $r = 1$.
+Thus:
+:$n = 2^3 \times 3^2 \times 5^c \times 7^d \times 11^e$
+where $c = 1$ or $c = 2$.
+Suppose $c = 2$.
+Then:
+{{begin-eqn}}
+{{eqn | l = 2^5 \times 3^2 \times 5 \times 7^d \times 11^e
+ | o = <
+ | r = 2^3 \times 3^2 \times 5^2 \times 7^d \times 11^e
+ | c = because $4 = 2^2 < 5$
+}}
+{{eqn | ll= \leadsto
+ | l = \tau \left({2^5 \times 3^2 \times 5 \times 7^d \times 11^e}\right)
+ | o = <
+ | r = \tau \left({2^3 \times 3^2 \times 5^2 \times 7^d \times 11^e}\right)
+ | c = as $2^3 \times 3^2 \times 5^2 \times 7^d \times 11^e$ is [[Definition:Highly Composite Number|highly composite]]
+}}
+{{eqn | ll= \leadsto
+ | l = \tau \left({2^5 \times 5}\right) \times \tau \left({3^2 \times 7^d \times 11^e}\right)
+ | o = <
+ | r = \tau \left({2^3 \times 5^2}\right) \times \tau \left({3^2 \times 7^d \times 11^e}\right)
+ | c = [[Tau Function is Multiplicative]]
+}}
+{{eqn | ll= \leadsto
+ | l = \tau \left({2^5 \times 5}\right)
+ | o = <
+ | r = \tau \left({2^3 \times 5^2}\right)
+ | c = simplifying
+}}
+{{eqn | ll= \leadsto
+ | l = \left({5 + 1}\right) \left({1 + 1}\right)
+ | o = <
+ | r = \left({3 + 1}\right) \left({2 + 1}\right)
+ | c = {{Defof|Tau Function}}
+}}
+{{eqn | ll= \leadsto
+ | l = 12
+ | o = <
+ | r = 12
+ | c = which is an absurdity
+}}
+{{end-eqn}}
+So $c = 1$ and so by [[Prime Decomposition of Highly Composite Number]]:
+:$n = 2^3 \times 3^2 \times 5 \times 7$
+or:
+:$n = 2^3 \times 3^2 \times 5 \times 7 \times 11$
+Thus it has been established that these are the only possible values of $n$ greater than $2520$ which may fit the criteria for $n$ and $2 n$ to be consecutive [[Definition:Highly Composite Number|highly composite numbers]].
+But the first of these is $2^3 \times 3^2 \times 5 \times 7 = 2520$ which fails through not being greater than $2520$.
+Thus we consider:
+:$n = 2^3 \times 3^2 \times 5 \times 7 \times 11 = 27 \, 720$
+We have that:
+{{begin-eqn}}
+{{eqn | l = \tau \left({27 \, 720}\right)
+ | r = \tau \left({2^3 \times 3^2 \times 5 \times 7 \times 11}\right)
+ | c =
+}}
+{{eqn | r = \left({3 + 1}\right) \left({2 + 1}\right) \left({1 + 1}\right) \left({1 + 1}\right) \left({1 + 1}\right)
+ | c =
+}}
+{{eqn | r = 96
+ | c =
+}}
+{{end-eqn}}
+It is now necessary to show that there are no [[Definition:Highly Composite Number|highly composite numbers]] between $27 \, 720$ and $2 \times 27 \, 720 = 55 \, 440$.
+That is, all numbers $n$ such that $27 \, 720 < n < 55 \, 440$ are to be shown to have $\tau \left({n}\right) < 96$.
+But:
+{{begin-eqn}}
+{{eqn | l = \tau \left({45 \, 360}\right)
+ | r = \tau \left({2^4 \times 3^4 \times 5 \times 7}\right)
+ | c =
+}}
+{{eqn | r = \left({4 + 1}\right) \left({4 + 1}\right) \left({1 + 1}\right) \left({1 + 1}\right)
+ | c =
+}}
+{{eqn | r = 100
+ | c =
+}}
+{{end-eqn}}
+So $45 \, 360$ has a higher $\tau$ than $27 \, 720$ and so the next higher [[Definition:Highly Composite Number|highly composite number]] than $27 \, 720$ is less than twice it.
+The result follows by [[Proof by Contradiction]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Modified Kaprekar Process on 4-Digit Number terminates in 2538}
+Tags: Kaprekar's Process
+
+\begin{theorem}
+Let $n$ be a [[Definition:Digit|$4$-digit]] [[Definition:Natural Number|number]].
+Let $n$ be operated on by the [[Definition:Modified Kaprekar Process|modified Kaprekar process]].
+The eventual result is always $2538$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|I find these processes dull. Does anyone else want to go into a detailed analysis of this and other similar?}}
+\end{proof}<|endoftext|>
+\section{Fibonacci Number equal to Sum of Sequence of Cubes}
+Tags: Fibonacci Numbers, Cube Numbers, 2584
+
+\begin{theorem}
+The following [[Definition:Fibonacci Number|Fibonacci number]] can be expressed as the [[Definition:Integer Addition|sum]] of a [[Definition:Integer Sequence|sequence]] of [[Definition:Cube Number|cubes]]:
+:$F_{18} = 2584 = 7^3 + 8^3 + 9^3 + 10^3$
+{{expand|Any more?}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 2584
+ | r = 343 + 512 + 729 + 1000
+ | c =
+}}
+{{eqn | r = 7^3 + 8^3 + 9^3 + 10^3
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Dudeney's Property of 2592}
+Tags: Recreational Mathematics, 2592
+
+\begin{theorem}
+:$2592 = 2^5 \times 9^2$
+It is the only [[Definition:Natural Number|number]] $n$ that has the property that:
+:$n = \sqbrk {abcd} = a^b \times c^d$
+where $\sqbrk {abcd}$ denotes the [[Definition:Decimal Notation|decimal representation]] of $n$.
+\end{theorem}
+
+\begin{proof}
+First we verify that $2592$ does indeed satisfy the given property.
+{{begin-eqn}}
+{{eqn | l = 2592
+ | r = 2^5 \times 3^4
+ | c = [[Definition:Prime Decomposition|Prime Decomposition]] of $2592$
+}}
+{{eqn | r = 2^5 \times \paren {3^2}^2
+ | c =
+}}
+{{eqn | r = 2^5 \times 9^2
+ | c =
+}}
+{{end-eqn}}
+{{qed|lemma}}
+It remains to be shown that this is the only such number.
+Because $\sqbrk {abcd} = a^b \times c^d$, neither $a^b$ nor $c^d$ can have more than $4$ [[Definition:Digit|digits]].
+Hence, for each [[Definition:Digit|digit]], the highest [[Definition:Integer Power|power]] is:
+{{begin-eqn}}
+{{eqn | l = 1^9
+ | r = 1
+}}
+{{eqn | l = 2^9
+ | r = 512
+}}
+{{eqn | l = 3^8
+ | r = 6561
+}}
+{{eqn | l = 4^6
+ | r = 4096
+}}
+{{eqn | l = 5^5
+ | r = 3125
+}}
+{{eqn | l = 6^5
+ | r = 7776
+}}
+{{eqn | l = 7^4
+ | r = 2401
+}}
+{{eqn | l = 8^4
+ | r = 4096
+}}
+{{eqn | l = 9^4
+ | r = 6561
+}}
+{{end-eqn}}
+Neither $a$ or $c$ can be [[Definition:Zero Digit|zero]], or that would make $\sqbrk {abcd} = 0$.
+Suppose $a = 1$ or $b = 0$.
+Then:
+:$\sqbrk {abcd} = c^d$
+Apart from the above [[Definition:Integer Power|powers]] which have $4$ [[Definition:Digit|digits]], we also have:
+{{begin-eqn}}
+{{eqn | l = 3^7
+ | r = 2187
+}}
+{{eqn | l = 4^5
+ | r = 1024
+}}
+{{eqn | l = 6^4
+ | r = 1296
+}}
+{{end-eqn}}
+Hence, by inspection, it is seen that none of these fit the pattern $\sqbrk {1bcd}$ or $\sqbrk {a0cd}$.
+Similarly, suppose $c = 1$ or $d = 0$.
+Then:
+:$\sqbrk {abcd} = a^b$
+Again, by inspection, it is seen that none of the above $4$-[[Definition:Digit|digit]] [[Definition:Integer Power|powers]] fit the pattern $\sqbrk {ab1d}$ or $\sqbrk {abc0}$.
+Suppose either $a^b$ or $c^d$ has $4$ [[Definition:Digit|digits]].
+Then the other is less than $10$, giving:
+{{begin-eqn}}
+{{eqn | l = n^1
+ | r = n
+}}
+{{eqn | l = 2^2
+ | r = 4
+}}
+{{eqn | l = 2^3
+ | r = 8
+}}
+{{eqn | l = 3^2
+ | r = 9
+}}
+{{end-eqn}}
+We try multiplying these by all the above $4$-[[Definition:Digit|digit]] [[Definition:Integer Power|powers]] such that their [[Definition:Product|product]] is $4$ [[Definition:Digit|digits]] (there are not many).
+First note that $1^1$ can be ruled out as none of these $4$-[[Definition:Digit|digit]] [[Definition:Integer Power|powers]] either begins or ends with $11$.
+{{begin-eqn}}
+{{eqn | l = 3^7 \times 2^1
+ | r = 2187 \times 2
+}}
+{{eqn | r = 4374
+}}
+{{eqn | l = 3^7 \times 3^1
+ | r = 2187 \times 3
+}}
+{{eqn | r = 6561
+}}
+{{eqn | l = 3^7 \times 4^1
+ | r = 3^7 \times 2^2
+}}
+{{eqn | r = 2187 \times 4
+}}
+{{eqn | r = 8748
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 4^5 \times 2^1
+ | r = 1024 \times 2
+}}
+{{eqn | r = 2048
+}}
+{{eqn | l = 4^5 \times 3^1
+ | r = 1024 \times 3
+}}
+{{eqn | r = 3072
+}}
+{{eqn | l = 4^5 \times 4^1
+ | r = 4^5 \times 2^2
+}}
+{{eqn | r = 1024 \times 4
+}}
+{{eqn | r = 4096
+}}
+{{eqn | l = 4^5 \times 5^1
+ | r = 1024 \times 5
+}}
+{{eqn | r = 5120
+}}
+{{eqn | l = 4^5 \times 6^1
+ | r = 1024 \times 6
+}}
+{{eqn | r = 6144
+ | c = a near miss: $6^1 \times 4^5 = 6144$
+}}
+{{eqn | l = 4^5 \times 7^1
+ | r = 1024 \times 7
+}}
+{{eqn | r = 7168
+}}
+{{eqn | l = 4^5 \times 8^1
+ | r = 4^5 \times 2^3
+}}
+{{eqn | r = 1024 \times 8
+}}
+{{eqn | r = 8192
+}}
+{{eqn | l = 4^5 \times 9^1
+ | r = 4^5 \times 3^2
+}}
+{{eqn | r = 1024 \times 9
+}}
+{{eqn | r = 9216
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 4^6 \times 2^1
+ | r = 4096 \times 2
+}}
+{{eqn | r = 8192
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 5^5 \times 2^1
+ | r = 3125 \times 2
+}}
+{{eqn | r = 6250
+}}
+{{eqn | l = 5^5 \times 3^1
+ | r = 3125 \times 3
+}}
+{{eqn | r = 9375
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6^4 \times 2^1
+ | r = 1296 \times 2
+}}
+{{eqn | r = 2592
+}}
+{{eqn | l = 6^4 \times 3^1
+ | r = 1296 \times 3
+}}
+{{eqn | r = 3888
+}}
+{{eqn | l = 6^4 \times 4^1
+ | r = 6^4 \times 2^2
+}}
+{{eqn | r = 1296 \times 4
+}}
+{{eqn | r = 5184
+}}
+{{eqn | l = 6^4 \times 5^1
+ | r = 1296 \times 5
+}}
+{{eqn | r = 6480
+}}
+{{eqn | l = 6^4 \times 6^1
+ | r = 1296 \times 6
+}}
+{{eqn | r = 7776
+}}
+{{eqn | l = 6^4 \times 7^1
+ | r = 1296 \times 7
+}}
+{{eqn | r = 9072
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 7^4 \times 2^1
+ | r = 2401 \times 2
+}}
+{{eqn | r = 4802
+}}
+{{eqn | l = 7^4 \times 3^1
+ | r = 2401 \times 3
+}}
+{{eqn | r = 7203
+}}
+{{eqn | l = 7^4 \times 4^1
+ | r = 7^4 \times 2^2
+}}
+{{eqn | r = 2401 \times 4
+}}
+{{eqn | r = 9604
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 8^4 \times 2^1
+ | r = 4096 \times 2
+}}
+{{eqn | r = 8192
+}}
+{{end-eqn}}
+So none of the above $4$-[[Definition:Digit|digit]] [[Definition:Integer Power|powers]] multiplied by any of these single-digit powers satisfies the condition.
+We have that $\sqbrk {abcd}$ cannot end in $0$, as that can also be ruled out by inspection of the $4$-[[Definition:Digit|digit]] [[Definition:Integer Power|powers]].
+Similarly, because $a \ne 1$, $\sqbrk {abcd} > 2000$.
+Also, it cannot be the case that both $a^b$ and $c^d$ are smaller than $44$, as $44^2 < 2000$.
+{{finish}}
+{{Namedfor|Henry Ernest Dudeney|cat = Dudeney}}
+\end{proof}<|endoftext|>
+\section{2601 as Sum of 3 Squares in 12 Different Ways}
+Tags: Sums of Squares, 2601
+
+\begin{theorem}
+$2601$ can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Square Number|squares]] in $12$ different ways.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 2601
+ | r = 51^2
+ | c =
+}}
+{{eqn | r = 1^2 + 10^2 + 50^2
+ | c =
+}}
+{{eqn | r = 2^2 + 14^2 + 49^2
+ | c =
+}}
+{{eqn | r = 10^2 + 10^2 + 49^2
+ | c =
+}}
+{{eqn | r = 14^2 + 14^2 + 47^2
+ | c =
+}}
+{{eqn | r = 1^2 + 22^2 + 46^2
+ | c =
+}}
+{{eqn | r = 14^2 + 17^2 + 46^2
+ | c =
+}}
+{{eqn | r = 1^2 + 34^2 + 38^2
+ | c =
+}}
+{{eqn | r = 14^2 + 31^2 + 38^2
+ | c =
+}}
+{{eqn | r = 3^2 + 36^2 + 36^2
+ | c =
+}}
+{{eqn | r = 24^2 + 27^2 + 36^2
+ | c =
+}}
+{{eqn | r = 17^2 + 34^2 + 34^2
+ | c =
+}}
+{{eqn | r = 22^2 + 31^2 + 34^2
+ | c =
+}}
+{{end-eqn}}
+That there are no more can be determined by exhaustion.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Reversal of Number Multiplied by 11}
+Tags: Reversals, 11, Reversal of Number Multiplied by 11
+
+\begin{theorem}
+Let $n \in \N$ be a [[Definition:Natural Number|number]] for which, when written in [[Definition:Decimal Notation|decimal notation]], no two adjacent [[Definition:Digit|digits]] [[Definition:Integer Addition|total]] to more than $9$.
+Let $n'$ denote the [[Definition:Reversal|reversal]] of $n$.
+Then $n \times 11$ is the [[Definition:Reversal|reversal]] of $n' \times 11$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Trivial but tedious, and it does not interest me.}}
+\end{proof}<|endoftext|>
+\section{Reduction Formula for Integral of Power of Tangent}
+Tags: Primitives involving Tangent Function
+
+\begin{theorem}
+For all $n \in \Z_{> 1}$:
+
+:$\displaystyle \int \map {\tan^n} x \rd x = \frac {\map {\tan^{n - 1} } x} {n - 1} - \int \map {\tan^{n - 2} } x \rd x$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \map {\tan^n} x \rd x
+ | r = \int \map {\tan^{n - 2} } x \, \map {\tan^2} x \rd x
+}}
+{{eqn | r = \int \map {\tan^{n - 2} } x \paren {\map {\sec^2} x - 1} \rd x
+ | c = [[Sum of Squares of Sine and Cosine]]
+}}
+{{eqn | r = \int \map {\tan^{n - 2} } x \, \map {\sec^2} x \rd x - \int \map {\tan^{n - 2} } x \rd x
+ | c = [[Linear Combination of Integrals]]
+}}
+{{end-eqn}}
+Let:
+{{begin-eqn}}
+{{eqn | l = t
+ | r = \map \tan x
+}}
+{{eqn | ll= \leadsto
+ | l = \frac {\d t} {\d x}
+ | r = \map {\sec^2} x
+ | c = [[Derivative of Tangent Function]]
+}}
+{{end-eqn}}
+Then:
+
+{{begin-eqn}}
+{{eqn | l = \int \map {\tan^{n - 2} } x \, \map {\sec^2} x \rd x - \int \map {\tan^{n - 2} } x \rd x
+ | r = \int t^{n - 2} \rd t - \int \map {\tan^{n - 2} } x \rd x
+ | c = [[Integration by Substitution]]
+}}
+{{eqn | r = \frac {t^{n - 1} } {n - 1} - \int \map {\tan^{n - 2} } x \rd x
+ | c = [[Primitive of Power]]
+}}
+{{eqn | r = \frac {\map {\tan^{n - 1} } x} {n - 1} - \int \map {\tan^{n - 2} } x \rd x
+ | c = substituting back $t \to \map \tan x$
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers Appearing 8 Times in Pascal's Triangle}
+Tags: Pascal's Triangle
+
+\begin{theorem}
+The number $3003$ is the smallest [[Definition:Positive Integer|integer]] to appear $8$ times in [[Definition:Pascal's Triangle|Pascal's triangle]].
+No other number below $2^{23}$ appears as often.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 3003
+ | m = \frac {3003!} {3002! \times 1!}
+ | mo= =
+ | r = \dbinom {3003} 1
+ | c =
+}}
+{{eqn | m = \frac {78!} {76! \times 2!}
+ | mo= =
+ | r = \dbinom {78} 2
+ | c =
+}}
+{{eqn | m = \frac {15!} {10! \times 5!}
+ | mo= =
+ | r = \dbinom {15} 5
+ | c =
+}}
+{{eqn | m = \frac {14!} {8! \times 6!}
+ | mo= =
+ | r = \dbinom {14} 6
+ | c =
+}}
+{{eqn | m = \frac {14!} {6! \times 8!}
+ | mo= =
+ | r = \dbinom {14} 8
+ | c =
+}}
+{{eqn | m = \frac {15!} {5! \times 10!}
+ | mo= =
+ | r = \dbinom {15} {10}
+ | c =
+}}
+{{eqn | m = \frac {78!} {2! \times 76!}
+ | mo= =
+ | r = \dbinom {78} {76}
+ | c =
+}}
+{{eqn | m = \frac {3003!} {1! \times 3002!}
+ | mo= =
+ | r = \dbinom {3003} {3002}
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that there are no other occurrences of $3003$, and that there are no other numbers less than $2^23$ with this property.}}
+\end{proof}<|endoftext|>
+\section{Squares of 3...34}
+Tags: Recreational Mathematics
+
+\begin{theorem}
+The following pattern holds:
+{{begin-eqn}}
+{{eqn | l = 4^2
+ | r = 16
+}}
+{{eqn | l = 34^2
+ | r = 1156
+}}
+{{eqn | l = 334^2
+ | r = 111 \, 556
+}}
+{{eqn | l = 3334^2
+ | r = 11 \, 115 \, 556
+}}
+{{eqn | l = 33334^2
+ | r = 1 \, 111 \, 155\, 556
+}}
+{{end-eqn}}
+and so on.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Simple but tedious.}}
+\end{proof}<|endoftext|>
+\section{3367 Multiplied by 2-Digit Number}
+Tags: Recreational Mathematics, 3367
+
+\begin{theorem}
+In order to [[Definition:Integer Multiplication|multiply]] $3367$ by a $2$-[[Definition:Digit|digit]] [[Definition:Integer|integer]] $\sqbrk {xy}$:
+:[[Definition:Integer Division|divide]] the $6$-[[Definition:Digit|digit]] [[Definition:Integer|integer]] $\sqbrk {xyxyxy}$ by $3$.
+\end{theorem}
+
+\begin{proof}
+We have that:
+:$10101 = 3367 \times 3$
+Then:
+:$10101 \times \sqbrk {xy} = \sqbrk {xyxyxy}$
+The result follows.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integer as Sum of 2 Cubes in 3 Ways}
+Tags: Sums of Cubes
+
+\begin{theorem}
+$4104$ is the smallest [[Definition:Natural Number|natural number]] which can be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Cube Number|cubes]] in $3$ different ways:
+{{begin-eqn}}
+{{eqn | l = 4104
+ | r = 16^3 + 2^3
+ | c =
+}}
+{{eqn | r = 15^3 + 9^3
+ | c =
+}}
+{{eqn | r = \paren {-12}^3 + 18^3
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = 16^3 + 2^3
+ | r = 4096 + 8
+ | c =
+}}
+{{eqn | r = 4104
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 15^3 + 9^3
+ | r = 3375 + 729
+ | c =
+}}
+{{eqn | r = 4104
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \paren {-12}^3 + 18^3
+ | r = -1728 + 5832
+ | c =
+}}
+{{eqn | r = 4104
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown there are no smaller}}
+\end{proof}<|endoftext|>
+\section{Composite Fibonacci Numbers with Prime Index}
+Tags: Fibonacci Numbers, Prime Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Composite Number|composite]] [[Definition:Fibonacci Number|Fibonacci numbers]] with a [[Definition:Prime Number|prime index]] begins:
+:$4181, 1 \, 346 \, 269, 24 \, 157 \, 817, 165 \, 580 \, 141, \ldots$
+{{OEIS|A050937}}
+The corresponding [[Definition:Integer Sequence|sequence]] of [[Definition:Prime Number|prime indices]] begins:
+:$19, 31, 37, 41, 53, 59, 61, 67, 71, 73, 79, \ldots$
+{{OEIS|A038672}}
+\end{theorem}
+
+\begin{proof}
+By observation:
+{{begin-eqn}}
+{{eqn | l = F_{19}
+ | r = 4181
+ | c =
+}}
+{{eqn | r = 37 \times 113
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = F_{31}
+ | r = 1 \, 346 \, 269
+ | c =
+}}
+{{eqn | r = 557 \times 2417
+ | c =
+}}
+{{end-eqn}}
+{{finish}}
+\end{proof}<|endoftext|>
+\section{Closed Subset is Upper in Lower Topology}
+Tags: Topological Order Theory
+
+\begin{theorem}
+Let $T = \left({S, \preceq, \tau}\right)$ be a [[Definition:Transitive Relation|transitive]] [[Definition:Relational Structure with Topology|relational structure]] with [[Definition:Lower Topology|lower topology]].
+Let $A \subseteq S$ such that
+:$A$ is [[Definition:Closed Set (Topology)|closed]].
+Then $A$ is [[Definition:Upper Set|upper]].
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Closed Set (Topology)|closed set]]:
+:$S \setminus A$ is [[Definition:Open Set (Topology)|open]].
+By [[Open Subset is Lower in Lower Topology]]:
+:$S \setminus A$ is [[Definition:Lower Set|lower]].
+Thus by [[Complement of Lower Set is Upper Set]] and [[Relative Complement of Relative Complement]]:
+:$A$ is [[Definition:Upper Set|upper]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Square Pyramidal Number also Square}
+Tags: Pyramidal Numbers, Square Numbers, 4900
+
+\begin{theorem}
+$4900$ is the only [[Definition:Square Pyramidal Number|square pyramidal number]] which is also [[Definition:Square Number|square]]:
+:$4900 = 70^2 = \displaystyle \sum_{k \mathop = 1}^{24} k^2 = \dfrac {24 \paren {24 + 1} \paren {2 \times 24 + 1} } 6$
+\end{theorem}<|endoftext|>
+\section{Mapping Preserves Non-Empty Infima implies Mapping is Continuous in Lower Topological Lattice}
+Tags: Topological Order Theory
+
+\begin{theorem}
+Let $T = \left({S, \preceq, \tau}\right)$ and $Q = \left({X, \preceq', \tau'}\right)$ be [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Lower Topology|lower topologies]].
+Let $f: S \to X$ be a [[Definition:Mapping|mapping]] such that
+:for all [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subsets]] $Y$ of $S$: $f$ [[Definition:Mapping Preserves Infimum/Subset|preserves the infimum]] of $Y$.
+Then $f$ is [[Definition:Continuous (Topology)|continuous mapping]].
+\end{theorem}
+
+\begin{proof}
+Define $B = \left\{ {\complement_X\left({x^{\succeq'}}\right): x \in X}\right\}$
+We will prove that
+:$\forall A \in B: f^{-1}\left[{\complement_X\left({A}\right)}\right]$ is [[Definition:Closed Set (Topology)|closed]].
+Let $A \in B$.
+By definition of $B$:
+:$\exists x \in X: A = \complement_X\left({x^{\succeq'} }\right)$
+By [[Relative Complement of Relative Complement]]:
+:$\complement_X\left({A}\right) = x^{\succeq'}$
+By [[Infimum of Upper Closure of Element]]:
+:$\inf \left({\complement_X\left({A}\right)}\right) = x$
+Suppose that the case: $f^{-1}\left[{\complement_X\left({A}\right)}\right] = \varnothing$ holds.
+Thus by [[Empty Set is Closed in Topological Space]]:
+:$f^{-1}\left[{\complement_X\left({A}\right)}\right]$ is [[Definition:Closed Set (Topology)|closed]].
+{{qed|lemma}}
+Suppose that the case: $f^{-1}\left[{\complement_X\left({A}\right)}\right] \ne \varnothing$ holds.
+By assumption:
+:$f$ [[Definition:Mapping Preserves Infimum/Subset|preserves the infimum]] of $f^{-1}\left[{\complement_X\left({A}\right)}\right]$
+By definitions of [[Definition:Mapping Preserves Infimum/Subset|mapping preserves the infimum]] and [[Definition:Complete Lattice|complete lattice]]:
+:$f\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right) = \inf\left({f\left[{f^{-1}\left[{\complement_X\left({A}\right)}\right]}\right]}\right)$
+By [[Image of Preimage under Mapping]]:
+:$f\left[{f^{-1}\left[{\complement_X\left({A}\right)}\right]}\right] \subseteq x^{\succeq'}$
+By [[Infimum of Subset]] and definition of [[Definition:Complete Lattice|complete lattice]]:
+:$x \preceq' f\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right)$
+We will prove that
+:$f^{-1}\left[{\complement_X\left({A}\right)}\right] = {\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right)}^\succeq$
+Let $a \in f^{-1}\left[{\complement_X\left({A}\right)}\right]$
+By definitions of [[Definition:Infimum of Set|infimum]] and [[Definition:Lower Bound of Set|lower bound]]:
+:$\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right) \preceq a$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$a \in {\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right)}^\succeq$
+{{qed|lemma}}
+Let $a \in {\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right)}^\succeq$
+By assumption:
+:$f$ [[Definition:Mapping Preserves Infimum/Subset|preserves the infimum]] of $\left\{ {\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right), a}\right\}$
+By definitions of [[Definition:Mapping Preserves Infimum/Subset|mapping preserves the infimum]] and [[Definition:Complete Lattice| complete lattice]]:
+:$f\left({\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right) \wedge a}\right) = f\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right) \wedge f\left({a}\right)$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right) \preceq a$
+By [[Meet Precedes Operands]]:
+:$f\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right) = f\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right) \wedge f\left({a}\right)$
+By [[Preceding iff Meet equals Less Operand]]:
+:$f\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right) \preceq' f\left({a}\right)$
+By definition of [[Definition:Transitivity|transitivity]]:
+:$x \preceq' f\left({a}\right)$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$f\left({a}\right) \in x^{\succeq'}$
+Thus by definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
+:$a \in f^{-1}\left[{\complement_X\left({A}\right)}\right]$
+{{qed|lemma}}
+Thus by [[Complement of Upper Closure of Element is Open in Lower Topology]]:
+:$f^{-1}\left[{\complement_X\left({A}\right)}\right]$ is [[Definition:Closed Set (Topology)|closed]].
+{{qed|lemma}}
+We will prove that
+:$\forall A \in B: f^{-1}\left[{A}\right] \in \tau$
+Let $A \in B$.
+Then by previous:
+:$f^{-1}\left[{\complement_X\left({A}\right)}\right]$ is [[Definition:Closed Set (Topology)|closed]].
+By [[Complement of Preimage equals Preimage of Complement]]
+:$f^{-1}\left[{\complement_X\left({A}\right)}\right] = \complement_S\left({f^{-1}\left[{A}\right]}\right)$
+Thus by definition of [[Definition:Closed Set (Topology)|closed set]]
+:$f^{-1}\left[{A}\right] \in \tau$
+{{qed|lemma}}
+Thus [[Continuity Test using Sub-Basis]]:
+:$f$ is [[Definition:Continuous (Topology)|continuous mapping]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{5040 is Product of Consecutive Numbers in Two Ways}
+Tags: Factorials, 5040
+
+\begin{theorem}
+:$5040 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8 \times 7$
+\end{theorem}
+
+\begin{proof}
+Follows from [[Factorial as Product of Two Factorials]]:
+:$10! = 6! \times 7!$
+and so:
+:$\dfrac {10!} {6!} = 10 \times 9 \times 8 \times 7 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Products of Consecutive Integers in 2 Ways}
+Tags: Factorials
+
+\begin{theorem}
+The following [[Definition:Integer|integers]] are the [[Definition:Integer Multiplication|product]] of consecutive [[Definition:Integer|integers]] in $2$ ways:
+:$-720, 720, 5040$
+\end{theorem}
+
+\begin{proof}
+From [[720 is Product of Consecutive Numbers in Two Ways]]:
+:$720 = 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8$
+From [[5040 is Product of Consecutive Numbers in Two Ways]]:
+:$5040 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8 \times 7$
+Then:
+:$-720 = \left({-6}\right) \left({-5}\right) \left({-4}\right) \left({-3}\right) \left({-2}\right) = \left({-10}\right) \left({-9}\right) \left({-8}\right)$
+The same trick cannot be used for $-5040$ because there are four [[Definition:Divisor of Integer|divisors]] in $10 \times 9 \times 8 \times 7$, and negating them makes the [[Definition:Integer Multiplication|product]] [[Definition:Positive Integer|positive]].
+Hence the result.
+{{qed}}
+[[Category:Factorials]]
+htuoyjqsfoqjoxcauiwslkjcxp6eitx
+\end{proof}<|endoftext|>
+\section{8 Mutually Non-Attacking Rooks on Chessboard}
+Tags: 5282, Recreational Mathematics
+
+\begin{theorem}
+On a standard [[Definition:Chessboard|chessboard]], it is possible to arrange a maximum of $8$ [[Definition:Chess Rook|rooks]] so that no [[Definition:Chess Rook|rook]] is attacking any other [[Definition:Chess Rook|rook]].
+There are $5282$ such arrangements, up to rotation and reflection.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|No doubt we will eventually progress to chess problems of various styles.}}
+\end{proof}<|endoftext|>
+\section{Triangular Lucas Numbers}
+Tags: Lucas Numbers, Triangular Numbers
+
+\begin{theorem}
+The only [[Definition:Lucas Number|Lucas numbers]] which are also [[Definition:Triangular Number|triangular]] are:
+:$1, 3, 5778$
+{{OEIS|A248506}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1
+ | r = \dfrac {1 \times 2} 2
+}}
+{{eqn | l = 3
+ | r = \dfrac {2 \times 3} 2
+ | rr= = 2 + 1
+}}
+{{eqn | l = 5778
+ | r = \dfrac {107 \times 108} 2
+ | rr= = 2207 + 3571
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that these are the only ones.}}
+\end{proof}<|endoftext|>
+\section{Sum of Two Rational 4th Powers but not Two Integer 4th Powers}
+Tags: Fourth Powers, 5906
+
+\begin{theorem}
+$5906$ is the smallest [[Definition:Integer|integer]] which can be expressed as the [[Definition:Rational Addition|sum]] of two [[Definition:Rational Power|rational $4$th powers]], but not two [[Definition:Integer Power|integer $4$th powers]].
+\end{theorem}
+
+\begin{proof}
+:$5906 = \paren {\dfrac {149} {17} }^4 + \paren {\dfrac {25} {17} }^4$
+Suppose $5906$ is a [[Definition:Integer Addition|sum]] of two [[Definition:Integer Power|integer $4$th powers]].
+We have:
+:$9^4 = 6561 > 5906$
+which shows that no [[Definition:Integer Power|$4$th power]] greater than $8^4$ is in the [[Definition:Integer Addition|sum]].
+:$7^4 + 7^4 = 4802 < 5906$
+which shows that some [[Definition:Integer Power|$4$th power]] greater than $7^4$ is in the [[Definition:Integer Addition|sum]].
+So the [[Definition:Integer Addition|sum]] must contain $8^4$.
+We have:
+:$5906 - 8^4 = 1810$
+but $1810$ is not an [[Definition:Integer Power|integer $4$th power]].
+Therefore $5906$ is not a [[Definition:Integer Addition|sum]] of two [[Definition:Integer Power|integer $4$th powers]].
+{{finish|It remains to show that it is the smallest such. It is, acccording to A NEW CHARACTERIZATION OF THE INTEGER 5906 by A. Bremner and P. Morton}}
+\end{proof}<|endoftext|>
+\section{Subspace of Subspace is Subspace}
+Tags: Topological Subspaces
+
+\begin{theorem}
+Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
+Let $H \subseteq S$ and $\tau_H$ be the [[Definition:Subspace Topology|subspace topology]] on $H$.
+Let $K\subseteq H$.
+Then the [[Definition:Subspace Topology|subspace topology]] on $K$ induced by $\tau$ equals the [[Definition:Subspace Topology|subspace topology]] on $K$ induced by $\tau_H$.
+\end{theorem}
+
+\begin{proof}
+Let $\tau_K$ be the [[Definition:Subspace Topology|subspace topology]] on $K$ induced by $\tau$.
+Let $\tau’_K$ be the [[Definition:Subspace Topology|subspace topology]] on $K$ induced by $\tau_H$.
+Then
+{{begin-eqn}}
+{{eqn | l = V \in \tau’_K
+ | o = \leadstoandfrom
+ | r = \exists U’ \in \tau_H : V = U’ \cap K
+ | c = {{Defof|Subspace Topology}} $\tau’_K$
+}}
+{{eqn | o = \leadstoandfrom
+ | r = \exists U \in \tau : V = \paren {U \cap H} \cap K
+ | c = {{Defof|Subspace Topology}} $\tau_H$
+}}
+{{eqn | o = \leadstoandfrom
+ | r = \exists U \in \tau : V = U \cap \paren {H \cap K}
+ | c = [[Intersection is Associative]]
+}}
+{{eqn | o = \leadstoandfrom
+ | r = \exists U \in \tau : V = U \cap K
+ | c = [[Intersection with Subset is Subset]]
+}}
+{{eqn | o = \leadstoandfrom
+ | r = V \in \tau_K
+ | c = {{Defof|Subspace Topology}} $\tau_K$
+}}
+{{end-eqn}}
+[[Category:Topological Subspaces]]
+mrfeh7eskyccopbrjwm7cr98zmshv6h
+\end{proof}<|endoftext|>
+\section{Positive Integer Sum of 3 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 $3$ [[Definition:Fourth Power|fourth powers]] in $2$ different ways is $6578$:
+{{begin-eqn}}
+{{eqn | l = 1^4 + 2^4 + 9^4
+ | r = 1 + 16 + 6561
+ | c =
+}}
+{{eqn | r = 6578
+ | c =
+}}
+{{eqn | r = 81 + 2401 + 4096
+ | c =
+}}
+{{eqn | r = 3^4 + 7^4 + 8^4
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+The fact that this is the smallest can be demonstrated by calculation.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Mapping Preserves Infima implies Mapping is Continuous in Lower Topological Lattice}
+Tags: Topological Order Theory
+
+\begin{theorem}
+Let $T = \struct {S, \preceq, \tau}$ and $Q = \struct {X, \preceq', \tau'}$ be [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Lower Topology|lower topologies]].
+Let $f: S \to X$ be a [[Definition:Mapping|mapping]] such that
+:$f$ [[Definition:Mapping Preserves Infimum/All|preserves all infima]].
+Then $f$ is [[Definition:Continuous (Topology)|continuous mapping]].
+\end{theorem}
+
+\begin{proof}
+By assumption:
+:for all [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subsets]] $Y$ of $S$: $f$ [[Definition:Mapping Preserves Infimum/Subset|preserves the infimum]] of $Y$.
+Thus by [[Mapping Preserves Non-Empty Infima implies Mapping is Continuous in Lower Topological Lattice]]:
+:$f$ is [[Definition:Continuous (Topology)|continuous mapping]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Compact in Subspace is Compact in Topological Space}
+Tags: Compact Spaces, Topological Subspaces
+
+\begin{theorem}
+Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
+Let $K \subseteq S$ be a [[Definition:Subset|subset]].
+Let $\tau_K$ be the [[Definition:Subspace Topology|subspace topology]] on $K$.
+Let $T' = \struct {K, \tau_K}$ be the [[Definition:Topological Subspace|topological subspace]] of $T$ determined by $K$.
+Let $H \subseteq K$ be [[Definition:Compact Subspace|compact]] in $T'$.
+Then $H$ is [[Definition:Compact Subspace|compact]] in $T$.
+\end{theorem}
+
+\begin{proof}
+Suppose that $H$ is [[Definition:Compact Topological Subspace|compact]] in $T'$.
+Let $\set {W_i}_{i \mathop \in J}$ be an [[Definition:Open Cover|open cover]] of $H$ in $T$.
+Then $\displaystyle H \subseteq \bigcup_{i \mathop \in J} W_i$.
+Then:
+{{begin-eqn}}
+{{eqn | l = H
+ | r = H \cap K
+ | c = [[Intersection with Subset is Subset]]: from $H \subseteq K$
+}}
+{{eqn | o = \subseteq
+ | r = \paren {\bigcup_{i \mathop \in J} W_i} \cap K
+ | c = [[Set Intersection Preserves Subsets/Corollary|Set Intersection Preserves Subsets: Corollary]]: from $\displaystyle H \subseteq \bigcup_{i \in J} W_i$
+}}
+{{eqn | r = \bigcup_{i \mathop \in J} \paren {W_i \cap K}
+ | c = [[Intersection Distributes over Union/Family of Sets|Intersection Distributes over Union: Family of Sets]]
+}}
+{{end-eqn}}
+Since $W_i \cap K \in \tau_K$, $\set {W_i \cap K}_{i \mathop \in J}$ is an [[Definition:Open Cover|open cover]] of $H$ in $T'$.
+Since $H$ is [[Definition:Compact Topological Subspace|compact]] in $T'$, $\set {W_i \cap K}_{i \mathop \in J}$ has some [[Definition:Finite Subcover|finite subcover]] $\set {W_i \cap K}_{i \mathop = 1}^r$.
+Therefore:
+{{begin-eqn}}
+{{eqn | l = H
+ | o = \subseteq
+ | r = \set {W_i \cap K}_{i \mathop = 1}^r
+ | c = {{Defof|Finite Subcover}}
+}}
+{{eqn | o = \subseteq
+ | r = \paren {\bigcup_{i \mathop = 1}^r W_i} \cap K
+ | c = [[Intersection Distributes over Union/Family of Sets|Intersection Distributes over Union: Family of Sets]]
+}}
+{{eqn | r = \bigcup_{i \mathop = 1}^r W_i
+ | c = [[Intersection is Subset]]
+}}
+{{end-eqn}}
+So $\set {W_i}_{i \mathop = 1}^r$ is an [[Definition:Open Cover|open cover]] of $H$ in $T$, which is a [[Definition:Finite Subcover|finite subcover]].
+As $\set {W_i}_{i \mathop \in J}$ is arbitrary:
+Any [[Definition:Open Cover|open cover]] of $H$ has a [[Definition:Finite Subcover|finite subcover]] in $T$.
+So $H$ is [[Definition:Compact Subspace|compact]] in $T$.
+{{qed}}
+[[Category:Compact Spaces]]
+[[Category:Topological Subspaces]]
+n15rln3t7hfy09w6nb2fy174c5pehnf
+\end{proof}<|endoftext|>
+\section{If Infimum of Filtered Subset belongs to Element of Sub-Basis then Subset and Element Intersect implies Infimum of Subset belongs to Closure of Subset}
+Tags: Topological Order Theory
+
+\begin{theorem}
+Let $T = \left({S, \preceq, \tau}\right)$ be a [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Lower Topology|lower topology]].
+Let $B$ be an [[Definition:Analytic Sub-Basis|analytic sub-basis]] of $T$.
+Let $F$ be a [[Definition:Filtered Subset|filtered subset]] of $S$ such that
+:$\forall A \in B: \inf F \in A \implies F \cap A \ne \varnothing$
+Then $\inf F \in F^-$
+where $F^-$ denotes the [[Definition:Closure (Topology)|topological closure]] of $F$.
+\end{theorem}
+
+\begin{proof}
+We will prove that
+:$\forall A \in B, x \in F \cap A, y \in F: y \preceq x \implies y \in A$
+Let $A \in B$, $x \in F \cap A$, $y \in F$.
+By definition of [[Definition:Analytic Sub-Basis|sub-basis]]:
+:$A$ is [[Definition:Open Set (Topology)|open]].
+By [[Open Subset is Lower in Lower Topology]]:
+:$A$ is [[Definition:Lower Set|lower]].
+By definition of [[Definition:Set Intersection|intersection]]:
+:$x \in A$.
+Thus by definition of [[Definition:Lower Set|lower set]]:
+:$y \preceq x \implies y \in A$.
+{{qed|lemma}}
+Define $H := \left\{ {\bigcap G: G \subseteq B, G \text{ is finite} }\right\}$
+By definitions of [[Definition:Analytic Sub-Basis|sub-basis]] and [[Definition:Analytic Basis|basis]]:
+:$H$ is [[Definition:Analytic Basis|basis]] of $T$.
+We will prove that
+:$\forall A \in H: \inf F \in A \implies F \cap A \ne \varnothing$
+Let $A \in H$ such that
+:$\inf F \in A$
+By definition of $H$:
+:$\exists G \subseteq B: A = \bigcap G \land G$ is [[Definition:Finite Set|finite]].
+By definition of [[Definition:Intersection of Set of Sets|intersection]]:
+:$\forall C \in G: \inf F \in C \in B$
+By assumption:
+:$\forall C \in G: F \cap C \ne \varnothing$
+By definition of [[Definition:Non-Empty Set|non-empty set]]:
+:$\forall C \in G: \exists x: x \in F \cap C$
+By [[Axiom:Axiom of Choice|Axiom of Choice]]:
+:$\exists f:G \to F: \forall C \in G: f\left({C}\right) \in F \cap C$
+By [[Image of Mapping from Finite Set is Finite]]:
+:$f\left[{G}\right]$ is [[Definition:Finite Set|finite]].
+By [[Filtered iff Finite Subsets have Lower Bounds]]:
+:$\exists h \in F: \forall x \in f\left[{G}\right]: h \preceq x$
+Then
+:$\forall C \in G: h \in C$
+By definition of [[Definition:Intersection of Set of Sets|intersection]]:
+:$h \in \bigcap G$
+Thus by definitions of [[Definition:Set Intersection|intersection]] and [[Definition:Non-Empty Set|non-empty set]]:
+:$F \cap A \ne \varnothing$
+{{qed|lemma}}
+Thus by [[Characterization of Closure by Basis]]:
+:$\inf F \in F^-$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Square of Repdigit Number consisting of Instances of 6}
+Tags: Square Numbers, Repdigit Numbers
+
+\begin{theorem}
+The following pattern holds:
+{{begin-eqn}}
+{{eqn | l = 6^2
+ | r = 36
+}}
+{{eqn | l = 3 + 6
+ | r = 9
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 66^2
+ | r = 4356
+}}
+{{eqn | l = 43 + 56
+ | r = 99
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 666^2
+ | r = 443 \, 556
+}}
+{{eqn | l = 443 + 556
+ | r = 999
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 6666^2
+ | r = 44 \, 435 \, 556
+}}
+{{eqn | l = 4443 + 5556
+ | r = 9999
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 66666^2
+ | r = 4 \, 444 \, 355\, 556
+}}
+{{eqn | l = 44 \, 443 + 55 \, 556
+ | r = 99 \, 999
+}}
+{{end-eqn}}
+and so on.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Simple but tedious.}}
+\end{proof}<|endoftext|>
+\section{Square of Repdigit Number consisting of Instances of 3}
+Tags: Square Numbers, Repdigit Numbers
+
+\begin{theorem}
+The following pattern holds:
+{{begin-eqn}}
+{{eqn | l = 3^2
+ | r = 09
+}}
+{{eqn | l = 0 + 9
+ | r = 9
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 33^2
+ | r = 1089
+}}
+{{eqn | l = 10 + 89
+ | r = 99
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 333^2
+ | r = 110 \, 889
+}}
+{{eqn | l = 110 + 889
+ | r = 999
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 3333^2
+ | r = 11 \, 108 \, 889
+}}
+{{eqn | l = 1110 + 8889
+ | r = 9999
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 33 \, 333^2
+ | r = 1 \, 111 \, 088 \, 889
+}}
+{{eqn | l = 11 \, 110 + 88 \, 889
+ | r = 99 \, 999
+}}
+{{end-eqn}}
+and so on.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Simple but tedious.}}
+\end{proof}<|endoftext|>
+\section{Product with Repdigit can be Split into Parts which Add to Repdigit}
+Tags: Repdigit Numbers, Product with Repdigit can be Split into Parts which Add to Repdigit
+
+\begin{theorem}
+Let $n$ be a [[Definition:Positive Integer|positive integer]] with $d_1$ [[Definition:Digit|digits]].
+Let $m$ be a [[Definition:Repdigit Number|repdigit number]] with $d_2$ [[Definition:Digit|digits]] such that $d_2 > d_1$.
+Let $r$ consist of the result when the rightmost $d_2$ [[Definition:Digit|digits]] of $m n$ is cut off and added to the remaining left hand portion.
+Then $r$ is a [[Definition:Repdigit Number|repdigit number]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Straightforward but tedious.}}
+\end{proof}<|endoftext|>
+\section{Sum of Reciprocals of Squares of Odd Integers as Double Integral}
+Tags: Riemann Zeta Function, Sum of Reciprocals of Squares of Odd Integers as Double Integral
+
+\begin{theorem}
+:$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2} = \int_0^1 \int_0^1 \frac 1 {1 - x^2 y^2} \rd x \rd y$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2}
+ | r = \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1} \paren {2 n - 1} }
+}}
+{{eqn | r = \sum_{n \mathop = 1}^\infty \int_0^1 x^{2 n - 2} \rd x \int_0^1 y^{2 n - 2} \rd y
+}}
+{{eqn | r = \sum_{n \mathop = 1}^\infty \int_0^1 \int_0^1 \paren {x^2 y^2}^{n - 1} \rd x \rd y
+}}
+{{eqn | r = \int_0^1 \int_0^1 \sum_{n \mathop = 0}^\infty \paren {x^2 y^2}^n \rd x \rd y
+ | c = [[Fubini's Theorem]]
+}}
+{{eqn | r = \int_0^1 \int_0^1 \frac 1 {1 - x^2 y^2} \rd x \rd y
+ | c = [[Sum of Infinite Geometric Sequence]]
+}}
+{{end-eqn}}
+{{qed}}
+{{MissingLinks|The jump from the $1$st to $2$nd lines needs to be justified. So does the $2$nd to the $3$rd, now I come to think about it.}}
+\end{proof}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int_0^1 \int_0^1 \frac 1 {1 - x^2 y^2} \rd x \rd y
+ | r = \int_0^1 \int_0^1 \paren {1 + x^2 y^2 + x^4 y^4 + x^6 y^6 + \cdots} \rd x \rd y
+ | c = [[Sum of Infinite Geometric Sequence]]
+}}
+{{eqn | r = \int_0^1 \intlimits {y + \frac {x^2y^3} 3 + \frac {x^4y^5} 5 + \frac {x^6y^7} 7 + \cdots } {y \mathop = 0} {y \mathop = 1} \rd x
+}}
+{{eqn | r = \int_0^1 \paren {1 + \frac 1 3 x^2 + \frac 1 5 x^4 + \frac 1 7 x^6 + \cdots} \rd x
+}}
+{{eqn | r = \intlimits {x + \frac {x^3} 9 + \frac {x^5} {25} + \frac {x^7} {49} + \cdots } {x \mathop = 0} {x \mathop = 1}
+}}
+{{eqn | r = \paren {1 + \frac 1 9 + \frac 1 {25} + \frac 1 {49} + \cdots }
+}}
+{{eqn | r = \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2}
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Three Tri-Automorphic Numbers for each Number of Digits}
+Tags: Tri-Automorphic Numbers
+
+\begin{theorem}
+Let $d \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
+Then there exist exactly $3$ [[Definition:Tri-Automorphic Number|tri-automorphic numbers]] with exactly $d$ [[Definition:Digit|digits]].
+These [[Definition:Tri-Automorphic Number|tri-automorphic numbers]] all end in $2$, $5$ or $7$.
+\end{theorem}
+
+\begin{proof}
+Let $n$ be a [[Definition:Tri-Automorphic Number|tri-automorphic number]] with $d$ [[Definition:Digit|digits]].
+Let $n = 10 a + b$.
+Then:
+:$3 n^2 = 300a^2 + 60 a b + 3 b^2$
+As $n$ is [[Definition:Tri-Automorphic Number|tri-automorphic]], we have:
+:$(1): \quad 300 a^2 + 60 a b + 3 b^2 = 1000 z + 100 y + 10 a + b$
+and:
+:$(2): \quad 3 b^2 - b = 10 x$
+where $x$ is an [[Definition:Integer|integer]].
+This condition is only satisfied by $b = 2$, $b = 5$, or $b = 7$
+{{ProofWanted|Guess: Try proving for $n {{=}} 10 a + b$ and then by induction.}}
+Substituting $b = 2$ in equation $(1)$:
+:$a = 9$
+Substituting $b = 5$ in equation $(1)$:
+:$a = 7$
+Substituting $b = 7$ in equation $(1)$:
+:$a = 6$
+{{qed}}
+
+\end{proof}<|endoftext|>
+\section{Fourier Series/x squared over Minus Pi to Pi}
+Tags: Examples of Fourier Series
+
+\begin{theorem}
+:$\displaystyle x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$
+\end{theorem}
+
+\begin{proof}
+From [[Even Power is Even Function]], $x^2$ is an [[Definition:Even Function|even function]].
+By [[Fourier Series for Even Function over Symmetric Range]], we have:
+:$\displaystyle x^2 \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$
+where:
+{{begin-eqn}}
+{{eqn | l = a_n
+ | r = \frac 2 \pi \int_0^\pi x^2 \map \cos {n x} \rd x
+}}
+{{eqn | ll= \leadsto
+ | l = \frac 1 2 a_0
+ | r = \frac 2 {2 \pi} \int_0^\pi x^2 \rd x
+ | c = [[Cosine of Zero is One]]
+}}
+{{eqn | r = \frac 1 \pi \cdot \frac {\pi^3} 3
+ | c = [[Primitive of Power]], [[Fundamental Theorem of Calculus]]
+}}
+{{eqn | r = \frac {\pi^2} 3
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \frac 2 \pi \int_0^\pi x^2 \map \cos {n x} \rd x
+ | r = \frac 2 \pi \intlimits {\frac {2 x \cos n x} {n^2} + \paren {\frac {x^2} n - \frac 2 {n^3} } \sin n x} 0 \pi
+ | c = [[Primitive of x squared by Cosine of a x|Primitive of $x^2 \cos a x$]], [[Fundamental Theorem of Calculus]]
+}}
+{{eqn | r = \frac 2 \pi \intlimits {\frac {2 x \cos n x} {n^2} } 0 \pi
+ | c = [[Sine of Multiple of Pi]]
+}}
+{{eqn | r = \frac 2 \pi \cdot \frac {2 \pi \cos n \pi} {n^2}
+}}
+{{eqn | r = \frac {4 \cos n \pi} {n^2}
+}}
+{{eqn | r = \paren {-1}^n \frac 4 {n^2}
+ | c = [[Cosine of Multiple of Pi]]
+}}
+{{end-eqn}}
+Substituting for $a_n$ in $(1)$:
+:$\displaystyle x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$
+as required.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integer and its Double forming Pandigital Pair}
+Tags: Recreational Mathematics
+
+\begin{theorem}
+$6729$ and its double contain all the [[Definition:Digit|digits]] from $1$ to $9$ between them.
+\end{theorem}
+
+\begin{proof}
+$2 \times 6729 = 13 \, 458$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{4-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 $4$-[[Definition:Digit|digit]] [[Definition:Positive Integer|integers]] $m$ which need the largest number of iterations before reaching a [[Definition:Palindromic Number|palindromic number]] are:
+:$6999, 7998, 8997, 9996$
+all of which need $20$ iterations.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | n = 1
+ | o =
+ | m = 6999 + 9996
+ | mo= =
+ | r = 16995
+}}
+{{eqn | n = 2
+ | o =
+ | m = 16995 + 59961
+ | mo= =
+ | r = 76956
+}}
+{{eqn | n = 3
+ | o =
+ | m = 76956 + 65967
+ | mo= =
+ | r = 142923
+}}
+{{eqn | n = 4
+ | o =
+ | m = 142923 + 329241
+ | mo= =
+ | r = 472164
+}}
+{{eqn | n = 5
+ | o =
+ | m = 472164 + 461274
+ | mo= =
+ | r = 933438
+}}
+{{eqn | n = 6
+ | o =
+ | m = 933438 + 834339
+ | mo= =
+ | r = 1767777
+}}
+{{eqn | n = 7
+ | o =
+ | m = 1767777 + 7777671
+ | mo= =
+ | r = 9545448
+}}
+{{eqn | n = 8
+ | o =
+ | m = 9545448 + 8445459
+ | mo= =
+ | r = 17990907
+}}
+{{eqn | n = 9
+ | o =
+ | m = 17990907 + 70909971
+ | mo= =
+ | r = 88900878
+}}
+{{eqn | n = 10
+ | o =
+ | m = 88900878 + 87800988
+ | mo= =
+ | r = 176701866
+}}
+{{eqn | n = 11
+ | o =
+ | m = 176701866 + 668107671
+ | mo= =
+ | r = 844809537
+}}
+{{eqn | n = 12
+ | o =
+ | m = 844809537 + 735908448
+ | mo= =
+ | r = 1580717985
+}}
+{{eqn | n = 13
+ | o =
+ | m = 1580717985 + 5897170851
+ | mo= =
+ | r = 7477888836
+}}
+{{eqn | n = 14
+ | o =
+ | m = 7477888836 + 6388887747
+ | mo= =
+ | r = 13866776583
+}}
+{{eqn | n = 15
+ | o =
+ | m = 13866776583 + 38567766831
+ | mo= =
+ | r = 52434543414
+}}
+{{eqn | n = 16
+ | o =
+ | m = 52434543414 + 41434543425
+ | mo= =
+ | r = 93869086839
+}}
+{{eqn | n = 17
+ | o =
+ | m = 93869086839 + 93868096839
+ | mo= =
+ | r = 187737183678
+}}
+{{eqn | n = 18
+ | o =
+ | m = 187737183678 + 876381737781
+ | mo= =
+ | r = 1064118921459
+}}
+{{eqn | n = 19
+ | o =
+ | m = 1064118921459 + 9541298114601
+ | mo= =
+ | r = 10605417036060
+}}
+{{eqn | n = 20
+ | o =
+ | m = 10605417036060 + 06063071450601
+ | mo= =
+ | r = 16668488486661
+}}
+{{end-eqn}}
+which is [[Definition:Palindromic Number|palindromic]].
+$7998$ and its [[Definition:Reversal|reversal]] converge on the same sequence immediately:
+{{begin-eqn}}
+{{eqn | l = 7998 + 8997
+ | r = 16995
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that these are the longest.}}
+\end{proof}<|endoftext|>
+\section{Two Consecutive Integers each Product of Four Distinct Primes}
+Tags: Recreational Mathematics
+
+\begin{theorem}
+The [[Definition:Sequence|sequence]] of [[Definition:Ordered Pair|pairs]] of consecutive [[Definition:Positive Integer|positive integers]] which are each the [[Definition:Integer Multiplication|product]] of exactly $4$ [[Definition:Distinct|distinct]] [[Definition:Prime Number|prime numbers]] begins:
+:$\tuple {7314, 7315}, \tuple {8294, 8295}, \tuple {8645, 8646}, \tuple {9009, 9010}, \ldots$
+{{OEIS|A140078|order = first}}
+\end{theorem}<|endoftext|>
+\section{Fourier Series/Identity Function over Minus Pi to Pi}
+Tags: Fourier Series for Identity Function
+
+\begin{theorem}
+For $x \in \openint {-\pi} \pi$:
+:$\displaystyle x = 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin n x$
+\end{theorem}<|endoftext|>
+\section{Fourier Series/Fourth Power of x over Minus Pi to Pi}
+Tags: Examples of Fourier Series
+
+\begin{theorem}
+:$\displaystyle x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \cos n \pi \cos n x$
+\end{theorem}
+
+\begin{proof}
+Since $x^4 = \paren {-x}^4$, $x^4$ is an [[Definition:Even Function|even function]].
+By [[Fourier Series for Even Function over Symmetric Range]], the [[Definition:Fourier Series|Fourier series]] of $\map f x$ can be expressed as:
+:$x^4 \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty a_n \cos n x$
+where for all $n \in \Z_{> 0}$:
+{{begin-eqn}}
+{{eqn | l = a_n
+ | r = \dfrac 2 \pi \displaystyle \int_0^\pi x^4 \cos n x \ \d x
+}}
+{{eqn | r = \dfrac 2 \pi \paren {\intlimits {\frac {\sin n x} n x^4 + \frac {4 \cos n x} {n^2} x^3 - \frac {12 \sin n x} {n^3} x^2 - \frac {24 \cos n x} {n^4} x + \frac {24 \sin n x} {n^5} } {x \mathop = 0} {x \mathop = \pi} }
+ | c = [[Primitive of x fourth by Cosine of a x|Primitive of $x^4 \cos a x$]]
+}}
+{{eqn | r = \dfrac 2 \pi \paren {\frac {4 \pi^3 \cos n \pi} {n^2} - \frac {24 \pi \cos n \pi} {n^4} }
+ | c =
+}}
+{{eqn | r = \frac {8 n^2 \pi^2 - 48} {n^4} \cos n \pi
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = a_0
+ | r = \dfrac 2 \pi \int_0^\pi x^4 \ \d x
+}}
+{{eqn | r = \dfrac 2 \pi \intlimits {\frac {x^5} 5} {x \mathop = 0} {x \mathop = \pi}
+ | c = [[Primitive of Power]]
+}}
+{{eqn | r = \frac {2 \pi^4} 5
+}}
+{{end-eqn}}
+This gives:
+:$x^4 \sim \dfrac {\pi^4} 5 + \displaystyle \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \cos n \pi \cos n x$
+{{qed}}
+[[Category:Examples of Fourier Series]]
+4corhmbfraqiwoqpxj7c5d4d2uajbe6
+\end{proof}<|endoftext|>
+\section{Fourier Series/Absolute Value of x over Minus Pi to Pi}
+Tags: Fourier Series for Absolute Value Function, Fourier Series: Absolute Value of x over Minus Pi to Pi
+
+\begin{theorem}
+For $x \in \openint {-\pi} \pi$:
+:$\displaystyle \size x = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\map \cos {2 n - 1} x} {\paren {2 n - 1}^2}$
+\end{theorem}<|endoftext|>
+\section{Fourier Series/Pi minus x over 0 to 2 Pi}
+Tags: Examples of Fourier Series
+
+\begin{theorem}
+:$\displaystyle \pi - x = 2 \sum_{n \mathop = 1}^\infty \frac {\sin n x} n$
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Fourier Series over Range 2 Pi|Fourier series]]:
+:$\displaystyle \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
+where:
+{{begin-eqn}}
+{{eqn | l = a_n
+ | r = \dfrac 1 \pi \int_0^{2 \pi} \map f x \cos n x \rd x
+}}
+{{eqn | l = b_n
+ | r = \dfrac 1 \pi \int_0^{2 \pi} \map f x \sin n x \rd x
+}}
+{{end-eqn}}
+for all $n \in \Z_{>0}$.
+Thus:
+{{begin-eqn}}
+{{eqn | l = a_0
+ | r = \frac 1 \pi \int_0^{2 \pi} \map f x \rd x
+ | c = [[Cosine of Zero is One]]
+}}
+{{eqn | r = \frac 1 \pi \int_0^{2 \pi} \pi - x \rd x
+ | c = Definition of $f$
+}}
+{{eqn | r = \frac 1 \pi \intlimits {\pi x - \frac 1 2 x^2} 0 {2 \pi}
+ | c = [[Primitive of Power]]
+}}
+{{eqn | r = 2 \pi^2 - \frac 4 2 \pi^2
+ | c =
+}}
+{{eqn | r = 0
+}}
+{{end-eqn}}
+{{qed|lemma}}
+For $n > 0$:
+{{begin-eqn}}
+{{eqn | l = a_n
+ | r = \frac 1 \pi \int_0^{2 \pi} \map f x \cos n x \rd x
+ | c =
+}}
+{{eqn | r = \frac 1 \pi \int_0^{2 \pi} \paren {\pi - x} \cos n x \rd x
+ | c = Definition of $f$
+}}
+{{eqn | r = \pi \int_0^{2 \pi} \cos n x \rd x - \frac 1 \pi \int_0^{2 \pi} x \cos n x \rd x
+ | c =
+}}
+{{eqn | r = - \frac 1 \pi \int_0^{2 \pi} x \cos n x \rd x
+ | c = [[Integral over 2 pi of Cosine of n x|Integral over $2 \pi$ of $\cos n x$]]
+}}
+{{eqn | r = - \frac 1 \pi \intlimits {\frac {\cos n x} {n^2} + \frac {x \sin n x} n} 0 {2 \pi}
+ | c = [[Primitive of x by Cosine of a x|Primitive of $x \cos n x$]]
+}}
+{{eqn | r = - \frac 1 \pi \paren {\cos 0 - \cos 2 \pi}
+ | c = [[Sine of Integer Multiple of Pi]]
+}}
+{{eqn | r = 0
+ | c = [[Sine and Cosine are Periodic on Reals]]
+}}
+{{end-eqn}}
+{{qed|lemma}}
+Now for the $\sin n x$ terms:
+{{begin-eqn}}
+{{eqn | l = b_n
+ | r = \frac 1 \pi \int_0^{2 \pi} \map f x \sin n x \rd x
+ | c =
+}}
+{{eqn | r = \frac 1 \pi \int_0^{2 \pi} \paren {\pi - x} \sin n x \rd x
+ | c = Definition of $f$
+}}
+{{eqn | r = \int_0^{2 \pi} \sin n x \rd x - \frac 1 \pi \int_0^{2 \pi} x \sin n x \rd x
+ | c =
+}}
+{{eqn | r = - \frac 1 \pi \int_0^{2 \pi} x \sin n x \rd x
+ | c = [[Integral over 2 pi of Sine of n x|Integral over $2 \pi$ of $\sin n x$]]
+}}
+{{eqn | r = - \frac 1 \pi \intlimits {\frac {\sin n x} {n^2} - \frac {x \cos n x} n} 0 {2 \pi}
+ | c = [[Primitive of x by Sine of a x|Primitive of $x \sin n x$]]
+}}
+{{eqn | r = \frac 1 \pi \frac {2 \pi \cos 2 n \pi} n
+ | c = [[Sine of Multiple of Pi]]
+}}
+{{eqn | r = \frac 2 n
+ | c = [[Cosine of Multiple of Pi]]
+}}
+{{end-eqn}}
+Finally:
+{{begin-eqn}}
+{{eqn | l = \map f x
+ | o = \sim
+ | r = \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}
+ | c =
+}}
+{{eqn | r = \sum_{n \mathop = 1}^\infty \frac 2 n \sin n x
+ | c = substituting for $a_0$, $a_n$ and $b_n$ from above
+}}
+{{eqn | r = 2 \sum_{n \mathop = 1}^\infty \frac {\sin n x} n
+ | c = rearranging
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cube of 20 is Sum of Sequence of 4 Consecutive Cubes}
+Tags: Cube Numbers, 20, 8000
+
+\begin{theorem}
+:$20^3 = \displaystyle \sum_{k \mathop = 11}^{14} k^3$
+That is:
+:$20^3 = 11^3 + 12^3 + 13^3 + 14^3$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sum_{k \mathop = 1}^{14} k^3
+ | r = \paren {\dfrac {14 \paren {14 + 1} } 2}^2
+ | c = [[Sum of Sequence of Cubes]]
+}}
+{{eqn | r = 11 \, 025
+ | c =
+}}
+{{eqn | l = \sum_{k \mathop = 1}^{10} k^3
+ | r = \paren {\dfrac {10 \paren {10 + 1} } 2}^2
+ | c = [[Sum of Sequence of Cubes]]
+}}
+{{eqn | r = 3025
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \sum_{k \mathop = 11}^{14} k^3
+ | r = \sum_{k \mathop = 1}^{14} k^3 - \sum_{k \mathop = 1}^{10} k^3
+ | c =
+}}
+{{eqn | r = 11 \, 025 - 3025
+ | c =
+}}
+{{eqn | r = 8000
+ | c =
+}}
+{{eqn | r = 20^3
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Mapping is Continuous implies Mapping Preserves Filtered Infima in Lower Topological Lattice}
+Tags: Topological Order Theory
+
+\begin{theorem}
+Let $T = \left({S, \preceq, \tau}\right)$ and $Q = \left({X, \preceq', \tau'}\right)$ be [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattices]] with [[Definition:Lower Topology|lower topologies]].
+Let $f: S \to X$ be a [[Definition:Mapping|mapping]] such that
+:$f$ is a [[Definition:Continuous (Topology)|continuous mapping]].
+Then $f$ [[Definition:Mapping Preserves Infimum/Filtered|preserves filtered infima]].
+\end{theorem}
+
+\begin{proof}
+Define $B := \left\{ {\complement_S\left({x^\succeq}\right): x \in S}\right\}$
+By definition of [[Definition:Lower Topology|lower topology]]:
+:$B$ is an [[Definition:Analytic Sub-Basis|analytic sub-basis]].
+Let $F$ be a [[Definition:Filtered Subset|filtered subset]] of $S$ such that
+:$F$ admits an [[Definition:Infimum of Set|infimum]] in $T$.
+Thus by definition of [[Definition:Complete Lattice|complete lattice]]:
+:$f\left[{F}\right]$ admits an [[Definition:Infimum of Set|infimum]] in $Q$.
+We will prove that
+:$\forall A \in B: \inf F \in A \implies F \cap A \ne \varnothing$
+Let $A \in B$ be such that
+:$\inf F \in A$
+By definition of $B$:
+:$\exists x \in S: A = \complement_S\left({x^\succeq}\right)$
+By definition of [[Definition:Relative Complement|relative complement]]:
+:$\inf F \notin x^\succeq$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$x \npreceq \inf F$
+By definition of [[Definition:Infimum of Set|infimum]]:
+:$x$ is [[Definition:Negation|not]] [[Definition:Lower Bound of Set|lower bound]] for $F$.
+By definition of [[Definition:Lower Bound of Set|lower bound]]:
+:$\exists y \in F: x \npreceq y$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$y \notin x^\succeq$
+By definition of [[Definition:Relative Complement|relative complement]]:
+:$y \in A$
+By definitions of [[Definition:Set Intersection|intersection]] and [[Definition:Non-Empty Set|non-empty set]]:
+:$F \cap A \ne \varnothing$
+{{qed|lemma}}
+Then by [[If Infimum of Filtered Subset belongs to Element of Sub-Basis then Subset and Element Intersect implies Infimum of Subset belongs to Closure of Subset]]:
+:$\inf F \in F^-$
+We will prove that
+:$f$ is an [[Definition:Increasing Mapping|increasing mapping]].
+Let $x, y \in S$ be such that
+:$x \preceq y$
+By definition of [[Definition:Reflexivity|reflexivity]]:
+:$f\left({x}\right) \preceq' f\left({x}\right)$
+By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$f\left({x}\right) \in \left({f\left({x}\right)}\right)^{\succeq'}$
+By definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
+:$x \in f^{-1}\left[{\left({f\left({x}\right)}\right)^{\succeq'} }\right]$
+By [[Complement of Upper Closure of Element is Open in Lower Topology]]:
+:$\left({f\left({x}\right)}\right)^{\succeq'}$ is [[Definition:Closed Set (Topology)|closed]].
+By [[Continuity by Closed Sets]]:
+:$f^{-1}\left[{\left({f\left({x}\right)}\right)^{\succeq'} }\right]$ is [[Definition:Closed Set (Topology)|closed]].
+By [[Closed Subset is Upper in Lower Topology]]:
+:$f^{-1}\left[{\left({f\left({x}\right)}\right)^{\succeq'} }\right]$ is [[Definition:Upper Set|upper]].
+By definition of [[Definition:Upper Set|upper set]]:
+:$y \in f^{-1}\left[{\left({f\left({x}\right)}\right)^{\succeq'} }\right]$
+By definition of [[Definition:Preimage of Subset under Mapping|preimage of set]]:
+:$f\left({y}\right) \in \left({f\left({x}\right)}\right)^{\succeq'}$
+Thus by definition of [[Definition:Upper Closure of Element|upper closure of element]]:
+:$f\left({x}\right) \preceq' f\left({y}\right)$
+{{qed|lemma}}
+We will prove that
+:$f\left({\inf F}\right)$ is [[Definition:Lower Bound of Set|lower bound]] for $f\left[{F}\right]$
+By definition of [[Definition:Infimum of Set|infimum]]:
+:$f\left({\inf F}\right) \preceq' \inf\left({f\left[{F}\right]}\right)$
+We will prove that
+:$F \subseteq f^{-1}\left[{\left({\inf\left({f\left[{F}\right]}\right)}\right)^{\succeq'} }\right]$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{3 Numbers in A.P. whose 4th Powers are Sum of Four 4th Powers}
+Tags: Fourth Powers
+
+\begin{theorem}
+The following [[Definition:Ordered Triple|triplets]] of [[Definition:Integer|integers]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] with [[Definition:Common Difference|common difference]] of $60$ can all be expressed as the [[Definition:Integer Addition|sum]] of four [[Definition:Fourth Power|$4$th powers]]:
+:$\tuple {8373, 8433, 8493}, \tuple {8517, 8577, 8637}, \ldots$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 8373^4
+ | r = 4450^4 + 5500^4 + 5670^4 + 7123^4
+}}
+{{eqn | l = 8433^4
+ | r = 4730^4 + 4806^4 + 5230^4 + 7565^4
+}}
+{{eqn | l = 8493^4
+ | r = 524^4 + 4910^4 + 5925^4 + 7630^4
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 8517^4
+ | r = 1642^4 + 3440^4 + 6100^4 + 7815^4
+}}
+{{eqn | l = 8577^4
+ | r = 1050^4 + 2905^4 + 5236^4 + 8230^4
+}}
+{{eqn | l = 8637^4
+ | r = 3450^4 + 3695^4 + 5780^4 + 8012^4
+}}
+{{end-eqn}}
+{{qed}}
+The internal structure of these numbers reveals an interesting pattern:
+{{begin-eqn}}
+{{eqn | l = 8373
+ | r = 3 \times 2791
+}}
+{{eqn | l = 8433
+ | r = 3^2 \times 937
+}}
+{{eqn | l = 8493
+ | r = 3 \times 19 \times 149
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 8517
+ | r = 3 \times 17 \times 169
+}}
+{{eqn | l = 8577
+ | r = 3^3 \times 953
+}}
+{{eqn | l = 8637
+ | r = 3 \times 2879
+}}
+{{end-eqn}}
+\end{proof}<|endoftext|>
+\section{Intersection of Chain of Prime Ideals of Commutative Ring is Prime Ideal}
+Tags: Commutative Rings, Prime Ideals of Rings
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
+Let $\Spec R$ be the [[Definition:Spectrum of Ring|spectrum]] of $R$, [[Definition:Ordered Set|ordered]] by [[Definition:Inclusion Relation|inclusion]].
+Let $\set {P_\alpha}_{\alpha \mathop \in A}$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Chain (Set Theory)|chain]] of [[Definition:Prime Ideal of Ring|prime ideals]] of $\Spec R$.
+Let $\displaystyle P = \bigcap_{\alpha \mathop \in A} P_\alpha$ be their [[Definition:Set Intersection|intersection]].
+Then $P$ is a [[Definition:Prime Ideal of Ring|prime ideal]] of $R$.
+\end{theorem}
+
+\begin{proof}
+By [[Intersection of Ring Ideals is Ideal]], $P$ is an [[Definition:Ideal of Ring|ideal]] of $R$.
+We show that $P$ is [[Definition:Prime Ideal of Ring|prime]].
+Let $a, b\in R$ with $a, b \notin P$.
+We show that $a b \notin P$.
+Because $a \notin P$, there exists $\alpha \in A$ with $a \notin P_\alpha$.
+Because $b \notin P$, there exists $\beta \in A$ with $b \notin P_\beta$.
+Because $\set {P_\gamma}_{\gamma \mathop \in A}$ is [[Definition:Total Ordering|totally ordered]], $P_\alpha \subseteq P_\beta$ or $P_\beta \subseteq P_\alpha$.
+{{WLOG}}, we can assume $P_\alpha \subseteq P_\beta$, which gives us $a, b \notin P_\alpha$.
+Since $P_\alpha$ is an [[Definition:Prime Ideal of Ring|prime ideal]], $a b \notin P_\alpha$.
+Thus $a b \notin P$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Number of Different Ways to play First n Moves in Chess}
+Tags: Chess
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] formed from the number of ways to play the first $n$ moves in [[Definition:Chess|chess]] begins:
+:$20, 400, 8902, 197 \, 742, \ldots$
+{{OEIS|A007545}}
+The count for the fourth move is already ambiguous, as it depends on whether only legal moves count, or whether all moves, legal or illegal, are included.
+The count as given here does include illegal moves in addition to legal ones.
+\end{theorem}
+
+\begin{proof}
+There are $20$ ways to make the $1$st move by White:
+:Each of the $8$ [[Definition:Chess Pawn|pawns]] may be moved either $1$ or $2$ squares forward, making $16$ moves
+:Each of the $2$ [[Definition:Chess Knight|knights]] may be moved to either of $2$ squares before it, making $4$ moves.
+For each of those $20$ first moves by White, Black has the same $20$ options.
+Thus there are $20 \times 20$ possible different games after the $2$nd move.
+To count the $3$rd moves, one needs to consider cases.
+First note that after the $1$st move, whatever it was, there are $7$ [[Definition:Chess Pawn|pawns]] on the $2$nd rank, each of which can be moved $1$ or $2$ squares forward, making $14$ moves for each of those $400$ possibilities.
+Note that if a [[Definition:Chess Knight|knight]] was one of the pieces to have moved first, the [[Definition:Chess Pawn|pawn]] in the square behind where it ends up cannot move -- hence the $7$ [[Definition:Chess Pawn|pawns]] on the $2$nd rank that can move.
+Thus there are $400 \times 14 = 5600$ possible moves involving a so-far unmoved [[Definition:Chess Pawn|pawn]].
+For each of the $400$ positions, there are exactly $8$ which consist of two [[Definition:Chess Pawn|pawns]] in opposition on the $4$th and $5$th rank.
+There are also another $4 \times 20 = 80$ positions in which white moved a [[Definition:Chess Knight|knight]].
+For all other $400 - 88 = 312$ positions, the already-moved [[Definition:Chess Pawn|pawn]] has the option of moving another square forward.
+This gives another $312$ options for the $3$rd move.
+We now need to take into account the possibility that White may be able to capture a Black [[Definition:Chess Pawn|pawn]].
+{{finish|Each case needs to be investigated.}}
+\end{proof}<|endoftext|>
+\section{Largest Product of Pandigital Factors}
+Tags: Recreational Mathematics
+
+\begin{theorem}
+The largest [[Definition:Integer|integer]] that can be obtained by [[Definition:Integer Multiplication|multiplying]] $2$ [[Definition:Integer|integers]] which between them use all the [[Definition:Digit|digits]] from $1$ to $9$ is:
+:$843 \, 973 \, 902 = 9642 \times 87531$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|probably quite simple}}
+\end{proof}<|endoftext|>
+\section{Sequence of 9 Primes of form 4n+1}
+Tags: Prime Numbers
+
+\begin{theorem}
+The following [[Definition:Integer Sequence|sequence]] of $9$ consecutive [[Definition:Prime Number|prime numbers]] are all of the form $4 n + 1$:
+:$11 \, 593, 11 \, 597, 11 \, 617, 11 \, 621, 11 \, 633, 11 \, 657, 11 \, 677, 11 \, 681, 11 \, 689$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 11 \, 593
+ | r = 4 \times 2898 + 1
+}}
+{{eqn | l = 11 \, 597
+ | r = 4 \times 2899 + 1
+}}
+{{eqn | l = 11 \, 617
+ | r = 4 \times 2904 + 1
+}}
+{{eqn | l = 11 \, 621
+ | r = 4 \times 2905 + 1
+}}
+{{eqn | l = 11 \, 633
+ | r = 4 \times 2908 + 1
+}}
+{{eqn | l = 11 \, 657
+ | r = 4 \times 2914 + 1
+}}
+{{eqn | l = 11 \, 677
+ | r = 4 \times 2919 + 1
+}}
+{{eqn | l = 11 \, 681
+ | r = 4 \times 2920 + 1
+}}
+{{eqn | l = 11 \, 689
+ | r = 4 \times 2922 + 1
+}}
+{{end-eqn}}
+It remains to be noted that:
+:the [[Definition:Prime Number|prime number]] before $11 \, 593$ is $11 \, 587$ which is $4 \times 2897 - 1$
+:the [[Definition:Prime Number|prime number]] after $11 \, 689$ is $11 \, 699$ which is $4 \times 2925 - 1$
+confirming that they are not of the form $4 n + 1$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Pandigital Square}
+Tags: Square Numbers, Pandigital Integers, 11,826, 139,854,276
+
+\begin{theorem}
+The smallest [[Definition:Square Number|square number]] which contains all the [[Definition:Digit|digits]] from $1$ to $9$ is:
+:$11 \, 826^2 = 139 \, 854 \, 276$
+\end{theorem}
+
+\begin{proof}
+To streamline the argument, the term '''$9$-pan''' will be coined to mean '''$9$-[[Definition:Digit|digit]] [[Definition:Pandigital Number|pandigital number]] which excludes the [[Definition:Zero Digit|$0$ digit]]'''.
+Let $n$ be the smallest [[Definition:Positive Integer|positive integer]] whose [[Definition:Square (Algebra)|square]] is '''$9$-pan'''.
+First it is noted that the smallest '''$9$-pan''' is $123 \, 456 \, 789$.
+Hence any [[Definition:Square Number|square]] '''$9$-pan''' must be at least as large as that.
+Thus we can can say that:
+:$n \ge \ceiling {\sqrt {123 \, 456 \, 789} } = 11 \, 112$
+where $\ceiling {\, \cdot \,}$ denotes the [[Definition:Ceiling Function|ceiling function]].
+It remains to be demonstrated that no [[Definition:Positive Integer|positive integer]] between $11 \, 112$ and $11 \, 826$ has a '''$9$-pan''' [[Definition:Square (Algebra)|square]].
+{{ProofWanted|This is over $700$ numbers. The task can be filtered by, for example, disregarding all $n$ ending in $1$ and $9$ because their squares will both begin and end in $1$ at this low range of the $10000$s, and of course all $n$ ending in $0$ because their squares will end in $0$.}}
+\end{proof}<|endoftext|>
+\section{Locally Compact Space is Weakly Locally Compact}
+Tags: Locally Compact Spaces, Weakly Locally Compact Spaces
+
+\begin{theorem}
+Let $T = \struct{S, \tau}$ be a [[Definition:Locally Compact Space|locally compact topological space]].
+Then $T$ is [[Definition:Weakly Locally Compact Space|weakly locally compact]].
+\end{theorem}
+
+\begin{proof}
+Because $T$ is [[Definition:Locally Compact Space|locally compact]], every point of $S$ has a [[Definition:Neighborhood Basis|neighborhood basis]] consisting of [[Definition:Compact Topological Subspace|compact sets]].
+From [[Space is Neighborhood of all its Points]], $S$ is a [[Definition:Neighborhood of Point|neighborhood]] of each point of $S$.
+
+By assumption, each point has a [[Definition:Compact Topological Subspace|compact]] [[Definition:Neighborhood of Point|neighborhood]] contained in $S$.
+Thus $T$ is [[Definition:Weakly Locally Compact Space|weakly locally compact]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Space is Neighborhood of all its Points}
+Tags: Neighborhoods
+
+\begin{theorem}
+Let $T = \struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
+Let $x\in X$.
+Then $X$ is a [[Definition:Neighborhood of Point|neighborhood]] of $x$.
+\end{theorem}
+
+\begin{proof}
+By the definition of the [[Definition:Topology|topology]] $\tau$, $X$ is an [[Definition:Open Set (Topology)|open set]].
+From [[Set is Open iff Neighborhood of all its Points]], $X$ is a [[Definition:Neighborhood of Point|neighborhood]] of $x$.
+\end{proof}<|endoftext|>
+\section{Local Compactness is Preserved under Open Continuous Surjection}
+Tags: Locally Compact Spaces, Open Mappings
+
+\begin{theorem}
+Let $T_A = \left({S_A, \tau_A}\right)$ and $T_B = \left({S_B, \tau_B}\right)$ be [[Definition:Topological Space|topological spaces]].
+Let $\phi: T_A \to T_B$ be a [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]] which is also an [[Definition:Open Mapping|open mapping]] and a [[Definition:Surjection|surjection]].
+If $T_A$ is [[Definition:Locally Compact Space|locally compact]], then $T_B$ is also [[Definition:Locally Compact Space|locally compact]].
+\end{theorem}<|endoftext|>
+\section{Equivalence of Definitions of Noetherian Module}
+Tags: Noetherian Modules
+
+\begin{theorem}
+Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
+Let $M$ be an $A$-[[Definition:Module|module]].
+{{TFAE|def = Noetherian Module}}
+\end{theorem}<|endoftext|>
+\section{Largest Positive Integer not Sum of Distinct Cubes}
+Tags: Sums of Cubes, 12,758
+
+\begin{theorem}
+$12 \, 758$ is the largest [[Definition:Positive Integer|positive integer]] that cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct|distinct]] [[Definition:Cube Number|cubes]].
+\end{theorem}<|endoftext|>
+\section{Particular Point Space is Locally Compact}
+Tags: Locally Compact Spaces, Particular Point Topology
+
+\begin{theorem}
+Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
+Then $T$ is [[Definition:Locally Compact Space|locally compact]].
+\end{theorem}
+
+\begin{proof}
+Let $x \in S$.
+Consider the set $\set {p, x}$.
+From the definition of [[Definition:Particular Point Topology|particular point topology]], $\set {p, x}$ is [[Definition:Open Set (Topology)|open]] in $T$.
+By [[Finite Topological Space is Compact]], $\set {p, x}$ is [[Definition:Compact Topological Space|compact]].
+Let $N$ be a [[Definition:Neighborhood (Topology)|neighborhood]] of $x$.
+Then:
+:$\exists U \in \tau_p: x \in U \subseteq N$.
+From the definition of [[Definition:Particular Point Topology|particular point topology]], since $U \ne \O$, we must have $p \in U$.
+Therefore $\set {p, x} \subseteq U \subseteq N$.
+Since $N$ is arbitrary, $\set {\set {p, x}}$ is a [[Definition:Neighborhood Basis|neighborhood basis]] for $x$.
+The result follows from definition of a [[Definition:Locally Compact Space|locally compact space]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Product of Summations is Summation Over Cartesian Product of Products}
+Tags: Algebra
+
+\begin{theorem}
+This is a generalization of the distributive law:
+:$\displaystyle \prod_{a \mathop \in A} \sum_{b \mathop \in B_a} t_{a, b} = \sum_{c \mathop \in \prod_{a \mathop \in A} B_a} \prod_{a \mathop \in A} t_{a, c_a}$
+where the product of sets $\prod_{a \mathop \in A} B_a$ is taken to be a cartesian product.
+{{explain|In order to make this comprehensible, the precise nature of $A$ and $B$ need to be defined. Presumably $A$ is a set (of numbers?), and $B$ is a family of sets of numbers indexed by $A$? And the $t$s are elements of $B$ and are numbers? I'm having difficulty.}}
+\end{theorem}
+
+\begin{proof}
+For simplicity, let $A = \closedint 1 n$.
+This reduces the complexity without loss of generality, as if we wanted to use an arbitrary set we could store the actual elements in an $n$-tuple and index them.
+So we can think of $\closedint 1 n$ as representing the actual elements.
+Use induction on $n$:
+For $n = 1$:
+:$\displaystyle \sum_{b \mathop \in B_1} t_{1, b} = \sum_{c \mathop \in B_1} t_{1, c}$
+which is true, proving the case.
+Assume the formula is true for $n$, and prove it for $n + 1$.
+:$\displaystyle \paren {\prod_{a \mathop \in \closedint 1 n} \sum_{b \mathop \in B_a} t_{a, b} } \paren {\sum_{b \mathop \in B_{n + 1} } t_{n+1, b} } = \paren {\sum_{c \mathop \in \prod_{a \in \closedint 1 n} B_a} \prod_{a \mathop \in \closedint 1 n} t_{a, c_a} } \paren {\sum_{b \mathop \in B_{n + 1} } t_{n + 1, b} }$
+We need this result which we will prove below,
+:$\displaystyle \paren {\sum_{c \mathop \in \prod_{a \in \closedint 1 n} B_a} \prod_{a \mathop \in \closedint 1 n} t_{a, c_a} } \sum_{b \mathop \in X_{n + 1} } t_{n + 1, b} = \sum_{c \mathop \in \paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times X_{n + 1} } \prod_{a \mathop \in \closedint 1 {n + 1} } t_{a, c_a}$
+which gives:
+{{begin-eqn}}
+{{eqn | l = \paren {\prod_{a \mathop \in \closedint 1 n} \sum_{b \mathop \in B_a} t_{a, b} } \paren {\sum_{b \mathop \in B_{n + 1} } t_{n + 1, b} }
+ | r = \sum_{c \mathop \in \paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times B_{n + 1} } \prod_{a \mathop \in \closedint 1 {n + 1} } t_{a, c_a}
+ | c =
+}}
+{{eqn | r = \sum_{c \mathop \in \prod_{a \in \closedint 1 {n + 1} } B_a} \prod_{a \mathop \in \closedint 1 {n + 1} } t_{a, c_a}
+ | c =
+}}
+{{end-eqn}}
+This completes the induction case on $n$ while assuming:
+:$\displaystyle \paren {\sum_{c \mathop \in \prod_{a \in \closedint 1 n} B_a} \prod_{a \mathop \in \closedint 1 n} t_{a, c_a} } \sum_{b \mathop \in X_{n + 1} } t_{n + 1, b} = \sum_{c \mathop \in \paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times X_{n + 1} } \prod_{a \mathop \in \closedint 1 {n + 1} } t_{a, c_a}$
+To prove this, use induction on the size $X_{n + 1}$.
+If $X_{n + 1}$ has a single element:
+:$\displaystyle \paren {\sum_{c \mathop \in \prod_{a \in \closedint 1 n} B_a} \prod_{a \mathop \in \closedint 1 n} t_{a, c_a} } t_{n + 1, k} = \sum_{c \mathop \in \paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times \set k} \prod_{a \mathop \in \closedint 1 {n + 1} } t_{a, c_a}$
+which is true, proving the case for size $1$.
+Now assume it is true for $X_{n + 1}$ and $Y_{n + 1}$ we will prove it for $Z_{n + 1}$.
+This corresponds to the induction step if the size of $Y_{n + 1}$ is $1$.
+So all we are doing is proving a more general case.
+We do this because $X_{n + 1}$ and $Y_{n + 1}$ are then symmetrical, and so the proof is easier to understand.
+:$Z_{n + 1} = X_{n + 1} \cup Y_{n + 1}$
+{{explain|It is not clear from the above paragraph exactly what is being done}}
+Then:
+:$\displaystyle \sum_{b \mathop \in Z_{n + 1} } t_{n + 1, b} = \paren {\sum_{b \mathop \in X_{n + 1} } t_{n + 1, b} + \sum_{b \mathop \in Y_{n + 1} } t_{n + 1, b} }$
+Thus:
+{{begin-eqn}}
+{{eqn | o =
+ | r = \paren {\sum_{c \mathop \in \prod_{a \mathop \in \closedint 1 n} B_a} \prod_{a \mathop \in \closedint 1 n} t_{a, c_a} } \sum_{b \mathop \in Z_{n + 1} } t_{n + 1, b}
+ | c = from {{LHS}} of assumption
+}}
+{{eqn | r = \paren {\sum_{c \in \prod_{a \mathop \in \closedint 1 n} B_a} \prod_{a \mathop \in \closedint 1 n} t_{a, c_a} } \paren {\sum_{b \mathop \in X_{n + 1} } t_{n + 1, b} + \sum_{b \mathop \in Y_{n + 1} } t_{n + 1,b} }
+ | c = substituting for $Z_{n + 1}$
+}}
+{{eqn | r = \paren {\sum_{c \in \prod_{a \mathop \in \closedint 1 n} B_a} \prod_{a \mathop \in \closedint 1 n} t_{a, c_a} } \sum_{b \mathop \in X_{n + 1} } t_{n + 1, b} + \paren {\sum_{c \mathop \in \prod_{a \mathop \in \closedint 1 n} B_a} \prod_{a \mathop \in \closedint 1 n} t_{a, c_a} } \sum_{b \in Y_{n + 1} } t_{n + 1, b}
+ | c =
+}}
+{{eqn | r = \sum_{c \mathop \in \paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times X_{n + 1} } \prod_{a \mathop \in \closedint 1 {n + 1} } t_{a, c_a} + \sum_{c \mathop \in \paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times Y_{n + 1} } \prod_{a \mathop \in \closedint 1 {n + 1} } t_{a, c_a}
+ | c = using the assumption for $X$ and $Y$
+}}
+{{eqn | r = \sum_{c \mathop \in \paren {\paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times X_{n + 1} \mathop \cup \paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times Y_{n + 1} } } \prod_{a \mathop \in \closedint 1 {n + 1} } t_{a, c_a}
+ | c = merging the two summations into one
+}}
+{{end-eqn}}
+The cartesian product is:
+{{begin-eqn}}
+{{eqn | o =
+ | r = \paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times X_{n + 1} \cup \paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times Y_{n + 1}
+ | c =
+}}
+{{eqn | r = \paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times \paren {X_{n + 1} \cup Y_{n + 1} }
+ | c = using $R \times P \cup R \times Q = R \times \paren {P \cup Q}$
+}}
+{{eqn | r = \paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times Z_{n + 1}
+ | c = using $X_{n + 1} \cup Y_{n + 1} = Z_{n + 1}$
+}}
+{{end-eqn}}
+Substituting back in:
+:$\displaystyle \sum_{c \mathop \in \paren {\paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times X_{n + 1} \mathop \cup \paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times Y_{n + 1} } } \prod_{a \mathop \in \closedint 1 {n + 1} } t_{a,c_a} = \sum_{c \mathop \in \paren {\prod_{a \mathop \in \closedint 1 n} B_a} \times Z_{n+1} } \prod_{a \mathop \in \closedint 1 {n + 1} } t_{a, c_a}$
+which is the {{RHS}} of the assumption.
+{{qed}}
+[[Category:Algebra]]
+b7i6os0ezx8ki8ye9z0ufvomzmz3rgm
+\end{proof}<|endoftext|>
+\section{Largest Number not Sum of Squares of Distinct Primes}
+Tags: Prime Numbers, Sums of Squares
+
+\begin{theorem}
+The largest [[Definition:Positive Integer|positive integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of the [[Definition:Square (Algebra)|squares]] of [[Definition:Distinct|distinct]] [[Definition:Prime Number|prime numbers]] is $17 \, 163$.
+\end{theorem}<|endoftext|>
+\section{Volume of Smallest Rational Tetrahedron}
+Tags: Tetrahedra
+
+\begin{theorem}
+The only [[Definition:Rational Tetrahedron|rational tetrahedron]] whose [[Definition:Edge of Polyhedron|edge]] [[Definition:Length of Line|lengths]] are less than $157$ has:
+: [[Definition:Edge of Polyhedron|edges]] of [[Definition:Length of Line|length]] $117$, $80$, $53$, $52$, $51$, $84$
+: [[Definition:Face of Polyhedron|faces]] of [[Definition:Area|area]] $1800$, $1890$, $2016$, $1170$
+: [[Definition:Volume|volume]] of $18 \, 144$.
+\end{theorem}<|endoftext|>
+\section{Square and Tetrahedral Numbers}
+Tags: Square Numbers, Pyramidal Numbers
+
+\begin{theorem}
+The only [[Definition:Positive Integer|positive integers]] which are simultaneously [[Definition:Tetrahedral Number|tetrahedral]] and [[Definition:Square Number|square]] are:
+:$1, 4, 19 \, 600$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1
+ | r = \dfrac {1 \paren {1 + 1} \paren {1 + 2} } 6
+ | c = [[Closed Form for Tetrahedral Numbers]]
+}}
+{{eqn | r = 1^2
+ | c = {{Defof|Square Number}}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 4
+ | r = \dfrac {2 \paren {2 + 1} \paren {2 + 2} } 6
+ | c = [[Closed Form for Tetrahedral Numbers]]
+}}
+{{eqn | r = 2^2
+ | c = {{Defof|Square Number}}
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 19 \, 600
+ | r = \dfrac {48 \paren {48 + 1} \paren {48 + 2} } 6
+ | c = [[Closed Form for Tetrahedral Numbers]]
+}}
+{{eqn | r = 140^2
+ | c = {{Defof|Square Number}}
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that these are the only such instances.}}
+\end{proof}<|endoftext|>
+\section{Largest Integer not Sum of Two Abundant Numbers}
+Tags: Abundant Numbers
+
+\begin{theorem}
+The largest [[Definition:Integer|integer]] which is not the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Abundant Number|abundant numbers]] is $20 \, 161$.
+\end{theorem}
+
+\begin{proof}
+First we show that for $1 < k < 90$, $315 k$ is [[Definition:Abundant Number|abundant]].
+If $k$ is [[Definition:Divisor of Integer|divisible]] by $3, 5, 7$, note that:
+:$945, 1575, 2205$
+are all [[Definition:Abundant Number|abundant]], and $315 k$ is a multiple of at least one of them.
+Hence $315 k$ is [[Definition:Abundant Number|abundant]] by [[Multiple of Abundant Number is Abundant]].
+If $k$ is not [[Definition:Divisor of Integer|divisible]] by $3, 5, 7$:
+Let $p$ be a [[Definition:Prime Number|prime]] such that $p \divides k$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \frac {\map \sigma {315 p} } {315 p}
+ | r = \frac 1 {315 p} \paren {1 + 3 + 3^2} \paren {1 + 5} \paren {1 + 7} \paren {1 + p}
+}}
+{{eqn | r = \frac {208} {105} \paren {1 + \frac 1 p}
+}}
+{{eqn | o = >
+ | r = \frac {208} {105} \paren {1 + \frac 1 {90} }
+ | c = $p < 90$
+}}
+{{eqn | o = >
+ | r = 2
+}}
+{{end-eqn}}
+hence $315 p$ and $315 k$ are [[Definition:Abundant Number|abundant]].
+Since $88$ and $315$ are [[Definition:Coprime Integers|coprime]]:
+:$88 = 2^3 \times 11$
+:$315 = 3^2 \times 5 \times 7$
+By [[Largest Number not Expressible as Sum of Multiples of Coprime Integers]], all numbers greater than or equal to:
+:$\paren {88 - 1} \paren {315 - 1} = 27 \, 318$
+can be expressed as a sum of multiples of $88$ and $315$.
+Hence for $n \ge 27 \, 318 + 315 \times 2 = 27 \, 948$:
+:$\exists s, t \in \N: 90 > t \ge 2: n = 88 s + 315 t$
+and both $88 s$ and $315 t$ are [[Definition:Abundant Number|abundant]] for $s > 0$.
+For $s = 0$, $t \ge \dfrac {27 \, 948} {315} > 2 = \paren {2 - 1} \paren {3 - 1}$.
+By [[Largest Number not Expressible as Sum of Multiples of Coprime Integers]], $t - 5$ can be expressed as a sum of multiples of $2$ and $3$.
+Hence:
+:$\exists a, b \in \Z_{> 0}: 2 a + 3 b = t$
+This gives:
+:$n = 630 a + 945 b$
+and both $630 a$ and $945 b$ are [[Definition:Abundant Number|abundant]].
+We still need to find representations for $20 \, 162 < n < 27 \, 948$.
+We can check this via brute force.
+Using [[Largest Number not Expressible as Sum of Multiples of Coprime Integers/Generalization]], we can narrow down our search to numbers that are not [[Definition:Divisor of Integer|divisible]] by small [[Definition:Prime Number|primes]]:
+Since $\gcd \set {18, 20} = 2$, the largest multiple of $2$ not expressible as a sum of multiples of $18$ and $20$ is:
+:$\dfrac {18 \times 20} 2 - 18 - 20 = 142 < 20161$
+Since $\gcd \set {12, 945} = 3$, the largest multiple of $3$ not expressible as a sum of multiples of $12$ and $945$ is:
+:$\dfrac {12 \times 945} 3 - 12 - 945 = 2823 < 20161$
+Since $\gcd \set {20, 945} = 5$, the largest multiple of $5$ not expressible as a sum of multiples of $20$ and $945$ is:
+:$\dfrac {20 \times 945} 5 - 20 - 945 = 2815 < 20161$
+Since $\gcd \set {56, 945} = 7$, the largest multiple of $7$ not expressible as a sum of multiples of $56$ and $945$ is:
+:$\dfrac {56 \times 945} 7 - 56 - 945 = 6559 < 20161$
+All numbers involved above are [[Definition:Abundant Number|abundant]].
+Hence we only need to consider $n$ not [[Definition:Divisor of Integer|divisible]] by $2, 3, 5, 7$.
+{{finish|Brute force by computer}}
+\end{proof}<|endoftext|>
+\section{Smallest Integer using Three Words in English Description}
+Tags: Recreational Mathematics
+
+\begin{theorem}
+The smallest [[Definition:Integer|integer]] which uses exactly $3$ [[Definition:Word (Natural Language)|words]] in its standard (British) English description is:
+:$21 \, 000$: '''twenty-one thousand'''
+counting hyphenations as separate [[Definition:Word (Natural Language)|words]].
+\end{theorem}
+
+\begin{proof}
+All [[Definition:Integer|integers]] up to $100$ ('''one hundred''') use either $1$ or $2$ [[Definition:Word (Natural Language)|words]]:
+:'''one'''
+:'''sixty'''
+:'''seventeen'''
+:'''ninety-eight'''
+All [[Definition:Integer|integers]] of the form $100 n$ for $n = 1, 2, \ldots 9$ use exactly $2$ [[Definition:Word (Natural Language)|words]]:
+:'''one hundred'''
+:'''seven hundred'''
+:'''nine hundred'''
+In British English, the technique for describing [[Definition:Integer|integers]] from $101$ to $199$, and $201$ to $299$ and so on, is to use '''and''' between the number of hundreds and the rest:
+:'''one hundred and one'''
+:'''three hundred and thirteen'''
+:'''four hundred and twenty-six'''
+:'''seven hundred and seventy'''
+thus using either $4$ or $5$ words.
+All [[Definition:Integer|integers]] of the form $1000 n$ for $n = 1, 2, \ldots 10$ use exactly $2$ [[Definition:Word (Natural Language)|words]]:
+:'''two thousand'''
+:'''five thousand'''
+:'''eight thousand'''
+:'''twelve thousand'''
+:'''nineteen thousand'''
+:'''twenty thousand'''
+Similarly with hundreds, the technique for describing [[Definition:Integer|integers]] of the form $1000 m + n$ for $1 \le n \le 99$ is to use '''and''' between the number of thousands and the rest:
+:'''five thousand and eighteen'''
+:'''sixteen thousand and forty-eight'''
+:'''thirty-seven thousand and sixty'''
+thus using either $4$ or $5$ words.
+All other numbers between $1100$ and $20 \, 999$ trivially use more than $3$ [[Definition:Word (Natural Language)|words]]:
+:'''four thousand, eight hundred'''
+:'''sixteen thousand, one hundred and seventy-seven'''
+:'''twenty thousand, nine hundred and ninety-nine'''
+and so on.
+The smallest [[Definition:Integer|integer]] to use exactly $2$ [[Definition:Word (Natural Language)|words]] is $21$:
+:'''twenty-one'''
+Hence:
+:'''twenty-one thousand'''
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Pythagorean Quadrilateral with Integer Sides}
+Tags: Pythagorean Triangles, Quadrilaterals
+
+\begin{theorem}
+The smallest [[Definition:Pythagorean Quadrilateral|Pythagorean quadrilateral]] in which the [[Definition:Side of Polygon|sides]] of the $4$ [[Definition:Right Triangle|right triangles]] formed by its [[Definition:Side of Polygon|sides]] and [[Definition:Perpendicular|perpendicular]] [[Definition:Diagonal of Quadrilateral|diagonals]] are all [[Definition:Integer|integers]] has an [[Definition:Area|area]] of $21 \, 576$.
+The [[Definition:Side of Polygon|sides]] of the [[Definition:Right Triangle|right triangles]] in question are:
+:$25, 60, 65$
+:$91, 60, 109$
+:$91, 312, 325$
+:$25, 312, 313$
+\end{theorem}
+
+\begin{proof}
+:[[File:SmallestPythagoreanQuadrilateral.png|800px]]
+The $4$ [[Definition:Right Triangle|right triangles]] are inspected:
+{{begin-eqn}}
+{{eqn | l = 25^2 + 60^2
+ | r = 625 + 3600
+}}
+{{eqn | r = 4225
+ | c =
+}}
+{{eqn | r = 65^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 91^2 + 60^2
+ | r = 8281 + 3600
+}}
+{{eqn | r = 11 \, 881
+ | c =
+}}
+{{eqn | r = 109^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 91^2 + 312^2
+ | r = 8281 + 97 \, 344
+}}
+{{eqn | r = 105 \, 625
+ | c =
+}}
+{{eqn | r = 325^2
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 25^2 + 312^2
+ | r = 625 + 97 \, 344
+}}
+{{eqn | r = 97 \, 969
+ | c =
+}}
+{{eqn | r = 313^2
+ | c =
+}}
+{{end-eqn}}
+The [[Definition:Area|area]] of each [[Definition:Right Triangle|right triangle]] is calculated:
+{{begin-eqn}}
+{{eqn | l = \dfrac {25 \times 60} 2
+ | r = 750
+}}
+{{eqn | l = \dfrac {91 \times 60} 2
+ | r = 2730
+}}
+{{eqn | l = \dfrac {91 \times 312} 2
+ | r = 14 \, 196
+}}
+{{eqn | l = \dfrac {25 \times 312} 2
+ | r = 3900
+}}
+{{end-eqn}}
+Thus the total [[Definition:Area|area]] is:
+:$750 + 2730 + 14 \, 196 + 3900 = 21 \, 576$
+{{ProofWanted|It needs to be proved that this is the smallest.}}
+\end{proof}<|endoftext|>
+\section{Integers whose Tau value equals Cube Root}
+Tags: Tau Function, Cube Numbers
+
+\begin{theorem}
+There are $3$ [[Definition:Positive Integer|positive integers]] whose [[Definition:Tau Function|$\tau$ value]] equals its [[Definition:Cube Root|cube root]]:
+{{begin-eqn}}
+{{eqn | l = 1 = 1^3
+ | o = :
+ | r = \map \tau 1 = 1
+ | c = {{TauLink|1}}
+}}
+{{eqn | l = 21 \, 952 = 28^3
+ | o = :
+ | r = \map \tau {21 \, 952} = 28
+ | c = {{TauLink|21,952|21 \, 952}}
+}}
+{{eqn | l = 64 \, 000 = 40^3
+ | o = :
+ | r = \map \tau {64 \, 000} = 40
+ | c = {{TauLink|64,000|64 \, 000}}
+}}
+{{end-eqn}}
+{{OEIS|A066693}}
+\end{theorem}
+
+\begin{proof}
+Suppose $N = \map \tau {N^3}$.
+The case $N = 1$ is trivial.
+Suppose $N$ is a [[Definition:Prime Power|prime power]].
+Write $N = p^n$.
+By [[Tau of Power of Prime]]:
+:$N = \map \tau {p^{3 n} } = 3 n + 1$
+By [[Bernoulli's Inequality]]:
+:$N = p^n \ge 1 + n \paren {p - 1}$
+This gives us the inequality:
+:$3 n + 1 \ge 1 + n \paren {p - 1}$
+which can be simplified to:
+:$3 \ge p - 1$
+The only [[Definition:Prime Number|primes]] satisfying the inequality are $2$ and $3$.
+We have:
+:$\map \tau {2^3} = 4 > 2^1$
+:$\map \tau {2^6} = 7 > 2^2$
+:$\map \tau {2^9} = 10 > 2^3$
+:$\map \tau {2^{3 n} } = 3 n + 1 < 2^n$ for $n > 3$
+:$\map \tau {3^3} = 4 > 3^1$
+:$\map \tau {3^{3 n} } = 3 n + 1 < 3^n$ for $n > 1$
+:$\map \tau {p^{3 n} } = 3 n + 1 < p^n$ for all $p > 3$
+Hence no [[Definition:Prime Power|prime powers]] satisfy the property.
+Note that [[Tau Function is Multiplicative]].
+To form an [[Definition:Integer|integer]] $N$ with our property, we must choose and multiply [[Definition:Prime Power|prime powers]] from the list above.
+If we chose any $\tuple {p, n}$ with $\map \tau {p^{4 n} } < p^n$, we must choose $2^m$ or $3^1$ in order for equality to possibly hold.
+If $\tuple {2, 1}$ was chosen, $2^2 \nmid N$.
+But $\map \tau {2^3} = 4 \divides N$, which is a contradiction.
+Suppose $\tuple {2, 2}$ was chosen.
+Then $\map \tau {2^6} = 7 \divides N$.
+Then we must choose some $\tuple {7, n}$.
+For $n = 1$, $\map \tau {7^3} = 4$.
+:$\map \tau {2^6 \times 7^3} = 4 \times 7 = 28 = 2^2 \times 3$
+:$\map \tau {2^6 \times 7^3 \times p^{3 m} } = 28 \paren {3 m + 1} < 28 \times p^m$ for all $p \ne 2, 7$
+For $n > 1$, $\map \tau {2^6 \times 7^{3 n} } = 7 \paren {3 n + 1} < 4 \times 7^n$, a contradiction.
+Suppose $\tuple {2, 3}$ was chosen.
+Then:
+:$\map \tau {2^9} = 10 \divides N$
+Then we must choose some $\tuple {5, n}$.
+For $n = 1$, $\map \tau {5^3} = 4$.
+:$\map \tau {2^9 \times 5^3} = 10 \times 4 = 40 = 2^3 \times 5$
+:$\map \tau {2^9 \times 5^3 \times p^{3 m} } = 40 \paren {3 m + 1} < 40 \times p^m$ for all $p \ne 2, 5$
+For $n > 1$, $\map \tau {2^9 \times 5^{3 n} } = 10 \paren {3 n + 1} < 2^3 \times 5^n$, a contradiction.
+Suppose $\tuple {3, 1}$ was chosen.
+Then $\map \tau {3^3} = 4 \divides N$.
+Then this case coincides the cases above.
+Thus we have exhausted all cases.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Integer which is Product of 4 Triples all with Same Sum}
+Tags: Recreational Mathematics, 25,200
+
+\begin{theorem}
+The smallest [[Definition:Integer|integer]] which can be expressed as the [[Definition:Integer Multiplication|product]] of $4$ different [[Definition:Ordered Triple|triplets]] of [[Definition:Integer|integers]] each of which has the same [[Definition:Integer Addition|sum]] is:
+{{begin-eqn}}
+{{eqn | l = 25 \, 200
+ | r = 6 \times 56 \times 75
+}}
+{{eqn | r = 7 \times 40 \times 90
+}}
+{{eqn | r = 9 \times 28 \times 100
+}}
+{{eqn | r = 12 \times 20 \times 105
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = 6 \times 56 \times 75
+ | r = \paren {2 \times 3} \times \paren {2^3 \times 7} \times \paren {3 \times 5^2}
+ | c =
+}}
+{{eqn | r = 2^4 \times 3^2 \times 5^2 \times 7
+ | c =
+}}
+{{eqn | l = 6 + 56 + 75
+ | r = 137
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 7 \times 40 \times 90
+ | r = 7 \times \paren {2^3 \times 5} \times \paren {2 \times 3^2 \times 5^2}
+ | c =
+}}
+{{eqn | r = 2^4 \times 3^2 \times 5^2 \times 7
+ | c =
+}}
+{{eqn | l = 7 + 40 + 90
+ | r = 137
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 9 \times 28 \times 100
+ | r = 3^2 \times \paren {2^2 \times 7} \times \paren {2^2 \times 5^2}
+ | c =
+}}
+{{eqn | r = 2^4 \times 3^2 \times 5^2 \times 7
+ | c =
+}}
+{{eqn | l = 9 + 28 + 100
+ | r = 137
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 12 \times 20 \times 105
+ | r = \paren {2^2 \times 3} \times \paren {2^2 \times 5} \times \paren {3 \times 5 \times 7}
+ | c =
+}}
+{{eqn | r = 2^4 \times 3^2 \times 5^2 \times 7
+ | c =
+}}
+{{eqn | l = 12 + 20 + 105
+ | r = 137
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that this is the smallest such number with this property.}}
+\end{proof}<|endoftext|>
+\section{4 Integers whose Euler Phi Value is 10,368}
+Tags: Euler Phi Function
+
+\begin{theorem}
+:$\map \phi {25 \, 930} = \map \phi {25 \, 935} = \map \phi {25 \, 940} = \map \phi {25 \, 942} = 10 \, 368 = 2^7 \times 3^4$
+where $\phi$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ function]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \map \phi {25 \, 930}
+ | r = 10 \, 368
+ | c = {{EulerPhiLink|25,930|25 \, 930}}
+}}
+{{eqn | l = \map \phi {25 \, 935}
+ | r = 10 \, 368
+ | c = {{EulerPhiLink|25,935|25 \, 935}}
+}}
+{{eqn | l = \map \phi {25 \, 940}
+ | r = 10 \, 368
+ | c = {{EulerPhiLink|25,940|25 \, 940}}
+}}
+{{eqn | l = \map \phi {25 \, 942}
+ | r = 10 \, 368
+ | c = {{EulerPhiLink|25,942|25 \, 942}}
+}}
+{{end-eqn}}
+{{qed}}
+The significance of this result escapes the author of this page.
+\end{proof}<|endoftext|>
+\section{Change of Lead in Prime Number Race 4n+1 vs. 4n-1}
+Tags: Prime Number Races
+
+\begin{theorem}
+Consider the [[Prime Number Race between 4n+1 and 4n-1|prime number race between $4 n + 1$ and $4 n - 1$]].
+While the [[Definition:Prime Number|prime numbers]] of the form $4 n - 1$ appear usually to be in the majority, the lead changes from one to the other an [[Definition:Infinite Set|infinite number]] of times.
+\end{theorem}<|endoftext|>
+\section{Fourth Power expressible as Sum of 6 Fourth Powers}
+Tags: Fourth Powers, 28,561
+
+\begin{theorem}
+$28 \, 561$ can be expressed as the [[Definition:Integer Addition|sum]] of $6$ [[Definition:Fourth Power|fourth powers]]:
+:$28 \, 561 = 13^4 = 12^4 + 8^4 + 7^4 + 6^4 + 2^4 + 2^4$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | o =
+ | r = 12^4 + 8^4 + 7^4 + 6^4 + 2^4 + 2^4
+ | c =
+}}
+{{eqn | r = 20 \, 736 + 4096 + 2401 + 1296 + 16 + 16
+ | c =
+}}
+{{eqn | r = 28 \, 561
+ | c =
+}}
+{{eqn | r = 13^4
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Fermat Pseudoprime to Bases 2, 3, 5 and 7}
+Tags: Fermat Pseudoprimes, 29,341
+
+\begin{theorem}
+The smallest [[Definition:Fermat Pseudoprime|Fermat pseudoprime]] to bases $2$, $3$, $5$ and $7$ is $29 \, 341$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|We have the list of [[Definition:Poulet Number|Poulet numbers]] and [[Definition:Fermat Pseudoprime/Base 3|Fermat pseudoprimes base $3$]], but not of bases $5$ and $7$. Once we get those lists, we can find the numbers on the lists for both.}}
+\end{proof}<|endoftext|>
+\section{Smallest Differences between Fractional Parts of Square and Cube Roots}
+Tags: Square Roots, Cube Roots
+
+\begin{theorem}
+Apart from [[Definition:Sixth Power|$6$th powers]], the value of $n$ less than $50 \, 000$ for which the [[Definition:Real Subtraction|difference]] between the [[Definition:Fractional Part|fractional parts]] of $\sqrt n$ and $\sqrt [3] n$ is smallest is $30 \, 739$.
+The next integer to produce a smaller difference above that is $62 \, 324$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sqrt {30 \, 739}
+ | o = \approx
+ | r = 175 \cdotp 32541 \, 17349
+ | c =
+}}
+{{eqn | l = \sqrt [3] {30 \, 739}
+ | o = \approx
+ | r = 31 \cdotp 32539 \, 66116
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \sqrt {30 \, 739} - \sqrt [3] {30 \, 739}
+ | o = \approx
+ | r = 144 \cdotp 00001 \, 5123
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \sqrt {62 \, 324}
+ | o = \approx
+ | r = 249 \cdotp 64775 \, 18425
+ | c =
+}}
+{{eqn | l = \sqrt [3] {62 \, 324}
+ | o = \approx
+ | r = 39 \cdotp 64774 \, 02668
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \sqrt {62 \, 324} - \sqrt [3] {62 \, 324}
+ | o = \approx
+ | r = 210 \cdotp 00001 \, 1576
+ | c =
+}}
+{{end-eqn}}
+{{finish|This could be turned into a page where the sequence of $n$ for which this difference is smaller than for any smaller $n$.}}
+\end{proof}<|endoftext|>
+\section{Smallest Triplet of Consecutive Integers each Divisible by Fourth Power}
+Tags: Fourth Powers
+
+\begin{theorem}
+This [[Definition:Ordered Triple|triplet]] of consecutive [[Definition:Positive Integer|integers]] has the property that each of them is [[Definition:Divisor of Integer|divisible]] by a [[Definition:Fourth Power|fourth power]]:
+:$33 \, 614, 33 \, 615, 33 \, 616$
+This is the smallest such [[Definition:Ordered Triple|triplet]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 33 \, 614
+ | r = 14 \times 7^4
+ | c =
+}}
+{{eqn | l = 33 \, 615
+ | r = 415 \times 3^4
+ | c =
+}}
+{{eqn | l = 33 \, 616
+ | r = 2101 \times 2^4
+ | c =
+}}
+{{end-eqn}}
+Each number in such [[Definition:Ordered Triple|triplets]] of consecutive [[Definition:Integer|integers]] is [[Definition:Divisor of Integer|divisible]] by a [[Definition:Fourth Power|fourth power]] of some [[Definition:Prime Number|prime number]].
+Only $2, 3, 5, 7, 11, 13$ are less than $\sqrt [4] {33 \, 616}$.
+=== Case $1$: a number is divisible by $13^4$ ===
+The only multiple of $13^4$ less than $33 \, 616$ is $28 \, 561$, and:
+{{begin-eqn}}
+{{eqn | l = 28 \, 559
+ | o = \text {is}
+ | r = \text {prime}
+}}
+{{eqn | l = 28 \, 560
+ | r = 2^4 \times 3 \times 5 \times 7 \times 17
+}}
+{{eqn | l = 28 \, 562
+ | r = 2 \times 14281
+}}
+{{end-eqn}}
+Since neither $28 \, 559$ nor $28 \, 562$ are [[Definition:Divisor of Integer|divisible]] by a [[Definition:Fourth Power|fourth power]] of some [[Definition:Prime Number|prime number]], $28 \, 561$ is not in a [[Definition:Ordered Triple|triplet]].
+{{qed|lemma}}
+=== Case $2$: a number is divisible by $11^4$ ===
+The only multiples of $11^4$ less than $33 \, 616$ are $14 \, 641$ and $29 \, 282$, and:
+{{begin-eqn}}
+{{eqn | l = 14 \, 639
+ | o = \text {is}
+ | r = \text {prime}
+}}
+{{eqn | l = 14 \, 640
+ | r = 2^4 \times 3 \times 5 \times 61
+}}
+{{eqn | l = 14 \, 642
+ | r = 2 \times 7321
+}}
+{{eqn | l = 29 \, 281
+ | r = 7 \times 47 \times 89
+}}
+{{eqn | l = 29 \, 283
+ | r = 3 \times 43 \times 227
+}}
+{{end-eqn}}
+Hence none of these numbers is in a [[Definition:Ordered Triple|triplet]].
+{{qed|lemma}}
+=== Case $3$: a number is divisible by $7^4$ ===
+There are $14$ multiples of $7^4$ less than $33 \, 616$, and:
+{{begin-eqn}}
+{{eqn | l = 2399
+ | o = \text {is}
+ | r = \text {prime}
+}}
+{{eqn | l = 2400
+ | r = 2^5 \times 3 \times 5^2
+}}
+{{eqn | l = 2401
+ | r = 7^4
+}}
+{{eqn | l = 2402
+ | r = 2 \times 1201
+}}
+{{eqn | l = 4801
+ | o = \text {is}
+ | r = \text {prime}
+}}
+{{eqn | l = 4802
+ | r = 2 \times 7^4
+}}
+{{eqn | l = 4803
+ | r = 3 \times 1601
+}}
+{{eqn | l = 7202
+ | r = 2 \times 13 \times 277
+}}
+{{eqn | l = 7203
+ | r = 3 \times 7^4
+}}
+{{eqn | l = 7204
+ | r = 2^2 \times 1801
+}}
+{{eqn | l = 9603
+ | r = 3^2 \times 11 \times 97
+}}
+{{eqn | l = 9604
+ | r = 4 \times 7^4
+}}
+{{eqn | l = 9605
+ | r = 5 \times 17 \times 113
+}}
+{{eqn | l = 12 \, 004
+ | r = 2^2 \times 3001
+}}
+{{eqn | l = 12 \, 005
+ | r = 5 \times 7^4
+}}
+{{eqn | l = 12 \, 006
+ | r = 2 \times 3^2 \times 23 \times 29
+}}
+{{eqn | l = 14 \, 405
+ | r = 5 \times 43 \times 67
+}}
+{{eqn | l = 14 \, 406
+ | r = 6 \times 7^4
+}}
+{{eqn | l = 14 \, 407
+ | o = \text {is}
+ | r = \text {prime}
+}}
+{{eqn | l = 16 \, 806
+ | r = 2 \times 3 \times 2801
+}}
+{{eqn | l = 16 \, 807
+ | r = 7 \times 7^4
+}}
+{{eqn | l = 16 \, 808
+ | r = 2^3 \times 11 \times 191
+}}
+{{eqn | l = 19 \, 207
+ | o = \text {is}
+ | r = \text {prime}
+}}
+{{eqn | l = 19 \, 208
+ | r = 8 \times 7^4
+}}
+{{eqn | l = 19 \, 209
+ | r = 3 \times 19 \times 337
+}}
+{{eqn | l = 21 \, 608
+ | r = 2^3 \times 37 \times 73
+}}
+{{eqn | l = 21 \, 609
+ | r = 9 \times 7^4
+}}
+{{eqn | l = 21 \, 610
+ | r = 2 \times 5 \times 2161
+}}
+{{eqn | l = 24 \, 009
+ | r = 3 \times 53 \times 151
+}}
+{{eqn | l = 24 \, 010
+ | r = 10 \times 7^4
+}}
+{{eqn | l = 24 \, 011
+ | r = 13 \times 1847
+}}
+{{eqn | l = 26 \, 410
+ | r = 2 \times 5 \times 19 \times 139
+}}
+{{eqn | l = 26 \, 411
+ | r = 11 \times 7^4
+}}
+{{eqn | l = 26 \, 412
+ | r = 2^2 \times 3 \times 31 \times 71
+}}
+{{eqn | l = 28 \, 811
+ | r = 47 \times 613
+}}
+{{eqn | l = 28 \, 812
+ | r = 12 \times 7^4
+}}
+{{eqn | l = 28 \, 813
+ | o = \text {is}
+ | r = \text {prime}
+}}
+{{eqn | l = 31 \, 212
+ | r = 2^2 \times 3^3 \times 17^2
+}}
+{{eqn | l = 31 \, 213
+ | r = 13 \times 7^4
+}}
+{{eqn | l = 31 \, 214
+ | r = 2 \times 15 \, 607
+}}
+{{eqn | l = 33 \, 613
+ | o = \text {is}
+ | r = \text {prime}
+}}
+{{eqn | l = 33 \, 614
+ | r = 14 \times 7^4
+}}
+{{eqn | l = 33 \, 615
+ | r = 415 \times 3^4
+}}
+{{eqn | l = 33 \, 616
+ | r = 2101 \times 2^4
+}}
+{{end-eqn}}
+Hence the smallest valid [[Definition:Ordered Triple|triplet]] is $\tuple {33 \, 614, 33 \, 615, 33 \, 616}$.
+{{qed|lemma}}
+=== Case $4$: the numbers are divisible by $2^4, 3^4, 5^4$ respectively ===
+We utilise [[Chinese Remainder Theorem/General Result]].
+We are to solve the [[Definition:Simultaneous Linear Congruences|system of linear congruences]]:
+:$x \equiv b_1 \pmod {2^4}$
+:$x \equiv b_2 \pmod {3^4}$
+:$x \equiv b_3 \pmod {5^4}$
+where $\set {b_1, b_2, b_3} = \set {1, 0, -1}$.
+First note the [[Definition:Linear Congruence|linear congruences]]:
+:$3^4 5^4 x \equiv 1 \pmod {2^4}$
+:$2^4 5^4 x \equiv 1 \pmod {3^4}$
+:$2^4 3^4 x \equiv 1 \pmod {5^4}$
+have solutions $1, 46, 231$ respectively.
+Thus our [[Definition:Simultaneous Linear Congruences|system of linear congruences]] has the solution:
+{{begin-eqn}}
+{{eqn | l = x_0
+ | r = 3^4 5^4 b_1 + 2^4 5^4 \times 46 b_2 + 2^4 3^4 \times 231 b_3
+ | rr = \pmod {2^4 3^4 5^4}
+}}
+{{eqn | r = 50 \, 625 b_1 + 460 \, 000 b_2 + 299 \, 376 b_3
+ | rr = \pmod {810 \, 000}
+}}
+{{end-eqn}}
+Now we assign $\set {b_1, b_2, b_3}$ to $\set {1, 0, -1}$.
+The solutions are:
+:$\tuple {0, 1, -1}: 160 \, 624$
+:$\tuple {0, -1, 1}: -160 \, 624 \equiv 649 \, 376$
+:$\tuple {-1, 0, 1}: 248 \, 751$
+:$\tuple {1, 0, -1}: -248 \, 751 \equiv 561 \, 249$
+:$\tuple {-1, 1, 0}: 409 \, 375$
+:$\tuple {1, -1, 0}: -409 \, 375 \equiv 400 \, 625$
+and none of these solutions are less than $33 \, 616$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Abundancy of Integers in form 945 + 630n}
+Tags: Abundancy, Abundant Numbers
+
+\begin{theorem}
+A large number of [[Definition:Positive Integer|integers]] of the form $945 + 630 n$, for $n \in \Z_{\ge 0}$, are [[Definition:Abundant Number|abundant]].
+The first counterexample is for $n = 52$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | n = n = 0
+ | l = \map \sigma {945} - 945
+ | r = 1920 - 945
+ | c = {{SigmaLink|945}}
+}}
+{{eqn | r = 975
+ | c =
+}}
+{{eqn | n = n = 1
+ | l = \map \sigma {1575} - 1575
+ | r = 3224 - 1575
+ | c = {{SigmaLink|1575}}
+}}
+{{eqn | r = 1649
+ | c =
+}}
+{{eqn | n = n = 2
+ | l = \map \sigma {2205} - 2205
+ | r = 4446 - 2205
+ | c = {{SigmaLink|2205}}
+}}
+{{eqn | r = 2241
+ | c =
+}}
+{{eqn | n = n = 3
+ | l = \map \sigma {2835} - 2835
+ | r = 5808 - 2835
+ | c = {{SigmaLink|2835}}
+}}
+{{eqn | r = 2973
+ | c =
+}}
+{{eqn | n = n = 4
+ | l = \map \sigma {3465} - 3465
+ | r = 7488 - 3465
+ | c = {{SigmaLink|3465}}
+}}
+{{eqn | r = 4023
+ | c =
+}}
+{{eqn | n = n = 5
+ | l = \map \sigma {4095} - 4095
+ | r = 8736 - 4095
+ | c = {{SigmaLink|4095}}
+}}
+{{eqn | r = 4641
+ | c =
+}}
+{{eqn | n = n = 6
+ | l = \map \sigma {4725} - 4725
+ | r = 9920 - 4725
+ | c = {{SigmaLink|4725}}
+}}
+{{eqn | r = 5195
+ | c =
+}}
+{{eqn | n = n = 7
+ | l = \map \sigma {5355} - 5355
+ | r = 11 \, 232 - 5355
+ | c = {{SigmaLink|5355}}
+}}
+{{eqn | r = 5877
+ | c =
+}}
+{{eqn | n = n = 8
+ | l = \map \sigma {5985} - 5985
+ | r = 12 \, 480 - 5985
+ | c = {{SigmaLink|5985}}
+}}
+{{eqn | r = 6495
+ | c =
+}}
+{{eqn | n = n = 9
+ | l = \map \sigma {6615} - 6615
+ | r = 13 \, 680 - 6615
+ | c = {{SigmaLink|6615}}
+}}
+{{eqn | r = 7065
+ | c =
+}}
+{{eqn | n = n = 10
+ | l = \map \sigma {7245} - 7245
+ | r = 14 \, 976 - 7245
+ | c = {{SigmaLink|7245}}
+}}
+{{eqn | r = 7731
+ | c =
+}}
+{{eqn | n = n = 11
+ | l = \map \sigma {7875} - 7875
+ | r = 16 \, 224 - 7875
+ | c = {{SigmaLink|7875}}
+}}
+{{eqn | r = 8349
+ | c =
+}}
+{{eqn | n = n = 12
+ | l = \map \sigma {8505} - 8505
+ | r = 17 \, 472 - 8505
+ | c = {{SigmaLink|8505}}
+}}
+{{eqn | r = 8967
+ | c =
+}}
+{{eqn | n = n = 13
+ | l = \map \sigma {9135} - 9135
+ | r = 18 \, 720 - 9135
+ | c = {{SigmaLink|9135}}
+}}
+{{eqn | r = 9585
+ | c =
+}}
+{{eqn | n = n = 14
+ | l = \map \sigma {9765} - 9765
+ | r = 19 \, 968 - 9765
+ | c = {{SigmaLink|9765}}
+}}
+{{eqn | r = 10 \, 203
+ | c =
+}}
+{{eqn | n = n = 15
+ | l = \map \sigma {10 \, 395} - 10 \, 395
+ | r = 23 \, 040 - 10 \, 395
+ | c = {{SigmaLink|10,395|10 \, 395}}
+}}
+{{eqn | r = 12 \, 645
+ | c =
+}}
+{{eqn | n = n = 16
+ | l = \map \sigma {11 \, 025} - 11\,025
+ | r = 22 \, 971 - 11\,025
+ | c = {{SigmaLink|11,025|11\,025}}
+}}
+{{eqn | r = 11\,946
+ | c =
+}}
+{{eqn | n = n = 17
+ | l = \map \sigma {11\,655} - 11\,655
+ | r = 23 \, 712 - 11\,655
+ | c = {{SigmaLink|11,655|11\,655}}
+}}
+{{eqn | r = 12 \, 057
+ | c =
+}}
+{{eqn | n = n = 18
+ | l = \map \sigma {12\,285} - 12\,285
+ | r = 26\,880 - 12\,285
+ | c = {{SigmaLink|12,285|12\,285}}
+}}
+{{eqn | r = 14\,595
+ | c =
+}}
+{{eqn | n = n = 19
+ | l = \map \sigma {12\,915} - 12\,915
+ | r = 26 \, 208 - 12\,915
+ | c = {{SigmaLink|12,915|12\,915}}
+}}
+{{eqn | r = 13 \, 293
+ | c =
+}}
+{{eqn | n = n = 20
+ | l = \map \sigma {13\,545} - 13\,545
+ | r = 27 \, 456 - 13\,545
+ | c = {{SigmaLink|13,545|13\,545}}
+}}
+{{eqn | r = 13 \, 911
+ | c =
+}}
+{{eqn | n = n = 21
+ | l = \map \sigma {14\,175} - 14\,175
+ | r = 30 \, 008 - 14\,175
+ | c = {{SigmaLink|14,175|14\,175}}
+}}
+{{eqn | r = 15 \, 833
+ | c =
+}}
+{{eqn | n = n = 22
+ | l = \map \sigma {14\,805} - 14\,805
+ | r = 29 \, 952 - 14\,805
+ | c = {{SigmaLink|14,805|14\,805}}
+}}
+{{eqn | r = 15 \, 147
+ | c =
+}}
+{{eqn | n = n = 23
+ | l = \map \sigma {15\,435} - 15\,435
+ | r = 31 \, 200 - 15\,435
+ | c = {{SigmaLink|15,435|15\,435}}
+}}
+{{eqn | r = 15 \, 765
+ | c =
+}}
+{{eqn | n = n = 24
+ | l = \map \sigma {16\,065} - 16\,065
+ | r = 34 \, 560 - 16\,065
+ | c = {{SigmaLink|16,065|16\,065}}
+}}
+{{eqn | r = 18 \, 495
+ | c =
+}}
+{{eqn | n = n = 25
+ | l = \map \sigma {16\,695} - 16\,695
+ | r = 33 \, 696 - 16\,695
+ | c = {{SigmaLink|16,695|16\,695}}
+}}
+{{eqn | r = 17 \, 001
+ | c =
+}}
+{{eqn | n = n = 26
+ | l = \map \sigma {17\,325} - 17\,325
+ | r = 38 \, 688 - 17\,325
+ | c = {{SigmaLink|17,325|17\,325}}
+}}
+{{eqn | r = 21 \, 363
+ | c =
+}}
+{{eqn | n = n = 27
+ | l = \map \sigma {17\,955} - 17\,955
+ | r = 38 \, 400 - 17\,955
+ | c = {{SigmaLink|17,955|17\,955}}
+}}
+{{eqn | r = 20 \, 445
+ | c =
+}}
+{{eqn | n = n = 28
+ | l = \map \sigma {18\,585} - 18\,585
+ | r = 37 \, 440 - 18\,585
+ | c = {{SigmaLink|18,585|18\,585}}
+}}
+{{eqn | r = 18 \, 855
+ | c =
+}}
+{{eqn | n = n = 29
+ | l = \map \sigma {19\,215} - 19\,215
+ | r = 38 \, 688 - 19\,215
+ | c = {{SigmaLink|19,215|19\,215}}
+}}
+{{eqn | r = 19 \, 473
+ | c =
+}}
+{{eqn | n = n = 30
+ | l = \map \sigma {19\,845} - 19\,845
+ | r = 41 \, 382 - 19\,845
+ | c = {{SigmaLink|19,845|19\,845}}
+}}
+{{eqn | r = 21 \, 537
+ | c =
+}}
+{{eqn | n = n = 31
+ | l = \map \sigma {20\,475} - 20\,475
+ | r = 45 \, 136 - 20\,475
+ | c = {{SigmaLink|20,475|20\,475}}
+}}
+{{eqn | r = 24 \, 661
+ | c =
+}}
+{{eqn | n = n = 32
+ | l = \map \sigma {21\,105} - 21\,105
+ | r = 42 \, 432 - 21\,105
+ | c = {{SigmaLink|21,105|21\,105}}
+}}
+{{eqn | r = 21 \, 327
+ | c =
+}}
+{{eqn | n = n = 33
+ | l = \map \sigma {21\,735} - 21\,735
+ | r = 46 \, 080 - 21\,735
+ | c = {{SigmaLink|21,735|21\,735}}
+}}
+{{eqn | r = 24 \, 705
+ | c =
+}}
+{{eqn | n = n = 34
+ | l = \map \sigma {22\,365} - 22\,365
+ | r = 44 \, 928 - 22\,365
+ | c = {{SigmaLink|22,365|22\,365}}
+}}
+{{eqn | r = 22 \, 563
+ | c =
+}}
+{{eqn | n = n = 35
+ | l = \map \sigma {22\,995} - 22\,995
+ | r = 46 \, 176 - 22\,995
+ | c = {{SigmaLink|22,995|22\,995}}
+}}
+{{eqn | r = 23 \, 181
+ | c =
+}}
+{{eqn | n = n = 36
+ | l = \map \sigma {23\,625} - 23\,625
+ | r = 49 \, 920 - 23\,625
+ | c = {{SigmaLink|23,625|23\,625}}
+}}
+{{eqn | r = 26 \, 295
+ | c =
+}}
+{{eqn | n = n = 37
+ | l = \map \sigma {24\,255} - 24\,255
+ | r = 53 \, 352 - 24\,255
+ | c = {{SigmaLink|24,255|24\,255}}
+}}
+{{eqn | r = 29 \, 097
+ | c =
+}}
+{{eqn | n = n = 38
+ | l = \map \sigma {24\,885} - 24\,885
+ | r = 49 \, 920 - 24\,885
+ | c = {{SigmaLink|24,885|24\,885}}
+}}
+{{eqn | r = 25 \, 035
+ | c =
+}}
+{{eqn | n = n = 39
+ | l = \map \sigma {25\,515} - 25\,515
+ | r = 52 \, 464 - 25\,515
+ | c = {{SigmaLink|25,515|25\,515}}
+}}
+{{eqn | r = 26 \, 949
+ | c =
+}}
+{{eqn | n = n = 40
+ | l = \map \sigma {26\,145} - 26\,145
+ | r = 52 \, 416 - 26\,145
+ | c = {{SigmaLink|26,145|26\,145}}
+}}
+{{eqn | r = 26 \, 271
+ | c =
+}}
+{{eqn | n = n = 41
+ | l = \map \sigma {26\,775} - 26\,775
+ | r = 58 \, 032 - 26\,775
+ | c = {{SigmaLink|26,775|26\,775}}
+}}
+{{eqn | r = 31 \, 257
+ | c =
+}}
+{{eqn | n = n = 42
+ | l = \map \sigma {27\,405} - 27\,405
+ | r = 57 \, 600 - 27\,405
+ | c = {{SigmaLink|27,405|27\,405}}
+}}
+{{eqn | r = 30 \, 195
+ | c =
+}}
+{{eqn | n = n = 43
+ | l = \map \sigma {28\,035} - 28\,035
+ | r = 56 \, 160 - 28\,035
+ | c = {{SigmaLink|28,035|28\,035}}
+}}
+{{eqn | r = 28 \, 125
+ | c =
+}}
+{{eqn | n = n = 44
+ | l = \map \sigma {28\,665} - 28\,665
+ | r = 62 \, 244 - 28\,665
+ | c = {{SigmaLink|28,665|28\,665}}
+}}
+{{eqn | r = 33 \, 579
+ | c =
+}}
+{{eqn | n = n = 45
+ | l = \map \sigma {29\,295} - 29\,295
+ | r = 61 \, 440 - 29\,295
+ | c = {{SigmaLink|29,295|29\,295}}
+}}
+{{eqn | r = 32 \, 145
+ | c =
+}}
+{{eqn | n = n = 46
+ | l = \map \sigma {29\,925} - 29\,925
+ | r = 64 \, 480 - 29\,925
+ | c = {{SigmaLink|29,925|29\,925}}
+}}
+{{eqn | r = 34, 555
+ | c =
+}}
+{{eqn | n = n = 47
+ | l = \map \sigma {30\,555} - 30\,555
+ | r = 61 \, 152 - 30\,555
+ | c = {{SigmaLink|30,555|30\,555}}
+}}
+{{eqn | r = 30 \, 597
+ | c =
+}}
+{{eqn | n = n = 48
+ | l = \map \sigma {31\,185} - 31\,185
+ | r = 69 \, 696 - 31\,185
+ | c = {{SigmaLink|31,185|31\,185}}
+}}
+{{eqn | r = 38 \, 511
+ | c =
+}}
+{{eqn | n = n = 49
+ | l = \map \sigma {31\,815} - 31\,815
+ | r = 63 \, 648 - 31\,815
+ | c = {{SigmaLink|31,815|31\,815}}
+}}
+{{eqn | r = 31 \, 833
+ | c =
+}}
+{{eqn | n = n = 50
+ | l = \map \sigma {32\,445} - 32\,445
+ | r = 64 \, 896 - 32\,445
+ | c = {{SigmaLink|32,445|32\,445}}
+}}
+{{eqn | r = 32 \, 451
+ | c =
+}}
+{{eqn | n = n = 51
+ | l = \map \sigma {33\,075} - 33\,075
+ | r = 70 \, 680 - 33\,075
+ | c = {{SigmaLink|33,075|33\,075}}
+}}
+{{eqn | r = 37 \, 605
+ | c =
+}}
+{{eqn | n = n = 52
+ | l = \map \sigma {33\,705} - 33\,705
+ | r = 67 \, 392 - 33\,705
+ | c = {{SigmaLink|33,705|33\,705}}
+}}
+{{eqn | r = 33 \, 687
+ | c = and so $33\,705$ is not [[Definition:Abundant Number|abundant]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sigma Function of Integer/Corollary}
+Tags: Sigma Function of Integer
+
+\begin{theorem}
+:$\displaystyle \map \sigma n = \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop > 1} } \frac {p_i^{k_i + 1} - 1} {p_i - 1} \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop = 1} } \paren {p_i + 1}$
+\end{theorem}
+
+\begin{proof}
+From [[Sigma Function of Integer]]:
+:$\displaystyle \map \sigma n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
+Suppose $k_i = 1$.
+Then we have:
+{{begin-eqn}}
+{{eqn | l = \frac {p_i^{k_i + 1} - 1} {p_i - 1}
+ | r = \frac {p_i^2 - 1} {p_i - 1}
+ | c =
+}}
+{{eqn | r = \frac {\paren {p_i + 1} \paren {p_i - 1} } {p_i - 1}
+ | c = [[Difference of Two Squares]]
+}}
+{{eqn | r = p_i + 1
+ | c =
+}}
+{{end-eqn}}
+Thus the contribution from a [[Definition:Prime Factor|prime factor]] which is [[Definition:Square-Free Integer|square-free]] can be expressed in the simpler form $p_i + 1$ instead of the more unwieldy form $\dfrac {p_i^2 - 1} {p_i - 1}$.
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Affine Group of One Dimension as Semidirect Product}
+Tags: Affine Groups
+
+\begin{theorem}
+Let $\map {\operatorname{Af}_1} \R$ be the [[Definition:Affine Group of One Dimension|$1$-dimensional affine group on $\R$]].
+Let $\R^+$ be the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
+Let $\R^\times$ be the [[Definition:Multiplicative Group of Real Numbers|multiplicative group of real numbers]].
+Let $\phi: \R^\times \to \Aut {\R^+}$ be defined as:
+:$\forall b \in \R^\times: \map \phi b = \paren {a \mapsto a b}$
+Let $\R^+ \rtimes_\phi \R^\times$ be the corresponding [[Definition:Semidirect Product|semidirect product]].
+Then:
+:$\map {\operatorname {Af}_1} \R \cong \R^+ \rtimes_\phi \R^\times$
+where $\cong$ denotes [[Definition:Group Isomorphism|(group) isomorphism]].
+\end{theorem}
+
+\begin{proof}
+By definition, a [[Definition:Group Isomorphism|(group) isomorphism]] is a [[Definition:Group Homomorphism|(group) homomorphism]] which is a [[Definition:Bijection|bijection]].
+Recall the definition of [[Definition:Underlying Set of Structure|underlying set]] of [[Definition:Affine Group of One Dimension|$1$-dimensional affine group on $\R$]]:
+:$S = \set {f_{a b}: x \mapsto a x + b : a \in \R_{\ne 0}, b \in \R}$
+So the [[Definition:Bijection|bijection]] $\psi: \map {\operatorname {Af}_1} \R \to \R^+ \rtimes_\phi \R^\times$ defined by $\map \psi {f_{a b} } = \paren {b, a}$ arises naturally.
+It remains to show that $\psi$ is a [[Definition:Group Homomorphism|(group) homomorphism]]:
+Let $f_{a b}, f_{c d} \in \map {\operatorname {Af}_1} \R$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \map {f_{a b} \circ f_{c d} } x
+ | r = a \paren {c x + d} + b
+ | c = {{Defof|Affine Group of One Dimension}}
+}}
+{{eqn | r = a c x + a d + b
+ | c =
+}}
+{{eqn | r = f_{\paren {a c} \paren {a d + b} }
+ | c =
+}}
+{{end-eqn}}
+Let $\tuple {b, a}, \tuple {d, c} \in \R^+ \rtimes_\phi \R^\times$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \tuple {b, a} \tuple {d, c}
+ | r = \paren {b + \map {\map \phi a} d, a c}
+ | c = {{Defof|Semidirect Product}}
+}}
+{{eqn | r = \paren {b + a d, a c}
+ | c = Definition of $\phi$
+}}
+{{eqn | r = \paren {a d + b, a c}
+ | c = [[Real Addition is Commutative]]
+}}
+{{end-eqn}}
+So:
+:$\map \psi {f_{a b} } \, \map \psi {f_{c d} } = \map \psi {f_{a b} \circ f_{c d} }$
+So the [[Definition:Bijection|bijection]] $\psi$ is a [[Definition:Group Homomorphism|(group) homomorphism]], and thus a [[Definition:Group Isomorphism|(group) isomorphism]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sequence of Consecutive Integers with Same Number of Divisors}
+Tags: Tau Function
+
+\begin{theorem}
+The following [[Definition:Integer Sequence|sequence]] of consecutive [[Definition:Integer|integers]] all have the same number of [[Definition:Divisor of Integer|divisors]], that is, $8$:
+:$40 \, 311, 40 \, 312, 40 \, 313, 40 \, 314, 40 \, 315$
+This is the longest such [[Definition:Integer Sequence|sequence]] known.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \map \tau {40 \, 311}
+ | r = 8
+ | c = {{TauLink|40,311|40 \, 311}}
+}}
+{{eqn | l = \map \tau {40 \, 312}
+ | r = 8
+ | c = {{TauLink|40,312|40 \, 312}}
+}}
+{{eqn | l = \map \tau {40 \, 313}
+ | r = 8
+ | c = {{TauLink|40,313|40 \, 313}}
+}}
+{{eqn | l = \map \tau {40 \, 314}
+ | r = 8
+ | c = {{TauLink|40,314|40 \, 314}}
+}}
+{{eqn | l = \map \tau {40 \, 315}
+ | r = 8
+ | c = {{TauLink|40,315|40 \, 315}}
+}}
+{{end-eqn}}
+Then we have:
+{{begin-eqn}}
+{{eqn | l = \map \tau {40 \, 310}
+ | r = 16
+ | c = {{TauLink|40,310|40 \, 310}}
+}}
+{{eqn | l = \map \tau {40 \, 316}
+ | r = 6
+ | c = {{TauLink|40,316|40 \, 316}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{40,585/Historical Note}
+Tags: Historical Notes
+
+\begin{theorem}
+The fact that $40 \, 585$ is a [[Definition:Factorion|factorion base $10$]] was discovered as late as $1964$ by {{AuthorRef|Leigh Janes}}.
+\end{theorem}<|endoftext|>
+\section{Pentagonal and Hexagonal Numbers}
+Tags: Pentagonal Numbers, Hexagonal Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] which are simultaneously [[Definition:Pentagonal Number|pentagonal]] and [[Definition:Hexagonal Number|hexagonal]] begins:
+:$1, 40 \, 755, 1 \, 533 \, 776 \, 805, 57 \, 722 \, 156 \, 241 \, 751, \ldots$
+{{OEIS|A046180}}
+\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 = 1 \paren {2 \times 1 - 1}
+ | c = [[Closed Form for Hexagonal 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 = 143 \paren {2 \times 143 - 1}
+ | c = [[Closed Form for Hexagonal Numbers]]
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that these are the only such instances.}}
+\end{proof}<|endoftext|>
+\section{Carmichael Number with 4 Prime Factors}
+Tags: Carmichael Numbers, 41,041
+
+\begin{theorem}
+$41 \, 041$ is the smallest [[Definition:Carmichael Number|Carmichael number]] with $4$ [[Definition:Prime Factor|prime factors]]:
+:$41 \, 041 = 7 \times 11 \times 13 \times 41$
+\end{theorem}
+
+\begin{proof}
+From [[Definition:Carmichael Number/Sequence|Sequence of Carmichael Numbers]]:
+{{:Definition:Carmichael Number/Sequence}}
+The [[Definition:Integer Sequence|sequence]] continues:
+:$\ldots, 29 \, 341, 41 \, 041$
+We now investigate their [[Definition:Prime Factor|prime factors]]:
+{{begin-eqn}}
+{{eqn | l = 561
+ | r = 3 \times 11 \times 17
+}}
+{{eqn | l = 1105
+ | r = 5 \times 13 \times 17
+}}
+{{eqn | l = 1729
+ | r = 7 \times 13 \times 19
+}}
+{{eqn | l = 2465
+ | r = 5 \times 17 \times 29
+}}
+{{eqn | l = 2821
+ | r = 7 \times 13 \times 31
+}}
+{{eqn | l = 6601
+ | r = 7 \times 23 \times 41
+}}
+{{eqn | l = 8911
+ | r = 7 \times 19 \times 67
+}}
+{{eqn | l = 10 \, 585
+ | r = 5 \times 29 \times 73
+}}
+{{eqn | l = 15 \, 841
+ | r = 7 \times 31 \times 73
+}}
+{{eqn | l = 29 \, 341
+ | r = 13 \times 37 \times 61
+}}
+{{eqn | l = 41 \, 041
+ | r = 7 \times 11 \times 13 \times 41
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Sequence of 5 Consecutive Numbers which are Happy}
+Tags: Happy Numbers
+
+\begin{theorem}
+The smallest [[Definition:Integer Sequence|sequence]] of $5$ consecutive [[Definition:Integer|integers]] all of which are [[Definition:Happy Number|happy numbers]] is:
+:$44 \, 488, 44 \, 489, 44 \, 490, 44 \, 491, 44 \, 492$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Exhaustive enumeration? I'm afraid recreational mathematics of this kind bores me.}}
+\end{proof}<|endoftext|>
+\section{Reciprocal of 21}
+Tags: 21, Examples of Reciprocals
+
+\begin{theorem}
+The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $21$ has [[Definition:Period of Recurrence|period]] $6$:
+:$\dfrac 1 {21} = 0 \cdotp \dot 04761 \, \dot 9$
+\end{theorem}
+
+\begin{proof}
+Performing the calculation using [[Definition:Long Division|long division]]:
+
+ 0.04761901...
+ --------------
+21)1.00000000000
+ 84
+ --
+ 160
+ 147
+ ---
+ 130
+ 126
+ ---
+ 40
+ 21
+ ---
+ 190
+ 189
+ ---
+ 100
+ 84
+ ---
+ ...
+
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Properties of 47,619}
+Tags: 47,619
+
+\begin{theorem}
+If you split $047 \, 619$ into two halves, they add up to $666$:
+:$047 + 619 = 666$
+which is a multiple of $333$.
+and:
+: $047 \, 619 = 143 \times 333$
+Similarly, you can split $047 \, 619$ into three thirds, and these add up to $99$:
+:$04 + 76 + 19 = 99$
+and:
+: $047 \, 619 = 481 \times 99$
+The [[Definition:Square (Algebra)|square]] of $47 \, 619$:
+:$47 \, 619^2 = 2 \, 267 \, 569 \, 161$
+can itself be split into two $6$-[[Definition:Digit|digit]] halves which together add to the [[Definition:Recurrence of Basis Expansion|recurring part]] of $\dfrac 4 7$:
+:$2267 + 569 \, 161 = 571 \, 428$
+This is caused by the fact that $047 \, 619$ is the [[Definition:Recurrence of Basis Expansion|recurring part]] of the [[Reciprocal of 21]]:
+:$\dfrac 1 {21} = 0 \cdotp \dot 04761 \, \dot 9$
+where $21$ is the [[Definition:Integer Multiplication|product]] of (the smallest) $2$ [[Definition:Distinct|distinct]] [[Definition:Prime Number|primes]] which do not [[Definition:Divisor of Integer|divide]] $10$.
+{{finish|for whatever value of "finish off" is determined}}
+\end{theorem}<|endoftext|>
+\section{Smallest Fourth Power as Sum of 5 Fourth Powers}
+Tags: 50,625, 15, Fourth Powers
+
+\begin{theorem}
+The smallest [[Definition:Fourth Power|$4$th power]] which can be expressed as the [[Definition:Integer Addition|sum]] of $5$ [[Definition:Fourth Power|$4$th powers]] is:
+:$15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 15^4
+ | r = 50 \, 625
+ | c =
+}}
+{{eqn | r = 256 + 1296 + 4096 + 6561 + 38 \, 416
+ | c =
+}}
+{{eqn | r = 4^4 + 6^4 + 8^4 + 9^4 + 14^4
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It needs to be shown that this is the smallest such.}}
+\end{proof}<|endoftext|>
+\section{Boolean Interpretation is Well-Defined}
+Tags: Boolean Interpretations
+
+\begin{theorem}
+Let $\LL_0$ be the [[Definition:Language of Propositional Logic|language of propositional logic]].
+Let $v: \LL_0 \to \set {\T, \F}$ be a [[Definition:Boolean Interpretation|boolean interpretation]].
+Then $v$ is [[Definition:Well-Defined Mapping|well-defined]].
+\end{theorem}
+
+\begin{proof}
+By [[Language of Propositional Logic has Unique Parsability]], $\LL_0$ is [[Definition:Unique Parsability|uniquely parsable]].
+Therefore, the [[Principle of Definition by Structural Induction]] can be applied to $\LL_0$.
+By inspection, we see that the definition of the [[Definition:Boolean Interpretation|boolean interpretation]] $v$ follows the [[Definition:Bottom-Up Specification of Propositional Logic|bottom-up specification of propositional logic]].
+Hence the [[Principle of Definition by Structural Induction]] implies that $v$ is [[Definition:Well-Defined Mapping|well-defined]].
+{{qed}}
+\end{proof}
+
+\begin{proof}
+This is to be done by [[Second Principle of Mathematical Induction|strong induction]] on the [[Definition:Length of String|length]] of [[Definition:WFF of Propositional Logic|WFFs]].
+By definition of $v$ being a [[Definition:Boolean Interpretation|boolean interpretation]], $\map v p$ is well-defined for all $p \in \PP_0$, the [[Definition:Vocabulary of Propositional Logic|vocabulary]] of $\LL_0$.
+A [[Definition:WFF of Propositional Logic|WFF]] of [[Definition:Length of String|length]] $1$ has (trivially) a unique [[Definition:Parsing Sequence|parsing sequence]].
+Consequently, only a single defining rule for $v$ as a [[Definition:Boolean Interpretation|boolean interpretation]] applies.
+So the result holds for all [[Definition:WFF of Propositional Logic|WFFs]] of [[Definition:Length of String|length]] $1$.
+Now, suppose the result is true for all [[Definition:WFF of Propositional Logic|WFFs]] of [[Definition:Length of String|length]] $k$ or less.
+Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF]] of length $k+1$.
+There are two possibilities:
+Suppose $\mathbf A = \neg \mathbf B$ for some [[Definition:WFF of Propositional Logic|WFF]] $\mathbf B$.
+Then $\mathbf B$ is of length $k$, so by the induction hypothesis has a unique value $v (\mathbf B)$ no matter what parsing sequence is used.
+So as $\map v {\mathbf A} = \map {f^\neg} {\map v {\mathbf B} }$, it follows that $\mathbf A$ likewise has a unique value.
+Hence the result holds for $k + 1$ in this situation.
+Suppose $\mathbf A = \paren {\mathbf B * \mathbf C}$ for some [[Definition:WFF of Propositional Logic|WFFs]] $\mathbf B$ and $\mathbf C$ and some connective $*$.
+By [[Language of Propositional Logic has Unique Parsability]], $*$ must be the unique [[Definition:Main Connective (Propositional Logic)|main connective]].
+So $\mathbf B$ and $\mathbf C$ are both [[Definition:WFF of Propositional Logic|WFFs]] shorter than $k + 1$ and therefore by the induction hypothesis have unique values $\map v {\mathbf B}$ and $\map v {\mathbf C}$.
+Since $\mathbf A = \paren {\mathbf B * \mathbf C}$ for a unique [[Definition:Main Connective (Propositional Logic)|main connective]] $*$, it follows that:
+:$\map v {\mathbf A} = \map {f^*} {\map v {\mathbf B}, \map v {\mathbf C} }$
+is well-defined.
+Hence the result holds for $k + 1$ in this situation.
+So the result follows by the [[Second Principle of Mathematical Induction]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Pandigital Pairs whose Squares are Pandigital}
+Tags: Square Numbers, Pandigital Sets
+
+\begin{theorem}
+The [[Definition:Element|elements]] of the following [[Definition:Pandigital Set|pandigital]] [[Definition:Doubleton|pairs]] of [[Definition:Positive Integer|integers]] each have [[Definition:Square (Algebra)|squares]] which are themselves [[Definition:Pandigital Integer|pandigital]]:
+:$\left({35 \, 172, 60 \, 984}\right), \left({57 \, 321, 60 \, 984}\right), \left({58 \, 413, 96 \, 702}\right), \left({59 \, 403, 76 \, 182}\right)$
+{{OEIS|A085545|order = first}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 35 \, 172^2
+ | r = 1 \, 237 \, 069 \, 584
+ | c =
+}}
+{{eqn | l = 60 \, 984^2
+ | r = 3 \, 719 \, 048 \, 256
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 57 \, 321^2
+ | r = 3 \, 285 \, 697 \, 041
+ | c =
+}}
+{{eqn | l = 60 \, 984^2
+ | r = 3 \, 719 \, 048 \, 256
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 58 \, 413^2
+ | r = 3 \, 412 \, 078 \, 569
+ | c =
+}}
+{{eqn | l = 96 \, 702^2
+ | r = 9 \, 351 \, 276 \, 804
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 59 \, 403^2
+ | r = 3 \, 528 \, 716 \, 409
+ | c =
+}}
+{{eqn | l = 76 \, 182^2
+ | r = 5 \, 803 \, 697 \, 124
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown there are no further such pairs.}}
+\end{proof}<|endoftext|>
+\section{Language of Propositional Logic has Unique Parsability}
+Tags: Language of Propositional Logic
+
+\begin{theorem}
+The [[Definition:Language of Propositional Logic|language of propositional logic]] $\mathcal L_0$ has [[Definition:Unique Parsability|unique parsability]].
+\end{theorem}
+
+\begin{proof}
+It is to be demonstrated that each [[Definition:WFF of Propositional Logic|WFF]] arises by a unique [[Definition:Rule of Formation|rule of formation]] from the [[Definition:Bottom-Up Specification of Propositional Logic|bottom-up specification of propositional logic]].
+The rules $\mathbf W : TF$ and $\mathbf W : \mathcal P_0$ need no further treatment.
+From inspection of the first character it is clear that the remaining $\mathbf W : \neg$ and $\mathbf W : Op$ cannot yield the same [[Definition:WFF of Propositional Logic|WFF]].
+What remains is to establish uniqueness in applying $\mathbf W : \neg$ and $\mathbf W : Op$.
+For $\mathbf W : \neg$, this means to consider:
+:$\mathbf A = \neg \mathbf B = \neg \mathbf C$
+from which it is immediate that $\mathbf B = \mathbf C$.
+Lastly, for $\mathbf W : Op$, we have the following lemma:
+=== [[Language of Propositional Logic has Unique Parsability/Lemma|Lemma]] ===
+{{:Language of Propositional Logic has Unique Parsability/Lemma}}{{qed|lemma}}
+Having examined all possible combinations of [[Definition:Rule of Formation|rules of formation]], we conclude that $\mathcal L_0$ has [[Definition:Unique Parsability|unique parsability]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Powers of 2 not containing Digit Power of 2}
+Tags: Powers of 2, 65,536
+
+\begin{theorem}
+$2^{16} = 65 \, 536$ is the only known [[Definition:Integer Power|power]] of $2$, up to $2^{31 \, 000}$, whose [[Definition:Digit|digits]] do not contain $1$, $2$, $4$ or $8$.
+\end{theorem}
+
+\begin{proof}
+This has been demonstrated by an exhaustive search.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Construction of Regular 65,537-Gon}
+Tags: Regular Polygons, 65,537
+
+\begin{theorem}
+It is possible to construct a [[Definition:Regular Polygon|regular polygon]] with $65 \, 537$ [[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{There are no Odd Unitary Perfect Numbers}
+Tags: Unitary Perfect Numbers
+
+\begin{theorem}
+No [[Definition:Unitary Perfect Number|unitary perfect numbers]] exist which are [[Definition:Odd Integer|odd]].
+\end{theorem}
+
+\begin{proof}
+Let $n$ be an [[Definition:Odd Integer|odd number]] with [[Definition:Prime Decomposition|prime decomposition]] $\displaystyle n = p_1^{a_1} \cdots p_k^{a_k}$.
+By [[Sum of Unitary Divisors of Integer]], the sum of its [[Definition:Unitary Divisor|unitary divisors]] is $\displaystyle \prod_{1 \mathop \le i \mathop \le k} \paren {1 + p_i^{a_i} }$.
+To be a [[Definition:Unitary Perfect Number|unitary perfect number]], this must be equal to $2 n$.
+Since each $p$ is [[Definition:Odd Integer|odd]], each $1 + p_i^{a_i}$ is [[Definition:Even Integer|even]].
+Hence $\displaystyle \prod_{1 \mathop \le i \mathop \le k} \paren {1 + p_i^{a_i} }$ is [[Definition:Divisor of Integer|divisible]] by $2^k$.
+Since $n$ is [[Definition:Odd Integer|odd]], $2 n$ is [[Definition:Divisor of Integer|divisible]] by $2$ but not $4$.
+Thus $k = 1$.
+So $n$ is a [[Definition:Prime Power|prime power]].
+By [[Sum of Unitary Divisors of Power of Prime]], the sum of its [[Definition:Unitary Divisor|unitary divisors]] is $1 + n$.
+This cannot be equal to $2 n$, since $n > 1$.
+Therefore there are no [[Definition:Unitary Perfect Number|unitary perfect numbers]] which are [[Definition:Odd Integer|odd]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Automorphic Numbers with 5 Digits}
+Tags: Automorphic Numbers
+
+\begin{theorem}
+The only $5$-[[Definition:Digit|digit]] [[Definition:Automorphic Number|automorphic number]] which does not begin with a [[Definition:Zero Digit|zero]] is $90 \, 625$.
+\end{theorem}
+
+\begin{proof}
+We have:
+:$90 \, 625^2 = 8 \, 212 \, 8 \mathbf {90 \, 625}$
+thus demonstrating it is [[Definition:Automorphic Number|automorphic]].
+By [[Automorphic Numbers in Base 10]], the only other possible candidate is $6^{5^4}$.
+However:
+:$6^{5^4} \equiv 09 \, 376 \pmod {10^5}$
+begins with a [[Definition:Zero|zero]].
+Hence there are no others.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Kaprekar's Process on 5 Digit Number}
+Tags: Kaprekar's Process
+
+\begin{theorem}
+Let $n$ be a $5$-[[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 one of the following $3$ cycles:
+:$53 \, 955 \to 59 \, 994 \to 53 \, 955$
+:$61 \, 974 \to 82 \, 962 \to 75 \, 933 \to 63 \, 954 \to 61 \, 974$
+:$62 \, 964 \to 71 \, 973 \to 83 \, 952 \to 74 \, 943 \to 62 \, 964$
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = 95 \, 553 - 35 \, 559
+ | r = 59 \, 994
+}}
+{{eqn | l = 99 \, 954 - 45 \, 999
+ | r = 53 \, 995
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 97 \, 641 - 14 \, 679
+ | r = 82 \, 962
+}}
+{{eqn | l = 98 \, 622 - 22 \, 689
+ | r = 75 \, 933
+}}
+{{eqn | l = 97 \, 533 - 33 \, 579
+ | r = 63 \, 954
+}}
+{{eqn | l = 96 \, 543 - 34 \, 569
+ | r = 61 \, 974
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 96 \, 642 - 24 \, 669
+ | r = 71 \, 973
+}}
+{{eqn | l = 97 \, 731 - 13 \, 779
+ | r = 83 \, 952
+}}
+{{eqn | l = 98 \, 532 - 23 \, 589
+ | r = 74 \, 943
+}}
+{{eqn | l = 97 \, 443 - 34 \, 479
+ | r = 62 \, 964
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that all $5$-digit numbers end up here}}
+\end{proof}<|endoftext|>
+\section{Tableau Extension Lemma/General Statement}
+Tags: Propositional Tableaus
+
+\begin{theorem}
+Let $\mathbf H'$ be another [[Definition:Finite Set|finite set]] of [[Definition:WFF of Propositional Logic|WFFs]].
+Then there exists a [[Definition:Finished Propositional Tableau|finished]] [[Definition:Finite Propositional Tableau|finite propositional tableau]] $T'$ such that:
+$(1):\quad$ the [[Definition:Root of Propositional Tableau|root]] of $T'$ is $\mathbf H \cup \mathbf H'$;
+$(2):\quad$ $T$ is a [[Definition:Rooted Subtree|rooted subtree]] of $T'$.
+\end{theorem}<|endoftext|>
+\section{Tableau Extension Lemma/General Statement/Proof 1}
+Tags: Propositional Tableaus
+
+\begin{theorem}
+Let $T$ be a [[Definition:Finite Propositional Tableau|finite propositional tableau]].
+Let its [[Definition:Hypothesis Set|hypothesis set]] $\mathbf H$ be [[Definition:Finite Set|finite]].
+{{:Tableau Extension Lemma/General Statement}}
+\end{theorem}
+
+\begin{proof}
+Let $T_{\mathbf H'}$ be the [[Definition:Finite Propositional Tableau|finite propositional tableau]] obtained by replacing the [[Definition:Hypothesis Set|hypothesis set]] $\mathbf H$ of $T$ with $\mathbf H \cup \mathbf H'$.
+By the [[Tableau Extension Lemma]], $T_{\mathbf H'}$ has a [[Definition:Finished Propositional Tableau|finished]] [[Definition:Extension of Propositional Tableau|extension]] $T'$.
+By definition of [[Definition:Extension of Propositional Tableau|extension]], $T_{\mathbf H'}$ is a [[Definition:Rooted Subtree|rooted subtree]] of $T'$.
+But $T_{\mathbf H'}$ and $T$ are equal when considered as [[Definition:Rooted Tree|rooted trees]].
+The result follows.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Tableau Extension Lemma/General Statement/Proof 2}
+Tags: Propositional Tableaus
+
+\begin{theorem}
+Let $T$ be a [[Definition:Finite Propositional Tableau|finite propositional tableau]].
+Let its [[Definition:Hypothesis Set|hypothesis set]] $\mathbf H$ be [[Definition:Finite Set|finite]].
+{{:Tableau Extension Lemma/General Statement}}
+\end{theorem}
+
+\begin{proof}
+The proof uses [[Principle of Mathematical Induction|induction]] on the number $n$ of elements of $\mathbf H$.
+Suppose we are given the result for the case $n = 1$, that is, when $\mathbf H$ is a [[Definition:Singleton|singleton]].
+Suppose also that we are given the result for all sets $\mathbf H'$ with $n$ [[Definition:Element|elements]].
+Now, given a set $\mathbf H' = \left\{{\mathbf A_1, \ldots, \mathbf A_{n+1}}\right\}$ with $n+1$ [[Definition:Element|elements]].
+Let $T$ be a [[Definition:Finite Propositional Tableau|finite propositional tableau]].
+By induction hypothesis, there is a [[Definition:Finished Propositional Tableau|finished]] [[Definition:Finite Propositional Tableau|finite propositional tableau]] $T'$ containing $T$ as a [[Definition:Subgraph|subgraph]], and with [[Definition:Root of Propositional Tableau|root]] $\mathbf H \cup \left\{{\mathbf A_1, \ldots, \mathbf A_n}\right)$.
+Now apply the case $n = 1$ to this resulting [[Definition:Propositional Tableau|propositional tableau]] $T'$ and the set $\left\{{\mathbf A_{n+1}}\right\}$.
+This yields a [[Definition:Finished Propositional Tableau|finished]] [[Definition:Finite Propositional Tableau|finite propositional tableau]] $T''$ which:
+$(1):\quad$ has [[Definition:Root of Propositional Tableau|root]] $\mathbf H \cup \left\{{\mathbf A_1, \ldots, \mathbf A_n}\right\} \cup \left\{{\mathbf A_{n+1}}\right\} = \mathbf H \cup \mathbf H'$;
+$(2):\quad$ contains $T'$ as a [[Definition:Subgraph|subgraph]].
+But then $T''$ also contains $T$ as a [[Definition:Subgraph|subgraph]], proving the result for $\mathbf H'$.
+It thus only remains to take care of the base cases $n = 0$ and $n = 1$.
+First, the case $n = 0$.
+Let $T$ be a [[Definition:Finite Propositional Tableau|finite propositional tableau]].
+To find the [[Definition:Finite Propositional Tableau|finite propositional tableau]] $T'$ with the desired properties, we use some of the [[Definition:Propositional Tableau/Definition 2|tableau construction rules]], starting with $T$.
+Let $t$ be any [[Definition:Leaf Node|leaf node]] of $T$, and let $\Gamma_t$ be the [[Definition:Branch (Graph Theory)|branch]] from [[Leaf of Rooted Tree is on One Branch]].
+Let $n \left({\Gamma_t}\right)$ be the number of non-[[Definition:Basic WFF|basic WFFs]] that were not [[Definition:Used WFF|used]] to add any of the [[Definition:Node (Graph Theory)|nodes]] of $\Gamma_t$ to $T$.
+It is seen that for any application of the [[Definition:Propositional Tableau/Definition 2|tableau construction rules]] on $t$:
+:If $s$ is added by the rule, then $n \left({\Gamma_s}\right) \le n \left({\Gamma_t}\right)$.
+Moreover, it is seen that any rule reduces the total count $m \left({\Gamma_t}\right)$ of [[Definition:Logical Connective|logical connectives]] occurring in these non-[[Definition:Basic WFF|basic]], [[Definition:Used WFF|unused]] [[Definition:WFF of Propositional Logic|WFFs]] [[Definition:Occurrence along Branch|along]] $\Gamma_t$.
+In conclusion:
+:If $s$ is added by a rule, then $m \left({\Gamma_s}\right) < m \left({\Gamma_t}\right)$
+By the [[Method of Infinite Descent]] applied to $m \left({\Gamma_t}\right)$, only finitely many rules can be applied, starting from $t$.
+Since $T$ has only finitely many [[Definition:Leaf Node|leaves]] and corresponding [[Definition:Branch (Graph Theory)|branches]], only finitely many rules can be applied to $T$ in total.
+Let $T'$ be the [[Definition:Finite Propositional Tableau|finite propositional tableau]] resulting from applying all these possible rules.
+By construction of $T'$, it follows that every [[Definition:Branch (Graph Theory)|branch]] of $\Gamma$ is either [[Definition:Contradictory Branch|contradictory]] or [[Definition:Finished Branch of Propositional Tableau|finished]].
+That is, $T'$ is [[Definition:Finished Propositional Tableau|finished]].
+Finally, the last case, $n = 1$.
+Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
+Let $T$ be a [[Definition:Finite Propositional Tableau|finite propositional tableau]].
+First, using the case $n = 0$, extend $T$ to a [[Definition:Finished Propositional Tableau|finished]] [[Definition:Finite Propositional Tableau|finite propositional tableau]] $T'$.
+Again using the case $n = 0$, let $T_{\mathbf A}$ be a [[Definition:Finished Propositional Tableau|finished]] [[Definition:Finite Propositional Tableau|finite propositional tableau]] with [[Definition:Root of Propositional Tableau|root]] $\left\{{\mathbf A}\right\}$.
+Now add $\mathbf A$ to the [[Definition:Root of Propositional Tableau|root]] of $T'$.
+Then at every [[Definition:Leaf Node|leaf]] $t$ of $T'$, $\mathbf A$ is the only [[Definition:WFF of Propositional Logic|WFF]] that is not used yet.
+As far as the [[Definition:Propositional Tableau/Definition 2|rules for propositional tableaus]] are concerned, there is no difference between:
+:$t$ as a [[Definition:Leaf Node|leaf]] of $T'$, and
+:the [[Definition:Propositional Tableau|tableau]] consisting only of a [[Definition:Root of Propositional Tableau|root]] and with [[Definition:Hypothesis Set|hypothesis set]] $\mathbf A$.
+Therefore, the rules allow to "paste", as it were, the [[Finished Propositional Tableau|finished tableau]] $T_{\mathbf A}$ under every [[Definition:Leaf Node|leaf]] $t$ of $T'$.
+Denote the resulting [[Definition:Propositional Tableau|tableau]] with $T'_{\mathbf A}$.
+Then for any [[Definition:Branch (Graph Theory)|branch]] $\Gamma$ of $T'_{\mathbf A}$ and every non-[[Definition:Basic WFF|basic WFF]] $\mathbf B$ [[Definition:Occurrence along Branch|along]] it:
+:$\mathbf B$ is on $T'$, or:
+:$\mathbf B$ is on a copy of $T_{\mathbf A}$.
+In either case, the [[Definition:Finished Propositional Tableau|finished]] nature of these [[Definition:Propositional Tableau|tableaus]] implies that:
+:$\mathbf B$ is [[Definition:Used WFF|used]] at some [[Definition:Node (Graph Theory)|node]] of $\Gamma$
+Hence $\Gamma$ is [[Definition:Contradictory Branch|contradictory]] or [[Definition:Finished Branch of Propositional Tableau|finished]].
+In conclusion, $T'_{\mathbf A}$ is [[Definition:Finished Propositional Tableau|finished]], and contains $T$ as a [[Definition:Subgraph|subgraph]].
+The result follows from the [[Principle of Mathematical Induction]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Factors of Sums of Powers of 100,000}
+Tags: Composite Numbers
+
+\begin{theorem}
+All [[Definition:Integer|integers]] $n$ of the form:
+:$n = \displaystyle \sum_{k \mathop = 0}^m 10^{5 k}$ for $m \in \Z_{> 0}$
+are [[Definition:Composite Number|composite]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sum_{k \mathop = 0}^1 10^{5 k}
+ | r = 100 \, 001
+ | c =
+}}
+{{eqn | r = 11 \times 9091
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \sum_{k \mathop = 0}^2 10^{5 k}
+ | r = 10 \, 000 \, 100 \, 001
+ | c =
+}}
+{{eqn | r = 3 \times 31 \times 37 \times 2 \, 906 \, 161
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \sum_{k \mathop = 0}^3 10^{5 k}
+ | r = 1 \, 000 \, 010 \, 000 \, 100 \, 001
+ | c =
+}}
+{{eqn | r = 11 \times 101 \times 3541 \times 9091 \times 27961
+ | c =
+}}
+{{end-eqn}}
+Notice that:
+:$\displaystyle \sum_{k \mathop = 0}^m 10^{5 k} \times R_5 = R_{5 \paren {m + 1} }$
+where $R_i$ is the $i$th [[Definition:Repunit|repunit]].
+Suppose $p \divides m + 1$, where $p$ is a [[Definition:Prime Number|prime]] that is not 5.
+By [[Divisors of Repunit with Composite Index]]:
+:$R_p \divides R_{5 \paren {m + 1} }$
+By [[Prime not Divisor implies Coprime]], $p$ and $5$ are [[Definition:Coprime Integers|coprime]].
+By [[Condition for Repunits to be Coprime]], $R_p$ and $R_5$ are [[Definition:Coprime Integers|coprime]].
+By [[Euclid's Lemma]]:
+:$R_p \divides \dfrac {R_{5 \paren {m + 1} } } {R_5} = n$
+Suppose $25 \divides m + 1$.
+By [[Divisors of Repunit with Composite Index]]:
+:$R_5 \divides R_{25}$
+and:
+:$R_{25} \divides R_{5 \paren {m + 1} }$
+So we have:
+:$\dfrac {R_{25}} {R_5} \divides \dfrac {R_{5 \paren {m + 1} } } {R_5} = n$
+The final case is $m + 1 = 5$.
+This is the case $n = 100 \, 001 \, 000 \, 010 \, 000 \, 100 \, 001$.
+We have:
+:$n = 21 \, 401 \times 25 \, 601 \times 182 \, 521 \, 213 \, 001$
+Thus all cases are covered.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers whose Cube equals Sum of Sequence of that many Squares}
+Tags: Cube Numbers, Square Numbers, Numbers whose Cube equals Sum of Sequence of that many Squares
+
+\begin{theorem}
+The [[Definition:Integer|integers]] $m$ in the following [[Definition:Integer Sequence|sequence]] all have the property that $m^3$ is equal to the [[Definition:Integer Addition|sum]] of $m$ consecutive [[Definition:Square Number|squares]]:
+:$m^3 = \displaystyle \sum_{k \mathop = 1}^m \left({n + k}\right)^2$
+for some $n \in \Z_{\ge 0}$:
+:$0, 1, 47, 2161, 99 \, 359, 4 \, 568 \, 353, \ldots$
+{{OEIS|A189173}}
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | n = 1
+ | l = m^3
+ | r = \sum_{k \mathop = 1}^m \left({n + k}\right)^2
+ | c =
+}}
+{{eqn | r = \sum_{k \mathop = 1}^m \left({n^2 + 2 n k + k^2}\right)
+ | c =
+}}
+{{eqn | r = n^2 \sum_{k \mathop = 1}^m 1 + 2 n \sum_{k \mathop = 1}^m k + \sum_{k \mathop = 1}^m k^2
+ | c =
+}}
+{{eqn | r = m n^2 + 2 n \frac {m \left({m + 1}\right)} 2 + \frac {m \left({m + 1}\right) \left({2 m + 1}\right)} 6
+ | c = [[Closed Form for Triangular Numbers]], [[Sum of Sequence of Squares]]
+}}
+{{eqn | ll= \leadsto
+ | l = m^2
+ | r = n^2 + n \left({m + 1}\right) + \frac {\left({m + 1}\right) \left({2 m + 1}\right)} 6
+ | c =
+}}
+{{end-eqn}}
+Thus we have the [[Definition:Quadratic Equation|quadratic equation]]:
+:$n^2 + \left({m + 1}\right) n + \dfrac {\left({m + 1}\right) \left({2 m + 1}\right)} 6 - m^2 = 0$
+From [[Solution to Quadratic Equation]]:
+{{begin-eqn}}
+{{eqn | l = n
+ | r = \dfrac {-\left({m + 1}\right) \pm \sqrt{\left({m + 1}\right)^2 - 4 \left({\dfrac {\left({m + 1}\right) \left({2 m + 1}\right)} 6 - m^2}\right)} } 2
+ | c =
+}}
+{{eqn | r = - \dfrac {m + 1} 2 \pm \frac 1 2 \sqrt {m^2 + 2 m + 1 - 2 \left({\dfrac {2 m^2 + 3 m + 1} 3}\right) - 4 m^2}
+ | c =
+}}
+{{eqn | r = - \dfrac {m + 1} 2 \pm \frac 1 2 \sqrt {5 m^2 + 2 m + 1 - \dfrac {4 m^2 + 6 m + 2} 3}
+ | c =
+}}
+{{eqn | r = - \dfrac {m + 1} 2 \pm \frac 1 2 \sqrt {\frac {15 m^2 + 6 m + 3 - 4 m^2 - 6 m - 2 + 6 m^2} 3}
+ | c =
+}}
+{{eqn | r = - \dfrac {m + 1} 2 \pm \frac 1 2 \sqrt {\frac {11 m^2 + 1} 3}
+ | c =
+}}
+{{eqn | r = - \dfrac {m + 1} 2 \pm \frac 1 2 \sqrt {\frac {33 m^2 + 3} 9}
+ | c =
+}}
+{{eqn | r = - \dfrac {m + 1} 2 \pm \frac 1 6 \sqrt {33 m^2 + 3}
+ | c =
+}}
+{{end-eqn}}
+Let $t := +\sqrt {33 m^2 + 3}$.
+We are given that $m$ is an [[Definition:Integer|integer]].
+Let $n$ be an [[Definition:Integer|integer]].
+Then $t$ is a [[Definition:Rational Number|rational number]] which is the [[Definition:Square Root|square root]] of an [[Definition:Integer|integer]].
+Therefore $t$ is an [[Definition:Integer|integer]].
+Now let $t$ be an [[Definition:Integer|integer]].
+Then $3$ is a [[Definition:Divisor of Integer|divisor]] of $t^2$.
+Thus $3$ is a [[Definition:Divisor of Integer|divisor]] of $t$.
+It follows that $\dfrac t 3$ and $m + 1$ have the same [[Definition:Parity of Integer|parity]].
+Thus either $\dfrac {m + 1} 2$ and $\dfrac t 6$ are both [[Definition:Integer|integers]] or both [[Definition:Half-Integer|half-integers]].
+Hence $n$ is an [[Definition:Integer|integer]]
+Thus it has been demonstrated that $n$ is an [[Definition:Integer|integer]] {{iff}} $t$ is an [[Definition:Integer|integer]].
+Thus, finding the solutions of $(1)$ is equivalent to finding the solutions to the [[Definition:Diophantine Equation|Diophantine equation]]:
+:$(3): \quad t^2 - 33m^2 = 3$
+We first note the degenerate solution:
+: $t = 6, m = 1$
+Consider [[Pell's Equation]]:
+:$(4): \quad x^2 - 33 y^2 = 1$
+By working it out (or looking it up), the first positive solution to $(4)$ is:
+:$x = 23, y = 4$
+Thus all the solutions to $(4)$ are:
+:$x = 1, y = 0$
+and:
+:$x = \pm x_n, y = \pm y_n$
+where:
+:$(5): \quad x_n + y_n \sqrt {33} = \left({23 + 4 \sqrt {33} }\right)^n$
+for all [[Definition:Positive Integer|positive integers]] $n$.
+Using the solution of $(3)$:
+: $t = 6, m = 1$
+we can obtain another solution of $(3)$ by using:
+:$\left({6 + \sqrt {33} }\right) \left({x + y \sqrt {33} }\right) = t + m \sqrt {33}$
+where:
+:$(6): \quad t = 6 x + 33 y, m = x + 6 y$
+Thus:
+:$t - m \sqrt {33} = \left({6 - \sqrt {33} }\right) \left({x - y \sqrt {33} }\right)$
+from which:
+{{begin-eqn}}
+{{eqn | l = t^2 - 33 m^2
+ | r = \left({t - m \sqrt {33} }\right) \left({t + m \sqrt {33} }\right)
+ | c =
+}}
+{{eqn | r = \left({6 - \sqrt {33} }\right) \left({6 + \sqrt {33} }\right) \left({x - y \sqrt {33} }\right) \left({x + y \sqrt {33} }\right)
+ | c =
+}}
+{{eqn | r = \left({6^2 - 1 \times 33}\right) \left({x^2 - 33 y^2}\right)
+ | c =
+}}
+{{eqn | r = 3 \times 1
+ | c =
+}}
+{{eqn | r = 4
+ | c =
+}}
+{{end-eqn}}
+Thus it is demonstrated that $\left({t, m}\right)$ is a solution of $(3)$.
+Now let $\left({t, m}\right)$ be any solution of $(3)$.
+Let:
+:$x = 2 t - 11 m$
+:$y = \dfrac {6 m - t} 3$
+We have that:
+:$t^2 - 33 m^2 = 3$
+and so:
+: $3$ is a [[Definition:Divisor of Integer|divisor]] of $t^2$
+and so:
+: $3$ is a [[Definition:Divisor of Integer|divisor]] of $t$
+and so $x$ and $y$ are both [[Definition:Integer|integers]].
+$x$ and $y$ are seen to be solutions to $(4)$, and:
+:$t = 6 x + 33 y$
+:$m = x + 6 y$
+Thus from $(5)$ and $(6)$ it follows that the solutions of $(3)$ with $m > 1$ are obtained from $x = \pm x_n, y = \pm y_n$ in $(5)$.
+It follows further that all values of $m$ in such solutions are [[Definition:Odd Integer|odd]].
+The trivial solution $x = 1, y - 0$ of $(4)$ corresponds to $m = 1, t = 6$ of $(3)$.
+Thus we have that all the values of $m$ are given by:
+:$m_n = x_n + 6 y_n$
+where:
+:$x_n + y_n \sqrt {33} = \left({23 + 4 \sqrt {33} }\right)^n$
+We can set up a recursive process to calculate $\left({x_n, y_n}\right)$ of $(4)$ and the corresponding $\left({t_n, m_n}\right)$ of $(3)$ as follows:
+:$(7): \quad \left({x_n, y_n}\right) = \begin{cases}
+\left({23, 4}\right) & : n = 1 \\
+\left({23 x_{n - 1} + 132 y_{n - 1}, 23 y_{n - 1}, 4 x_{n - 1} }\right) & : n > 1
+\end{cases}$
+:$(8): \quad \left({t_n, m_n}\right) = \begin{cases}
+\left({6, 1}\right) & : n = 0 \\
+\left({23 t_{n - 1} + 132 m_{n - 1}, 23 t_{n - 1}, 4 m_{n - 1} }\right) : & n > 0
+\end{cases}$
+Using $(8)$, the values of $m$ for $n \ge 1$ are found to be:
+:$m_1 = 47, m_2 = 2161, m_3 = 99 \, 359, \ldots$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Points Defined by Adjacent Pairs of Digits of Reciprocal of 7 lie on Ellipse}
+Tags: Ellipses, 7
+
+\begin{theorem}
+Consider the [[Definition:Digit|digits]] that form the [[Definition:Recurring Part|recurring part]] of the [[Definition:Reciprocal|reciprocal]] of $7$:
+:$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$
+Take the [[Definition:Digit|digits]] in [[Definition:Ordered Pair|ordered pairs]], and treat them as [[Definition:Coordinate|coordinates]] of a [[Definition:Cartesian Plane|Cartesian plane]].
+It will be found that they all lie on an [[Definition:Ellipse|ellipse]]:
+:[[File:EllipseFromSeventh.png|400px]]
+\end{theorem}
+
+\begin{proof}
+:[[File:EllipseFromSeventhSolution.png|400px]]
+Let the points be labelled to simplify:
+:$A := \tuple {1, 4}$
+:$B := \tuple {2, 8}$
+:$C := \tuple {4, 2}$
+:$D := \tuple {8, 5}$
+:$E := \tuple {7, 1}$
+:$F := \tuple {5, 7}$
+Let $ABCDEF$ be considered as a [[Definition:Hexagon|hexagon]].
+We join the opposite points of $ABCDEF$:
+:$AF: \tuple {1, 4} \to \tuple {5, 7}$
+:$BC: \tuple {2, 8} \to \tuple {4, 2}$
+:$BE: \tuple {2, 8} \to \tuple {7, 1}$
+:$AD: \tuple {1, 4} \to \tuple {8, 5}$
+:$CD: \tuple {4, 2} \to \tuple {8, 5}$
+:$EF: \tuple {7, 1} \to \tuple {5, 7}$
+It is to be shown that the [[Definition:Intersection (Geometry)|intersections]] of:
+:$AF$ and $BC$
+:$BE$ and $AD$
+:$CD$ and $EF$
+all lie on the same [[Definition:Straight Line|straight line]].
+The result then follows from [[Pascal's Mystic Hexagram]].
+From [[Equation of Straight Line in Plane through Two Points]]:
+:$\dfrac {y - y_1} {x - x_1} = \dfrac {y_2 - y_1} {x_2 - x_1}$
+Thus:
+{{begin-eqn}}
+{{eqn | n = AF
+ | l = \frac {y - 4} {x - 1}
+ | r = \frac {7 - 4} {5 - 1}
+ | c =
+}}
+{{eqn | r = \frac 3 4
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 4 \paren {y - 4}
+ | r = 3 \paren {x - 1}
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = \dfrac 3 4 x + \dfrac {13} 4
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | n = BC
+ | l = \frac {y - 8} {x - 2}
+ | r = \frac {2 - 8} {4 - 2}
+ | c =
+}}
+{{eqn | r = -3
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y - 8
+ | r = -3 \paren {x - 2}
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = -3 x + 14
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | n = BE
+ | l = \frac {y - 8} {x - 2}
+ | r = \frac {1 - 8} {7 - 2}
+ | c =
+}}
+{{eqn | r = -\frac 7 5
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 5 \paren {y - 8}
+ | r = -7 \paren {x - 2}
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 5 y - 40
+ | r = -7 x + 14
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = -\frac 7 5 x + \frac {54} 5
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | n = AD
+ | l = \frac {y - 4} {x - 1}
+ | r = \frac {5 - 4} {8 - 1}
+ | c =
+}}
+{{eqn | r = \frac 1 7
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 7 \paren {y - 4}
+ | r = x - 1
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 7 y - 28
+ | r = x - 1
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = \frac x 7 + \frac {27} 7
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | n = CD
+ | l = \frac {y - 2} {x - 4}
+ | r = \frac {5 - 2} {8 - 4}
+ | c =
+}}
+{{eqn | r = \frac 3 4
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 4 \paren {y - 2}
+ | r = 3 \paren {x - 4}
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 4 y - 8
+ | r = 3 x - 12
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = \frac 3 4 x - 1
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | n = EF
+ | l = \frac {y - 1} {x - 7}
+ | r = \frac {7 - 1} {5 - 7}
+ | c =
+}}
+{{eqn | r = -3
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y - 1
+ | r = -3 x + 21
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = -3 x + 22
+ | c =
+}}
+{{end-eqn}}
+Evaluate the [[Definition:Intersection (Geometry)|intersection]] of $AF$ and $BC$:
+{{begin-eqn}}
+{{eqn | l = y
+ | r = \dfrac 3 4 x + \dfrac {13} 4
+ | c =
+}}
+{{eqn | l = y
+ | r = -3 x + 14
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \dfrac 3 4 x + \dfrac {13} 4
+ | r = -3 x + 14
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 3 x + 13
+ | r = -12 x + 56
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 15 x
+ | r = 43
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = x
+ | r = \dfrac {43} {15}
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = -3 \paren {\dfrac {43} {15} } + 14
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = \dfrac {-129 + 210} {15}
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = \dfrac {81} {15}
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = \dfrac {27} 5
+ | c =
+}}
+{{end-eqn}}
+So $AF$ and $BC$ [[Definition:Intersection (Geometry)|intersect]] at $\paren {\dfrac {43} {15}, \dfrac {27} 5}$.
+Evaluate the [[Definition:Intersection (Geometry)|intersection]] of $BE$ and $AD$:
+{{begin-eqn}}
+{{eqn | l = y
+ | r = -\frac 7 5 x + \frac {54} 5
+ | c =
+}}
+{{eqn | l = y
+ | r = \frac x 7 + \frac {27} 7
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = -\frac 7 5 x + \frac {54} 5
+ | r = \frac x 7 + \frac {27} 7
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = -49 x + 378
+ | r = 5 x + 135
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 54 x
+ | r = 243
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = x
+ | r = \dfrac 9 2
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = \frac 1 7 \paren {\frac 9 2} + \frac {27} 7
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = \dfrac {9 + 54} {14}
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = \dfrac {63} {14}
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = \dfrac 9 2
+ | c =
+}}
+{{end-eqn}}
+So $BE$ and $AD$ [[Definition:Intersection (Geometry)|intersect]] at $\paren {\dfrac 9 2, \dfrac 9 2}$.
+Evaluate the [[Definition:Intersection (Geometry)|intersection]] of $CD$ and $EF$:
+{{begin-eqn}}
+{{eqn | l = y
+ | r = \frac 3 4 x - 1
+ | c =
+}}
+{{eqn | l = y
+ | r = -3 x + 22
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \frac 3 4 x - 1
+ | r = -3 x + 22
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \frac 3 x - 4
+ | r = -12 x + 88
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 15 x
+ | r = 92
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = x
+ | r = \dfrac {92} {15}
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = -3 \paren {\dfrac {92} {15} } + 22
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = \dfrac {-92 + 110} 5
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = y
+ | r = \dfrac {18} 5
+ | c =
+}}
+{{end-eqn}}
+So $CD$ and $EF$ [[Definition:Intersection (Geometry)|intersect]] at $\paren {\dfrac {92} {15}, \dfrac {18} 5}$.
+It remains to be shown that those points of [[Definition:Intersection (Geometry)|intersection]]:
+:$\paren {\dfrac {43} {15}, \dfrac {27} 5}$, $\paren {\dfrac 9 2, \dfrac 9 2}$, $\paren {\dfrac {92} {15}, \dfrac {18} 5}$
+all lie on the same [[Definition:Straight Line|straight line]].
+From [[Equation of Straight Line in Plane through Two Points]]:
+:$\dfrac {y - y_1} {x - x_1} = \dfrac {y_2 - y_1} {x_2 - x_1}$
+Thus:
+{{begin-eqn}}
+{{eqn | l = \frac {y - \frac {27} 5} {x - \frac {43} {15} }
+ | r = \frac {\frac {18} 5 - \frac {27} 5} {\frac {92} {15} - \frac {43} {15} }
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \frac {5 y - 27} {15 x - 43}
+ | r = \frac {18 - 27} {92 - 43}
+ | c = simplifying
+}}
+{{eqn | ll= \leadsto
+ | l = \frac {5 y - 27} {15 x - 43}
+ | r = -\frac 9 {49}
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 245 y - 1323
+ | r = - 135 x + 387
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 245 y
+ | r = - 135 x + 1710
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = 49 y + 27 x
+ | r = 342
+ | c =
+}}
+{{end-eqn}}
+It remains to demonstrate that $\paren {\dfrac 9 2, \dfrac 9 2}$ lies on this line:
+{{begin-eqn}}
+{{eqn | l = 49 \dfrac 9 2 + 27 \dfrac 9 2
+ | r = \dfrac {441 + 243} 2
+ | c =
+}}
+{{eqn | r = \dfrac {684} 2
+ | c =
+}}
+{{eqn | r = 342
+ | c =
+}}
+{{end-eqn}}
+Bingo.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Continuous Function on Compact Subspace of Euclidean Space is Bounded}
+Tags: Analysis, Continuity
+
+\begin{theorem}
+Let $\R^n$ be the [[Definition:Euclidean Space|$n$-dimensional Euclidean space]].
+Let $S \subseteq \R^n$ be a [[Definition:Compact (Real Analysis)|compact subspace]] of $\R^n$.
+Let $f: S \to \R$ be a [[Definition:Continuous Mapping (Metric Spaces)|continuous function]].
+Then $f$ is [[Definition:Bounded Mapping|bounded]] in $\R$.
+\end{theorem}
+
+\begin{proof}
+An application of [[Continuous Function on Compact Space is Bounded]].
+\end{proof}<|endoftext|>
+\section{Points Defined by Adjacent Pairs of Digits of Reciprocal of 13 lie on Hyperbola}
+Tags: Hyperbolas, 13
+
+\begin{theorem}
+Consider the [[Definition:Digit|digits]] that form the [[Definition:Recurring Part|recurring part]] of the [[Definition:Reciprocal|reciprocal]] of $13$:
+:$\dfrac 1 {13} = 0 \cdotp \dot 07692 \dot 3$
+Take the [[Definition:Digit|digits]] in [[Definition:Ordered Pair|ordered pairs]], and treat them as [[Definition:Coordinate|coordinates]] of a [[Definition:Cartesian Plane|Cartesian plane]].
+It will be found that they all lie on a [[Definition:Hyperbola|hyperbola]]:
+:[[File:HyperbolaFromThirteenth.png|600px]]
+\end{theorem}
+
+\begin{proof}
+:[[File:HyperbolaFromThirteenthSolution.png|600px]]
+Let the points be labelled to simplify:
+:$A := \left({0, 7}\right)$
+:$B := \left({7, 6}\right)$
+:$C := \left({6, 9}\right)$
+:$D := \left({9, 2}\right)$
+:$E := \left({2, 3}\right)$
+:$F := \left({3, 0}\right)$
+{{finish|Just too tedious to contemplate.}}
+\end{proof}<|endoftext|>
+\section{Recurring Part of Fraction times Period gives 9-Repdigit}
+Tags: Repdigit Numbers
+
+\begin{theorem}
+Let a [[Definition:Strictly Positive Integer|(strictly) positive integer]] $n$ be such that the [[Definition:Decimal Expansion|decimal expansion]] of its [[Definition:Reciprocal|reciprocal]] has a [[Definition:Recurring Part|recurring part]] of [[Definition:Period of Recurrence|period]] $d$.
+Let $m$ be the [[Definition:Integer|integer]] formed from the $d$ [[Definition:Digit|digits]] of the [[Definition:Recurring Part|recurring part]].
+Then $m \times n$ is a $d$-[[Definition:Digit|digit]] [[Definition:Repdigit Number|repdigit number]] consisting of $9$s.
+{{mistake|Counterexample:1/6. Possibly true if n is not a multiple of 2 or 5}}
+\end{theorem}<|endoftext|>
+\section{Properties of 142,857}
+Tags: 142,857
+
+\begin{theorem}
+This page gathers together some properties of $142 \, 857$ which arise through its being the [[Definition:Digit|digits]] of the [[Definition:Recurring Part|recurring part]] of the [[Definition:Reciprocal|reciprocal]] of $7$.
+[[Definition:Integer Multiplication|Multiplication]] of $142 \, 857$ by numbers higher than $7$ produces a similar pattern to when you [[Definition:Integer Multiplication|multiply]] it by a single [[Definition:Digit|digit]], but with added complications.
+For example:
+{{begin-eqn}}
+{{eqn | l = 142 \, 857 \times 12
+ | r = 1 \, 714 \, 284
+ | c =
+}}
+{{end-eqn}}
+This becomes $714 \, 285$ when you take the $1$ off the front and add it to the back.
+The exception is when you [[Definition:Integer Multiplication|multiply]] it by $7$ or a [[Definition:Integer Multiple|multiple]] of $7$:
+{{begin-eqn}}
+{{eqn | l = 142 \, 857 \times 7
+ | r = 999 \, 999
+ | c =
+}}
+{{eqn | l = 142 \, 857 \times 14
+ | r = 1 \, 999 \, 998
+ | c =
+}}
+{{end-eqn}}
+From [[Recurring Part of Fraction times Period gives 9-Repdigit]], it is seen that this property is shared of all numbers formed from the [[Definition:Digit|digits]] of the [[Definition:Recurring Part|recurring part]] of a recurring fraction.
+If you divide $142 \, 857$ into two equal parts and add them, you get $999$:
+:$142 + 857 = 999$
+Thus by [[Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999]], $142 \, 857$ is [[Definition:Divisor of Integer|divisible]] by $999$:
+:$142 \, 857 = 143 \times 999$
+Also, we have:
+:$999 \, 999 = 1001 \times 999$
+and so $999 \, 999$ is [[Definition:Divisor of Integer|divisible]] by $999$.
+But as $999 \, 999 = 7 \times 142 \, 857$ we have that $999 \, 999$ is [[Definition:Divisor of Integer|divisible]] by $7$.
+Thus it follows from [[Euclid's Lemma]] that $142 \, 857$ is [[Definition:Divisor of Integer|divisible]] by $999$.
+{{finish|A lot of material from this chapter of Wells has been skipped, because it's just not very interesting. I leave it open for someone else to complete, if they want to.}}
+\end{theorem}<|endoftext|>
+\section{Reciprocal of 142,857}
+Tags: 142,857, Examples of Reciprocals
+
+\begin{theorem}
+:$\dfrac 1 {142 \, 857} = 0 \cdotp \dot 00000 \, \dot 7$
+\end{theorem}
+
+\begin{proof}
+Performing the calculation using [[Definition:Long Division|long division]]:
+
+ 0.000007000007...
+ ------------------
+142857)1.000000000000...
+ 1 000000
+ 999999
+ --------
+ 1000000
+ 999999
+ -------
+
+\end{proof}<|endoftext|>
+\section{Quotient of Group by Itself}
+Tags: Quotient Groups
+
+\begin{theorem}
+Let $G$ be a [[Definition:Group|group]].
+Let $G / G$ be the [[Definition:Quotient Group|quotient group]] of $G$ by itself.
+Then:
+:$G / G \cong \set e$
+That is, the [[Definition:Quotient Group|quotient]] of a [[Definition:Group|group]] by itself is [[Definition:Group Isomorphism|isomorphic]] to the [[Definition:Trivial Group|trivial group]].
+\end{theorem}
+
+\begin{proof}
+Let the [[Definition:Group Homomorphism|homomorphism]] $\phi: G \to \set e$ be defined as:
+:$\forall g \in G: \map \phi g = e$
+Then:
+:$\map \ker \phi = G$
+and:
+:$\Img \phi = \set e$
+By the [[First Isomorphism Theorem]]:
+:$G / \map \ker \phi \cong \Img \phi$
+Hence the result:
+:$G / G \cong \set e$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999}
+Tags: Recreational Mathematics, Divisibility Tests, Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999
+
+\begin{theorem}
+Let $n$ be an [[Definition:Integer|integer]] which has at least $3$ [[Definition:Digit|digits]] when expressed in [[Definition:Decimal Notation|decimal notation]].
+Let the [[Definition:Digit|digits]] of $n$ be divided into groups of $3$, counting from the right, and those groups added.
+Then the result is equal to a [[Definition:Multiple of Integer|multiple]] of $999$ {{iff}} $n$ is [[Definition:Divisor of Integer|divisible]] by $999$.
+\end{theorem}
+
+\begin{proof}
+The mistake is either ''and conversely'' or ''equal to $999$'', since $999 \, 999$ is an easy counterexample.
+Here we will show that the result is equal to '''a multiple of''' $999$ {{iff}} $n$ is [[Definition:Divisor of Integer|divisible]] by $999$.
+Write $n = \displaystyle \sum_{i \mathop = 0}^k a_i 10^{3 i}$, where $0 \le a_i < 1000$.
+This divides the [[Definition:Digit|digits]] of $n$ into groups of $3$.
+Then the statement is equivalent to:
+:$999 \divides n \iff 999 \divides \displaystyle \sum_{i \mathop = 0}^k a_i$
+This statement is true since:
+{{begin-eqn}}
+{{eqn | l = n
+ | r = \sum_{i \mathop = 0}^k a_i 10^{3 i}
+}}
+{{eqn | r = \sum_{i \mathop = 0}^k a_i 1000^i
+}}
+{{eqn | o = \equiv
+ | r = \sum_{i \mathop = 0}^k a_i 1^i
+ | rr = \pmod {999}
+ | c = [[Congruence of Powers]]
+}}
+{{eqn | o = \equiv
+ | r = \sum_{i \mathop = 0}^k a_i
+ | rr = \pmod {999}
+}}
+{{end-eqn}}
+{{qed}}
+=== [[Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999/Examples|Examples]] ===
+{{:Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999/Examples}}
+\end{proof}<|endoftext|>
+\section{Number which is Sum of Subfactorials of Digits}
+Tags: Subfactorials, 148,349
+
+\begin{theorem}
+The only [[Definition:Integer|integer]] which is the [[Definition:Integer Addition|sum]] of the [[Definition:Subfactorial|subfactorials]] of its [[Definition:Digit|digits]] is $148 \, 349$:
+:$148 \, 349 = \mathop !1 + \mathop !4 + \mathop !8 + \mathop !3 \mathop + \mathop !4 \mathop + \mathop !9$
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = 148 \, 349
+ | r = 0 + 9 + 14 \, 833 + 2 + 9 + 133 \, 496
+ | c =
+}}
+{{eqn | r = \mathop !1 + \mathop !4 + \mathop !8 + \mathop !3 \mathop + \mathop !4 \mathop + \mathop !9
+ | c =
+}}
+{{end-eqn}}
+A computer search can verify solutions under $10^6$ (that is, with no more than $6$ digits) in seconds.
+Let $n$ be a $k$-[[Definition:Digit|digit]] number, for $k \ge 7$.
+Then the [[Definition:Integer Addition|sum]] of the [[Definition:Subfactorial|subfactorials]] of its [[Definition:Digit|digits]] is not more than $\mathop !9 \times k$.
+But we have:
+{{begin-eqn}}
+{{eqn | l = n
+ | o = \ge
+ | r = 10^{k - 1}
+}}
+{{eqn | r = 10^6 \times 10^{k - 7}
+}}
+{{eqn | o = \ge
+ | r = 10^6 \times \paren {1 + 9 \paren {k - 7} }
+ | c = [[Bernoulli's Inequality]]
+}}
+{{eqn | o = >
+ | r = 7 \times \mathop !9 \times \paren {9 k - 62}
+ | c = $7 \times \mathop !9 = 934472$
+}}
+{{eqn | r = \mathop !9 \paren {63 k - 62 \times 7}
+}}
+{{eqn | o = >
+ | r = \mathop !9 \times k
+ | c = $k \ge 7$
+}}
+{{eqn | o = \ge
+ | r = \text{sum of the subfactorials of digits of } n
+}}
+{{end-eqn}}
+So no more numbers have this property.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integers Representable as Product of both 3 and 4 Consecutive Integers}
+Tags: Number Theory
+
+\begin{theorem}
+There are $3$ [[Definition:Integer|integers]] which can be expressed as both $x \paren {x + 1} \paren {x + 2} \paren {x + 3}$ for some $x$, and $y \paren {y + 1} \paren {y + 2}$ for some $y$:
+:$24, 120, 175 \, 560$
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = 24
+ | r = 1 \times 2 \times 3 \times 4
+ | c =
+}}
+{{eqn | r = 2 \times 3 \times 4
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 120
+ | r = 2 \times 3 \times 4 \times 5
+ | c =
+}}
+{{eqn | r = 4 \times 5 \times 6
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 175 \, 560
+ | r = 55 \times 56 \times 57
+ | c =
+}}
+{{eqn | r = 19 \times 20 \times 21 \times 22
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown there are no more.}}
+\end{proof}<|endoftext|>
+\section{Squares whose Digits form Consecutive Increasing Integers}
+Tags: Square Numbers, Recreational Mathematics
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] whose [[Definition:Square (Algebra)|squares]] have a [[Definition:Decimal Notation|decimal representation]] consisting of the concatenation of $2$ consecutive increasing [[Definition:Integer|integers]] begins:
+:$428, 573, 727, 846, 7810, 36 \, 365, 63 \, 636, 326 \, 734, \ldots$
+{{OEIS|A030467}}
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = 428^2
+ | r = 183 \, 184
+ | c =
+}}
+{{eqn | l = 573^2
+ | r = 328 \, 329
+ | c =
+}}
+{{eqn | l = 727^2
+ | r = 528 \, 529
+ | c =
+}}
+{{eqn | l = 846^2
+ | r = 715 \, 716
+ | c =
+}}
+{{eqn | l = 7810^2
+ | r = 6099 \, 6100
+ | c =
+}}
+{{eqn | l = 36 \, 365^2
+ | r = 13224 \, 13225
+ | c =
+}}
+{{eqn | l = 63 \, 636^2
+ | r = 40495 \, 40496
+ | c =
+}}
+{{eqn | l = 326 \, 734^2
+ | r = 106755 \, 106756
+ | c =
+}}
+{{end-eqn}}
+They can be determined by inspection.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Fifth Power which is Sum of 6 Fifth Powers}
+Tags: Fifth Powers
+
+\begin{theorem}
+The smallest [[Definition:Fifth Power|fifth power]] which is the [[Definition:Integer Addition|sum]] of $6$ [[Definition:Fifth Power|fifth powers]] is $12^5 = 248 \, 832$:
+:$12^5 = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5$
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = 12^5
+ | r = 248 \, 832
+ | c =
+}}
+{{eqn | r = 1024 + 3125 + 7776 + 16 \, 807 + 59 \, 049 + 161 \, 051
+ | c =
+}}
+{{eqn | r = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that this is the smallest.}}
+\end{proof}<|endoftext|>
+\section{Prime Numbers Embedded in Digits of Pi}
+Tags: Pi, Prime Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Prime Number|prime numbers]] that can be found starting from the beginning of the [[Definition:Decimal Expansion|decimal expansion]] of [[Definition:Pi|$\pi$ (pi)]] begins:
+:$3, 31, 314 \, 159, 31 \, 415 \, 926 \, 535 \, 897 \, 932 \, 384 \, 626 \, 433 \, 832 \, 795 \, 028 \, 841, \ldots$
+{{OEIS|A005042}}
+\end{theorem}
+
+\begin{proof}
+By inspection.
+\end{proof}<|endoftext|>
+\section{Properties of Family of 333,667 and Related Numbers}
+Tags: Recreational Mathematics, 333,667
+
+\begin{theorem}
+This page reports on certain properties, difficult to classify, of the number $333 \, 667$, and patterns arising.
+\end{theorem}<|endoftext|>
+\section{333,667 is Only Prime whose Reciprocal is of Period 9}
+Tags: 333,667, Examples of Reciprocals
+
+\begin{theorem}
+The only [[Definition:Prime Number|prime number]] whose [[Definition:Reciprocal|reciprocal]] has a [[Definition:Period of Recurrence|period]] of $9$ is $333 \, 667$:
+:$\dfrac 1 {333 \, 667} = 0 \cdotp \dot 00000 \, 299 \dot 7$
+\end{theorem}
+
+\begin{proof}
+By [[Definition:Long Division|long division]]:
+
+ 0.000002997000002...
+ ---------------------
+333667)1.000000000000000000
+ 667334
+ --------
+ 3326660
+ 3003003
+ -------
+ 3236570
+ 3003003
+ -------
+ 2335670
+ 2335669
+ -------
+ 1000000
+ 667334
+ -------
+ ......
+
+It remains to be shown that $333 \, 667$ is the only [[Definition:Prime Number|prime number]] with the required property.
+We have that:
+:$333 \, 667 \nmid 10$
+From [[Period of Reciprocal of Prime]], the [[Definition:Period of Recurrence|period]] of such a [[Definition:Prime Number|prime]] is the [[Definition:Multiplicative Order of Integer|order of $10$ modulo $p$]].
+That is, the smallest [[Definition:Integer|integer]] $d$ such that:
+:$10^d \equiv 1 \pmod p$
+From the above [[Definition:Long Division|long division]] we know that the [[Definition:Period of Recurrence|period]] of $\dfrac 1 {333 \, 667}$ is $9$, so $10^9 \equiv 1 \pmod {333 \, 667}$.
+The only other possible [[Definition:Prime Number|primes]] $p$ whose [[Definition:Reciprocal|reciprocals]] might have a [[Definition:Period of Recurrence|period]] of $9$ must also satisfy:
+:$10^9 \equiv 1 \pmod p$
+that is:
+:$p \divides 10^9 - 1$
+Consider:
+{{begin-eqn}}
+{{eqn | l = 10^9 - 1
+ | r = 999 \, 999 \, 999
+ | c =
+}}
+{{eqn | r = 3^4 \times 37 \times 333 \, 667
+ | c = prime factorization
+}}
+{{end-eqn}}
+Therefore the only other possible [[Definition:Prime Number|primes]] whose [[Definition:Reciprocal|reciprocals]] might have a [[Definition:Period of Recurrence|period]] of $9$ are $3$ and $37$.
+From [[Period of Reciprocal of 37 has Length 3]]:
+:$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$
+and trivially:
+:$\dfrac 1 3 = 0 \cdotp \dot 3$
+which has a [[Definition:Period of Recurrence|period]] of $1$.
+As required, the only [[Definition:Prime Number|prime number]] whose [[Definition:Reciprocal|reciprocal]] has a [[Definition:Period of Recurrence|period]] of $9$ is $333 \, 667$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cube which can be Represented as Sum of 3, 4, 5, 6, 7 or 8 Cubes}
+Tags: Cube Numbers, 351,120
+
+\begin{theorem}
+:$351 \, 120^3$ can be represented as the [[Definition:Integer Addition|sum]] of $3$, $4$, $5$, $6$, $7$ or $8$ [[Definition:Cube Number|cubes]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 351120^3
+ | r = 175560^3 + 234080^3 + 292600^3
+}}
+{{eqn | r = 2 \times 87780^3 + 204820^3 + 321860^3
+}}
+{{eqn | r = 2 \times 87780^3 + 175560^3 + 2 \times 263340^3
+}}
+{{eqn | r = 3 \times 117040^3 + 3 \times 234080^3
+}}
+{{eqn | r = 2 \times 58520^3 + 117040^3 + 3 \times 175560^3 + 292600^3
+}}
+{{eqn | r = 8 \times 175560^3
+}}
+{{end-eqn}}
+These representations are not necessarily unique.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Prime Gaps of 100}
+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 $100$:
+:$\tuple {396 \, 733, 396 \, 833}, \ldots$
+{{expand|Only know the first pair so far. Research needed to find the next one(s).}}
+\end{theorem}
+
+\begin{proof}
+Demonstrated by listing the [[Definition:Prime Gap|prime gaps]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Property of 490,689}
+Tags: Cube Numbers, 490,689
+
+\begin{theorem}
+The number $490 \, 689$ can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Cube Number|cubes]] in $2$ different ways:
+:$490 \, 689 = 4^3 + 60^3 + 65^3 = 8^3 + 25^3 \times 78^3$
+while at the same time the [[Definition:Integer Multiplication|products]] of the contributory [[Definition:Cube Root|cube roots]] of each of those $2$ ways are equal:
+:$4 \times 60 \times 65 = 8 \times 25 \times 78$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 490 \, 689
+ | r = 64 + 216 \, 000 + 274 \, 625
+ | c =
+}}
+{{eqn | r = 4^3 + 60^3 + 65^3
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 490 \, 689
+ | r = 512 + 15 \, 625 + 474 \, 552
+ | c =
+}}
+{{eqn | r = 8^3 + 25^3 + 78^3
+ | c =
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = 15 \, 600
+ | r = 2^4 \times 3 \times 5^2 \times 13
+ | c =
+}}
+{{eqn | r = 2^2 \times \left({2^2 \times 3 \times 5}\right) \times \left({5 \times 13}\right)
+ | c =
+}}
+{{eqn | r = 4 \times 60 \times 65
+ | c =
+}}
+{{eqn | r = 2^3 \times 5^2 \times \left({2 \times 3 \times 13}\right)
+ | c =
+}}
+{{eqn | r = 8 \times 25 \times 78
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{510,510 is Product of 4 Consecutive Fibonacci Numbers}
+Tags: Fibonacci Numbers, Primorials, 510,510
+
+\begin{theorem}
+$510 \, 510$ Can be expressed as the [[Definition:Integer Multiplication|product]] of $4$ [[Definition:Distinct|distinct]] [[Definition:Fibonacci Number|Fibonacci numbers]]:
+:$510 \, 510 = 13 \times 21 \times 34 \times 55$
+and is also the $7$th [[Definition:Primorial|primorial]]:
+:$510 \, 510 = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17$
+\end{theorem}
+
+\begin{proof}
+By observation:
+{{begin-eqn}}
+{{eqn | l = 510 \, 510
+ | r = 13 \times 21 \times 34 \times 55
+ | c =
+}}
+{{eqn | r = 13 \times \paren {3 \times 7} \times \paren ({2 \times 17} \times \paren {5 \times 11}
+ | c =
+}}
+{{eqn | r = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Tableau Confutation contains Finite Tableau Confutation}
+Tags: Propositional Tableaus
+
+\begin{theorem}
+Let $\mathbf H$ be a [[Definition:Countable Set|countable set]] of [[Definition:WFF of Propositional Logic|WFFs of propositional logic]].
+Let $T$ be a [[Definition:Tableau Confutation|tableau confutation]] of $\mathbf H$.
+Then there exists a [[Definition:Finite Tree|finite]] [[Definition:Rooted Subtree|rooted subtree]] of $T'$ that is also a [[Definition:Tableau Confutation|tableau confutation]] of $\mathbf H'$.
+\end{theorem}
+
+\begin{proof}
+For each [[Definition:Node (Graph Theory)|node]] $v \in T$, let $p (v)$ be the [[Definition:Path (Graph Theory)|path]] from $v$ to $r_T$, the [[Definition:Root Node|root]] of $T$.
+This [[Definition:Path (Graph Theory)|path]] is unique by [[Path in Tree is Unique]].
+Let $\mathcal V$ be the [[Definition:Rooted Subtree|subtree]] of $T$ consisting those [[Definition:Node (Graph Theory)|nodes]] $v$ of $T$ such that $p (v)$ is not [[Definition:Contradictory Branch|contradictory]].
+Suppose that $\mathcal V$ were [[Definition:Infinite Set|infinite]].
+Then by [[König's Tree Lemma]], $\mathcal V$ has an [[Definition:Infinite Branch|infinite branch]] $\Gamma$.
+Since $\mathcal V \subseteq T$, it follows that $\Gamma$ is also a [[Definition:Branch (Graph Theory)|branch]] of $T$.
+However, by construction, it is impossible that $\Gamma$ is [[Definition:Contradictory Branch|contradictory]].
+This contradicts that $T$ is a [[Definition:Tableau Confutation|tableau confutation]].
+Hence $\mathcal V$ is [[Definition:Finite Set|finite]].
+Next, define a [[Definition:Finite Propositional Tableau|finite propositional tableau]] $T'$ by:
+:$v \in T' \iff \pi (v) \in \mathcal V$
+that is, the [[Definition:Rooted Tree|rooted tree]] formed by $\mathcal V$ and all its [[Definition:Child Node|children]].
+Then by construction, for each [[Definition:Leaf Node|leaf node]] $v$ of $T'$, we have that $v \notin \mathcal V$.
+That is, that $p (v)$ is a [[Definition:Contradictory Branch|contradictory branch]] of $T'$.
+By [[Leaf of Rooted Tree is on One Branch]], every [[Definition:Branch (Graph Theory)|branch]] of $T'$ is [[Definition:Contradictory Branch|contradictory]].
+Hence $T'$ is a [[Definition:Tableau Confutation|tableau confutation]] of $\mathbf H'$, as desired.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Set of 7 Anagrams which are Square}
+Tags: Square Numbers
+
+\begin{theorem}
+The following [[Definition:Integer|integers]] are all [[Definition:Anagram|anagrams]], and all [[Definition:Square Number|square]]:
+:$1 \, 048 \, 576, 1 \, 056 \, 784, 1 \, 085 \, 764, 5 \, 740 \, 816, 5 \, 764 \, 801, 6 \, 754 \, 801, 7 \, 845 \, 601$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1 \, 048 \, 576
+ | r = 1024^2
+}}
+{{eqn | l = 1 \, 056 \, 784
+ | r = 1028^2
+}}
+{{eqn | l = 1 \, 085 \, 764
+ | r = 1042^2
+}}
+{{eqn | l = 5 \, 740 \, 816
+ | r = 2396^2
+}}
+{{eqn | l = 5 \, 764 \, 801
+ | r = 2401^2
+}}
+{{eqn | l = 6 \, 754 \, 801
+ | r = 2599^2
+}}
+{{eqn | l = 7 \, 845 \, 601
+ | r = 2801^2
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Burnside's Lemma}
+Tags: Group Actions
+
+\begin{theorem}
+Let $G$ be a [[Definition:Finite Group|finite group]] [[Definition:Group Action|acting]] on a [[Definition:Set|set]] $X$.
+Let $X / G$ be the [[Definition:Set of Orbits|set of orbits]] under this action.
+For $x \in X$, let $\Stab x$ be the [[Definition:Stabilizer|stabilizer]] of $x$ by $G$.
+For $g \in G$, let $X^g$ denotes the set of all elements in $X$ which is fixed by $g$, that is:
+:$X^g := \set {x \in X: g x = x}$
+Then:
+:$\displaystyle \size {X / G} = \frac 1 {\order G} \sum_{g \mathop \in G} \size {X^g}$
+In words, the number of [[Definition:Orbit (Group Theory)|orbits]] equals the average number of [[Definition:Fixed Point|fixed elements]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \frac 1 {\order G} \sum_{g \mathop \in G} \size {X^g}
+ | r = \frac 1 {\order G} \sum_{g \mathop \in G} \size {\set {x \in X: g x = x} }
+ | c = by definition
+}}
+{{eqn | r = \frac 1 {\order G} \sum_{x \mathop \in X} \size {\set {g \in G: g x = x} }
+ | c = Same summation, different indexing
+}}
+{{eqn | r = \frac 1 {\order G} \sum_{x \mathop \in X} \order {\Stab x}
+ | c = {{Defof|Stabilizer}}
+}}
+{{eqn | r = \frac 1 {\order G} \sum_{x \mathop \in X} \frac {\order G} {\order {\Orb x} }
+ | c = [[Orbit-Stabilizer Theorem]]
+}}
+{{eqn | r = \sum_{x \mathop \in X} \frac 1 {\order {\Orb x} }
+ | c =
+}}
+{{eqn | r = \sum_{\Orb x \mathop \in X / G} \paren {\sum_{x \mathop \in \Orb x} \frac 1 {\order {\Orb x} } }
+ | c =
+}}
+{{eqn | r = \sum_{\Orb x \mathop \in X / G} 1
+ | c =
+}}
+{{eqn | r = \order {X / G}
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Cunningham Chain of the First Kind of Length 7}
+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]] $7$ is:
+:$\left({1 \, 122 \, 659, 2 \, 245 \, 319, 4 \, 490 \, 639, 8 \, 981 \, 279, 17 \, 962 \, 559, 35 \, 925 \, 119, 71 \, 850 \, 239}\right)$
+\end{theorem}
+
+\begin{proof}
+Let $C$ denote the [[Definition:Sequence|sequence]] in question.
+We have that:
+:$\dfrac {1 \, 122 \, 659 - 1} 2 = 561 \, 329 = 83 \times 6763$
+and so is not [[Definition:Prime Number|prime]].
+Thus $1 \, 122 \, 659$ is not a [[Definition:Safe Prime|safe prime]], as is required for $C$ to be a [[Definition:Cunningham Chain of the First Kind|Cunningham chain of the first kind]].
+Then:
+{{begin-eqn}}
+{{eqn | l = 2 \times 561 \, 329 + 1
+ | r = 1 \, 122 \, 659
+ | c = which is the $87 \, 359$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 2 \times 1 \, 122 \, 659 + 1
+ | r = 2 \, 245 \, 319
+ | c = which is the $165 \, 760$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 2 \times 2 \, 245 \, 319 + 1
+ | r = 4 \, 490 \, 639
+ | c = which is the $315 \, 347$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 2 \times 4 \, 490 \, 639 + 1
+ | r = 8 \, 981 \, 279
+ | c = which is the $601 \, 286$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 2 \times 8 \, 981 \, 279 + 1
+ | r = 17 \, 962 \, 559
+ | c = which is the $1 \, 149 \, 096$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 2 \times 17 \, 962 \, 559 + 1
+ | r = 35 \, 925 \, 119
+ | c = which is the $2 \, 199 \, 933$rd [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 2 \times 35 \, 925 \, 119 + 1
+ | r = 71 \, 850 \, 239
+ | c = which is the $4 \, 220 \, 407$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 2 \times 71 \, 850 \, 239 + 1
+ | r = 143 \, 700 \, 479
+ | c =
+}}
+{{eqn | r = 13 \times 47 \times 479 \times 491
+ | c = and so is not [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+Establishing that this is indeed the smallest such [[Definition:Cunningham Chain of the First Kind|Cunningham chain of the first kind]] of [[Definition:Length of Sequence|length]] $7$ can be done by a computer search.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Square of Repunit times Sum of Digits}
+Tags: Repunits, Square Numbers
+
+\begin{theorem}
+The following pattern emerges:
+{{begin-eqn}}
+{{eqn | l = 121 \times \paren {1 + 2 + 1}
+ | r = 22^2
+}}
+{{eqn | l = 12 \, 321 \times \paren {1 + 2 + 3 + 2 + 1}
+ | r = 333^2
+}}
+{{eqn | l = 1 \, 234 \, 321 \times \paren {1 + 2 + 3 + 4 + 3 + 2 + 1}
+ | r = 4444^2
+}}
+{{end-eqn}}
+and so on, up until $999 \, 999 \, 999^2$ after which the pattern breaks down.
+\end{theorem}
+
+\begin{proof}
+From [[Square of Repunit]]:
+{{begin-eqn}}
+{{eqn | l = 121
+ | r = 11^2
+}}
+{{eqn | l = 12 \, 321
+ | r = 111^2
+}}
+{{eqn | l = 1 \, 234 \, 321
+ | r = 1111^2
+}}
+{{end-eqn}}
+and so on.
+Then from [[1+2+...+n+(n-1)+...+1 = n^2]]:
+{{begin-eqn}}
+{{eqn | l = 1 + 2 + 1
+ | r = 2^2
+}}
+{{eqn | l = 1 + 2 + 3 + 2 + 1
+ | r = 3^2
+}}
+{{eqn | l = 1 + 2 + 3 + 4 + 3 + 2 + 1
+ | r = 4^2
+}}
+{{end-eqn}}
+and so on.
+Then:
+{{begin-eqn}}
+{{eqn | l = 11^2 \times 2^2
+ | r = 22^2
+}}
+{{eqn | l = 111^2 \times 3^2
+ | r = 333^2
+}}
+{{eqn | l = 1111^2 \times 4^2
+ | r = 4444^2
+}}
+{{end-eqn}}
+The pattern breaks down after $9$:
+:$1 \, 111 \, 111 \, 111^2 = 1 \, 234 \, 567 \, 900 \, 987 \, 654 \, 321$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Factorisation of Quintic x^5 - x + n into Irreducible Quadratic and Irreducible Cubic}
+Tags: Polynomial Theory
+
+\begin{theorem}
+The [[Definition:Quintic Polynomial|quintic]] $x^5 - x + n$ can be [[Definition:Factorization|factorized]] into the [[Definition:Product of Polynomials|product]] of an [[Definition:Irreducible Polynomial|irreducible]] [[Definition:Quadratic Polynomial|quadratic]] and an an [[Definition:Irreducible Polynomial|irreducible]] [[Definition:Cubic Polynomial|cubic]] {{iff}} $n$ is in the set:
+:$\set {\pm 15, \pm 22 \, 440, \pm 2 \, 759 \, 640}$
+\end{theorem}
+
+\begin{proof}
+We have that:
+{{begin-eqn}}
+{{eqn | l = x^5 - x \pm 15
+ | r = \paren {x^2 \pm x + 3} \paren {x^3 \mp x^2 \mp 2 x \pm 5}
+}}
+{{eqn | l = x^5 - x \pm 22440
+ | r = \paren {x^2 \mp 12 x + 55} \paren {x^3 \pm 12 x^2 + 89 x \pm 408}
+}}
+{{eqn | l = x^5 - x \pm 2 \, 759 \, 640
+ | r = \paren {x^2 \pm 12 x + 377} \paren {x^3 \mp 12 x^2 - 233 x \pm 7320}
+}}
+{{end-eqn}}
+{{ProofWanted}}
+\end{proof}<|endoftext|>
+\section{Factorial as Product of Consecutive Factorials}
+Tags: Factorials, Factorial as Product of Consecutive Factorials
+
+\begin{theorem}
+The only [[Definition:Factorial|factorials]] which are the product of consecutive [[Definition:Factorial|factorials]] are:
+{{begin-eqn}}
+{{eqn | l = 0!
+ | r = 0! \times 1!
+ | c =
+}}
+{{eqn | l = 1!
+ | r = 0! \times 1!
+ | c =
+}}
+{{eqn | l = 2!
+ | r = 1! \times 2!
+ | c =
+}}
+{{eqn | r = 0! \times 1! \times 2!
+ | c =
+}}
+{{eqn | l = 10!
+ | r = 6! \times 7!
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+Suppose $m, n \in \N$ and $m < n$.
+Write $\map F {n, m} = n! \paren {n + 1}! \cdots m!$.
+Suppose we have $\map F {n, m} > r!$ for some $r \in \N$.
+Suppose further that there is a [[Definition:Prime Number|prime]] $p$ where $m < p \le r$.
+We claim that $\map F {n, m}$ cannot be a [[Definition:Factorial|factorial]] of any number.
+{{AimForCont}} $\map F {n, m} = s!$ for some $s \in \N$.
+Since $s! > r!$, we must have $r! \divides s!$.
+Since $p \le r$, we must have $p \divides r!$.
+Thus we have $p \divides s!$.
+However, since $n, n + 1, \dots, m < p$, we must have $p \nmid k!$ for each $n \le k \le m$.
+Thus $p \nmid \map F {n, m} = s!$, which is a contradiction.
+Hence $\map F {n, m}$ cannot be a [[Definition:Factorial|factorial]] of some number.
+We have the following [[Definition:Lemma|lemmata]]:
+=== [[Factorial as Product of Consecutive Factorials/Lemma 1|Lemma 1]] ===
+{{:Factorial as Product of Consecutive Factorials/Lemma 1}}{{qed|lemma}}
+=== [[Factorial as Product of Consecutive Factorials/Lemma 2|Lemma 2]] ===
+{{:Factorial as Product of Consecutive Factorials/Lemma 2}}{{qed|lemma}}
+We also have:
+:$\forall n \in \N: n > 1 \implies \exists p: 2 n < p < 3 n$
+{{finish|The result above is shown by M. El Bachraoui in their paper Primes in the Interval [2n, 3n]. I do not have the time to comprehend their proof yet}}
+=== Case $1$: $m$ is an even number larger than $2$ ===
+Write $m = 2 k$, where $k > 1$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \map F {n, m}
+ | o = \ge
+ | r = \paren {m - 1}! \, m!
+}}
+{{eqn | r = \paren {2 k - 1}! \, \paren {2 k}!
+}}
+{{eqn | o = >
+ | r = \paren {3 k - 1}!
+ | c = [[Factorial as Product of Consecutive Factorials/Lemma 1|Lemma 1]]
+}}
+{{end-eqn}}
+There is a [[Definition:Prime Number|prime]] $p$ where $m = 2 k < p \le 3 k - 1$.
+Therefore $\map F {n, m}$ is not a [[Definition:Factorial|factorial]] of some number.
+=== Case $2$: $m$ is an odd number larger than $11$ ===
+Write $m = 2 k - 1$, where $k \ge 7$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \map F {n, m}
+ | o = \ge
+ | r = \paren {m - 1}! \, m!
+}}
+{{eqn | r = \paren {2 k - 2}! \, \paren {2 k - 1}!
+}}
+{{eqn | o = >
+ | r = \paren {3 k - 1}!
+ | c = [[Factorial as Product of Consecutive Factorials/Lemma 2|Lemma 2]]
+}}
+{{end-eqn}}
+There is a [[Definition:Prime Number|prime]] $p$ where $m = 2 k - 1 < 2 k < p \le 3 k - 1$.
+Therefore $\map F {n, m}$ is not a [[Definition:Factorial|factorial]] of some number.
+=== Case $3$: Particular values of $m$ ===
+The cases above leaves us with $m = 1, 2, 3, 5, 7, 9, 11$ to check.
+We have:
+{{begin-eqn}}
+{{eqn | l = 10! \times 11!
+ | o = >
+ | r = 13!
+ | c = and $13$ is a [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 8! \times 9!
+ | o = >
+ | r = 11!
+ | c = and $11$ is a [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 6! \times 7!
+ | r = 10!
+}}
+{{eqn | l = 5! \times 6! \times 7!
+ | o = >
+ | r = 11!
+ | c = and $11$ is a [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 4! \times 5!
+ | r = 2880
+ | c = is not a [[Definition:Factorial|factorial]] of some number
+}}
+{{eqn | l = 3! \times 4! \times 5!
+ | o = >
+ | r = 7!
+ | c = and $7$ is a [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = \map F {n, 3}
+ | r = 12
+ | c = is not a [[Definition:Factorial|factorial]] of some number
+}}
+{{eqn | l = \map F {n, 2}
+ | r = 2!
+}}
+{{eqn | l = \map F {0, 1}
+ | r = 1!
+}}
+{{eqn | r = 0!
+}}
+{{end-eqn}}
+Thus there are no more.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers whose Fourth Root equals Number of Divisors}
+Tags: Fourth Powers, Tau Function
+
+\begin{theorem}
+There are $4$ [[Definition:Positive Integer|positive integers]] whose [[Definition:Root (Analysis)|$4$th root]] equals the number of its [[Definition:Divisor of Integer|divisors]]:
+{{begin-eqn}}
+{{eqn | l = 1
+ | r = 1^4
+ | c =
+}}
+{{eqn | l = 625
+ | r = 5^4
+ | c =
+}}
+{{eqn | l = 6561
+ | r = 9^4
+ | c =
+}}
+{{eqn | l = 4 \, 100 \, 625
+ | r = 45^4
+ | c =
+}}
+{{end-eqn}}
+{{OEIS|A143026}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \map \tau 1
+ | r = 1
+ | c = {{TauLink|1}}
+}}
+{{eqn | l = \map \tau {625}
+ | r = 5
+ | c = {{TauLink|625}}
+}}
+{{eqn | l = \map \tau {6561}
+ | r = 9
+ | c = {{TauLink|6561}}
+}}
+{{eqn | l = \map \tau {4 \, 100 \, 625}
+ | r = 45
+ | c = {{TauLink|4,100,625|4 \, 100 \, 625}}
+}}
+{{end-eqn}}
+Suppose $N = \map \tau {N^4}$.
+By [[Tau Function Odd Iff Argument is Square]], $N$ must be [[Definition:Odd Integer|odd]].
+The case $N = 1$ is trivial.
+Suppose $N$ is a [[Definition:Prime Power|prime power]].
+Write $N = p^n$.
+By [[Tau of Power of Prime]]:
+:$N = \map \tau {p^{4 n} } = 4 n + 1$
+By [[Bernoulli's Inequality]]:
+:$N = p^n \ge 1 + n \paren {p - 1}$
+This gives us the inequality:
+:$4 n + 1 \ge 1 + n \paren {p - 1}$
+which can be simplified to:
+:$4 \ge p - 1$
+The only [[Definition:Odd Prime|odd primes]] satisfying the inequality are $3$ and $5$.
+We have:
+:$\map \tau {3^4} = 5 > 3^1$
+:$\map \tau {3^8} = 9 = 3^2$
+:$\map \tau {3^{4 n} } = 4 n + 1 < 3^n$ for $n > 2$
+:$\map \tau {5^4} = 5 = 5^1$
+:$\map \tau {5^{4 n} } = 4 n + 1 < 5^n$ for $n > 1$
+:$\map \tau {p^{4 n} } = 4 n + 1 < p^n$ for any $p > 5$
+Hence $625$ and $6561$ are the only [[Definition:Prime Power|prime powers]] satisfying the property.
+Note that [[Tau Function is Multiplicative]].
+To form an [[Definition:Integer|integer]] $N$ with our property, we must choose and multiply [[Definition:Prime Power|prime powers]] from the list above.
+The product $625 \times 6561 = 4 \, 100 \, 625$ gives equality.
+If we chose any $\tuple {p, n}$ with $\map \tau {p^{4 n} } < p^n$, we must choose $3^4$ in order for equality to possibly hold.
+Then $\map \tau {3^4} = 5 \divides N$, so a tuple $\tuple {5, n}$ must be chosen.
+If $\tuple {5, 1}$ was chosen, $5^2 \nmid N$.
+But $\map \tau {3^4 \times 5^4} = 25 \divides N$, which is a contradiction.
+Suppose $\tuple {5, n}$ with $n \ge 2$ was chosen.
+Then $\map \tau {3^4 \times 5^{4 n} } = 20 n + 1 < 3 \times 5^n$, a contradiction.
+Thus we have exhausted all cases.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Solution to Equation p^p times q^q = r^r}
+Tags: Diophantine Equations
+
+\begin{theorem}
+Consider the [[Definition:Diophantine Equation|Diophantine equation]]:
+:$p^p \times q^q = r^r$
+Its smallest solution is:
+{{begin-eqn}}
+{{eqn | l = p
+ | r = 12^6
+ | rr= = 2 \, 985 \, 984
+}}
+{{eqn | l = q
+ | r = 6^8
+ | rr= = 1 \, 679 \, 616
+}}
+{{eqn | l = r
+ | r = 2^{11} \times 3^7
+ | rr= = 4 \, 478 \, 976
+}}
+{{end-eqn}}
+\end{theorem}<|endoftext|>
+\section{Existence of Matrix Logarithm}
+Tags:
+
+\begin{theorem}
+Let $T$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]].
+Then there exists a real matrix $S$ such that $e^S = T$ {{iff}}:
+:$(1): \quad T$ is not a [[Definition:Singular Matrix|singular matrix]]
+and:
+:$(2): \quad $for every negative [[Definition:Eigenvalue|eigenvalue]] $\lambda$ of $T$ and for every positive integer $k$, the [[Definition:Jordan Form|Jordan form]] of $T$ has an even number of $k \times k$ blocks associated with $\lambda$.
+pcu6qfvqi59x43t8w6ubk68j3p2rko5
+\end{theorem}<|endoftext|>
+\section{Weak Existence of Matrix Logarithm}
+Tags: Matrix Algebra, Matrix Logarithms
+
+\begin{theorem}
+Let $T$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]].
+Let $\norm {T - I} < 1$ in the [[Definition:Norm on Bounded Linear Transformation|norm on bounded linear operators]], where $I$ the [[Definition:Identity Matrix|identity matrix]].
+Then there is a [[Definition:Square Matrix|square matrix]] $S$ such that:
+:$e^S = T$
+where $e^S$ is the [[Definition:Matrix Exponential|matrix exponential]].
+\end{theorem}
+
+\begin{proof}
+Define:
+:$\ds S = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n \paren {T - I}^n$
+$S$ converges since $\norm {T - I} < 1$.
+We have that $\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n \norm {T - I}^n$ is the [[Definition:Newton-Mercator Series|Newton-Mercator Series]].
+This converges since $\norm {T - I} < 1$.
+Hence the series for $S$ converges absolutely, and so $S$ is well defined.
+Using the series definition for the [[Properties of Matrix Exponential|matrix exponential]]:
+{{begin-eqn}}
+{{eqn | l = e^S
+ | r = I + S + \frac 1 {2!} S^2 + \frac 1 {3!} S^3 + \cdots
+}}
+{{eqn | r = I + \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n \paren {T - I}^n + \frac 1 {2!} \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n \paren {T - I}^n}^2 + \frac 1 {3!} \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n \paren {T - I}^n}^3 + \cdots
+}}
+{{eqn | ll= \leadsto
+ | l = e^S
+ | r = I + \paren {T - I} + c_2 \paren {T - I}^2 + c_3 \paren {T - I}^3 + \cdots
+ | c = grouping terms by powers of $T - I$
+}}
+{{eqn | r = T + c_2 \paren {T - I}^2 + c_3 \paren {T - I}^3 + \cdots
+}}
+{{end-eqn}}
+If $c_i = 0$ for $i \ge 2$, then $e^S = T$, and the result is shown.
+The [[Definition:Newton-Mercator Series|Newton-Mercator Series]] is a Taylor expansion for $\map \ln {1 + x}$.
+When combined with the [[Power Series Expansion for Exponential Function]], it gives:
+{{begin-eqn}}
+{{eqn | l = e^{\map \ln {1 + x} }
+ | r = 1 + \map \ln {1 + x} + \frac 1 {2!} \paren {\map \ln {1 + x} }^2 + \frac 1 {3!} \paren {\map \ln {1 + x} }^3 + \cdots
+}}
+{{eqn | r = 1 + \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n x^n + \frac 1 {2!} \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n x^n}^2 + \frac 1 {3!} \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n x^n}^3 + \cdots
+}}
+{{eqn | r = 1 + x + c_2 x^2 + c_3 x^3 + \cdots
+ | c = grouping terms by powers of $x$
+}}
+{{end-eqn}}
+But $e^{\map \ln {1 + x} } = 1 + x$.
+Thus:
+:$1 + x = 1 + x + c_2 x^2 + c_3 x^3 + \cdots \implies c_i = 0$
+for $i \ge 2$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Norm on Vector Space is Continuous Function}
+Tags: Norm Theory
+
+\begin{theorem}
+Let $V$ be a vector space with norm $\norm {\, \cdot \,}$.
+The function $\norm {\, \cdot \,}: V \to \R$ is continuous.
+\end{theorem}
+
+\begin{proof}
+Let $x_n \to x$ in $V$.
+We have:
+:$x_n \to x \implies \norm {x_n - x} \to 0$
+By the [[Reverse Triangle Inequality]]:
+:$\size {\norm {x_n} - \norm x} \le \norm {x_n - x}$
+Hence:
+:$\size {\norm {x_n} - \norm {x_n} } \to 0$
+{{mistake|Should the above read $\size {\norm {x_n} - \norm x} \to 0$?}}
+Thus:
+:$\norm {x_n} \to \norm x$
+{{finish|Explain why, with use of links, this shows continuity.}}
+[[Category:Norm Theory]]
+qnjnc1afwm0bbaokk4abjx9wel2yoyw
+\end{proof}<|endoftext|>
+\section{Largest Integer Not Expressible as Sum of Distinct 4th Powers}
+Tags: Fourth Powers, 5,134,240
+
+\begin{theorem}
+The largest [[Definition:Integer|integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct|distinct]] [[Definition:Fourth Power|$4$th powers]] is $5 \, 134 \, 240$.
+\end{theorem}<|endoftext|>
+\section{Smallest 5 Consecutive Primes in Arithmetic Sequence}
+Tags: Prime Numbers, Arithmetic Sequences
+
+\begin{theorem}
+The smallest $5$ consecutive [[Definition:Prime Number|primes]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] are:
+:$9 \, 843 \, 019 + 30 n$
+for $n = 0, 1, 2, 3, 4$.
+Note that while there are many longer [[Definition:Arithmetic Sequence|arithmetic sequences]] of far smaller [[Definition:Prime Number|primes]], those primes are not consecutive.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 9 \, 843 \, 019 + 0 \times 30
+ | r = 9 \, 843 \, 019
+ | c = which is the $654 \, 926$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 9 \, 843 \, 019 + 1 \times 30
+ | r = 9 \, 843 \, 049
+ | c = which is the $654 \, 927$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 9 \, 843 \, 019 + 2 \times 30
+ | r = 9 \, 843 \, 079
+ | c = which is the $654 \, 928$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 9 \, 843 \, 019 + 3 \times 30
+ | r = 9 \, 843 \, 109
+ | c = which is the $654 \, 929$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 9 \, 843 \, 019 + 4 \times 30
+ | r = 9 \, 843 \, 139
+ | c = which is the $654 \, 930$th [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+But note that $9 \, 843 \, 019 + 5 \times 30 = 9 \, 843 \, 169 = 7^2 \times 200 \, 881$ and so is not [[Definition:Prime Number|prime]].
+Inspection of tables of [[Definition:Prime Number|primes]] (or a computer search) will reveal that this is the smallest such sequence.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Properties of 12,345,679}
+Tags: 12,345,679, Recreational Mathematics
+
+\begin{theorem}
+$12 \, 345 \, 679$ has the following properties:
+{{begin-eqn}}
+{{eqn | l = 12 \, 345 \, 679 \times 1
+ | r = 12 \, 345 \, 679
+ | c = [[Definition:Digit|digit]] $8$ is missing
+}}
+{{eqn | l = 12 \, 345 \, 679 \times 2
+ | r = 24 \, 691 \, 358
+ | c = [[Definition:Digit|digit]] $7$ is missing
+}}
+{{eqn | l = 12 \, 345 \, 679 \times 3
+ | r = 37 \, 037 \, 037
+ | c =
+}}
+{{eqn | l = 12 \, 345 \, 679 \times 4
+ | r = 49 \, 382 \, 716
+ | c = [[Definition:Digit|digit]] $5$ is missing
+}}
+{{eqn | l = 12 \, 345 \, 679 \times 5
+ | r = 61 \, 728 \, 395
+ | c = [[Definition:Digit|digit]] $4$ is missing
+}}
+{{eqn | l = 12 \, 345 \, 679 \times 6
+ | r = 74 \, 074 \, 074
+ | c =
+}}
+{{eqn | l = 12 \, 345 \, 679 \times 7
+ | r = 86 \, 419 \, 753
+ | c = [[Definition:Digit|digit]] $2$ is missing
+}}
+{{eqn | l = 12 \, 345 \, 679 \times 8
+ | r = 98 \, 765 \, 432
+ | c = [[Definition:Digit|digit]] $1$ is missing
+}}
+{{eqn | l = 12 \, 345 \, 679 \times 9
+ | r = 111 \, 111 \, 111
+ | c =
+}}
+{{end-eqn}}
+In each [[Definition:Integer Multiplication|product]], the sequence $1$ to $9$, with the one given [[Definition:Digit|digit]] missing, can be read in order by cycling round it, skipping a fixed number of digits (counting an extra one when going from start to end), for example:
+:$2 \ (4691) \ 3 \ (58?2) \ 4 \ (6913) \ 5 \ (8?24) \ 6 \ (9135) \ 8 (?246) \ 9$
+{{expand|Add some mathematical analysis explaining this phenomenon}}
+\end{theorem}<|endoftext|>
+\section{Number of Ways to Tile Standard Chessboard with Dominoes}
+Tags: Recreational Mathematics
+
+\begin{theorem}
+The number of ways to [[Definition:Tiling|tile]] a [[Definition:Chessboard|standard chessboard]] with [[Definition:Domino|dominoes]] is $12 \, 988 \, 816$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Considerable definition work needed before we can start on this sort of problem.}}
+\end{proof}<|endoftext|>
+\section{Smallest Triplet of Primitive Pythagorean Triangles with Same Area}
+Tags: Specific Numbers, 13,123,110
+
+\begin{theorem}
+The smallest [[Definition:Set|set of $3$]] [[Definition:Primitive Pythagorean Triangle|primitive Pythagorean triangles]] which all have the same [[Definition:Area|area]] are:
+:the [[Pythagorean Triangle/Examples/4485-5852-7373|$4485-5852-7373$ triangle]]
+:the [[Pythagorean Triangle/Examples/3059-8580-9109|$3059-8580-9109$ triangle]]
+:the [[Pythagorean Triangle/Examples/1380-19,019-19,069|$1380-19 \, 019-19 \, 069$ triangle]].
+That area is $13 \, 123 \, 110$.
+\end{theorem}
+
+\begin{proof}
+We have that:
+:the [[Pythagorean Triangle/Examples/4485-5852-7373|$4485-5852-7373$ triangle $T_1$ is Pythagorean]]
+:the [[Pythagorean Triangle/Examples/3059-8580-9109|$3059-8580-9109$ triangle $T_2$ is Pythagorean]]
+:the [[Pythagorean Triangle/Examples/1380-19,019-19,069|$1380-19 \, 019-19 \, 069$ triangle $T_3$ is Pythagorean]].
+Then from [[Area of Triangle]], their [[Definition:Area|areas]] $A_1$, $A_2$ and $A_3$ respectively are given by:
+{{begin-eqn}}
+{{eqn | l = A_1
+ | r = \dfrac {4485 \times 5852} 2
+ | c =
+}}
+{{eqn | r = \dfrac {\paren {3 \times 5 \times 13 \times 23} \times \paren {2^2 \times 7 \times 11 \times 19} } 2
+ | c =
+}}
+{{eqn | r = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 19 \times 23
+ | c =
+}}
+{{eqn | r = 13 \, 123 \, 110
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = A_2
+ | r = \dfrac {3059 \times 8580} 2
+ | c =
+}}
+{{eqn | r = \dfrac {\paren {7 \times 19 \times 23} \times \paren {2^2 \times 3 \times 5 \times 11 \times 13} } 2
+ | c =
+}}
+{{eqn | r = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 19 \times 23
+ | c =
+}}
+{{eqn | r = 13 \, 123 \, 110
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = A_3
+ | r = \dfrac {1380 \times 19 \, 019} 2
+ | c =
+}}
+{{eqn | r = \dfrac {\paren {2^2 \times 3 \times 5 \times 23} \times \paren {7 \times 11 \times 13 \times 19} } 2
+ | c =
+}}
+{{eqn | r = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 19 \times 23
+ | c =
+}}
+{{eqn | r = 13 \, 123 \, 110
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown that this is the smallest such triple.}}
+\end{proof}<|endoftext|>
+\section{Sequence of 8 Consecutive Primes with Same Pattern of Differences as from 11}
+Tags: Prime Gaps
+
+\begin{theorem}
+The $8$ [[Definition:Prime Number|prime numbers]] starting at $15 \, 760 \, 091$ have the same [[Definition:Prime Gap|prime gaps]] as the $8$ [[Definition:Prime Number|primes]] starting at $11$.
+Those $8$ [[Definition:Prime Number|primes]] are:
+:$15 \, 760 \, 091, 15 \, 760 \, 093, 15 \, 760 \, 097, 15 \, 760 \, 099, 15 \, 760 \, 103, 15 \, 760 \, 109, 15 \, 760 \, 111, 15 \, 760 \, 117$
+Their [[Definition:Prime Gap|prime gaps]] are:
+:$2, 4, 2, 4, 6, 2, 6$
+The $8$ [[Definition:Prime Number|primes]] starting at $11$:
+:$11, 13, 17, 19, 23, 29, 31, 37$
+Their [[Definition:Prime Gap|prime gaps]] are:
+:$2, 4, 2, 4, 6, 2, 6$
+\end{theorem}<|endoftext|>
+\section{Smallest Even Integer whose Euler Phi Value is not the Euler Phi Value of an Odd Integer}
+Tags: 33,817,088, Euler Phi Function
+
+\begin{theorem}
+The smallest [[Definition:Even Integer|even integer]] whose [[Definition:Euler Phi Function|Euler $\phi$ value]] is shared by no [[Definition:Odd Integer|odd integer]] is $33 \, 817 \, 088$.
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = \phi \left({33 \, 817 \, 088}\right)
+ | r = 16 \, 842 \, 752
+ | c = {{EulerPhiLink|33,817,088|33 \, 817 \, 088}}
+}}
+{{eqn | r = 2^{16} \times 257
+ | c =
+}}
+{{end-eqn}}
+Consider the equation:
+:$(1): \quad \phi \left({x}\right) = 2^{16} \times 257$
+[[Definition:Even Integer|Even]] solutions to $(1)$ are of the form:
+:$x = 2^{9 - \epsilon_0 - 2 \epsilon_1 - 4 \epsilon_2} \times 3^{\epsilon_0} \times 5^{\epsilon_1} \times 17^{\epsilon_2} \times 257^2$
+where each $\epsilon_k$ equals $0$ or $1$.
+{{explain|Justify the above by showing the working.}}
+{{AimForCont}} there exists an [[Definition:Odd Integer|odd integer]] $k$ satisfying $(1)$:
+:$\phi \left({k}\right) = 2^{16} \times 257$
+Let $p$ be a [[Definition:Prime Factor|prime factor]] of $k$.
+Then as [[Euler Phi Function is Multiplicative]], either:
+:$p - 1 = 2^j$ for some $j \in \Z$ such that $1 \le j \le 16$
+or:
+:$p - 1 = 2^j \times 257$ for some $j \in \Z$ such that $1 \le j \le 16$
+But $2^j \times 257 + 1$ is [[Definition:Composite Number|composite]] for $1 \le j \le 16$:
+{{explain|Do the arithmetic to demonstrate the above}}
+So the only possible [[Definition:Prime Factor|prime factors]] of $k$ are the [[Definition:Fermat Prime|Fermat primes]]:
+:$3, 5, 17, 257, 65 \, 537$
+such that:
+: $257$ occurs with [[Definition:Multiplicity of Prime Factor|multiplicity]] $2$
+: all other [[Definition:Prime Factor|prime factors]] occurs with [[Definition:Multiplicity of Prime Factor|multiplicity]] $1$.
+Because of the size of $257^2 \times 67 \, 537$ it follows that $257$ and $65 \, 537$ cannot appear together.
+We have that:
+{{begin-eqn}}
+{{eqn | l = \phi \left({3^{\epsilon_0} \times 5^{\epsilon_1} \times 17^{\epsilon_2} \times 257^2}\right)
+ | r = 2^{8 + \epsilon_0 + 2 \epsilon_1 + 4 \epsilon_2} \times 257
+ | c =
+}}
+{{eqn | o = <
+ | r = 2^{16} \times 257
+ | c =
+}}
+{{end-eqn}}
+{{explain|expand the reasoning of the above}}
+Thus there can be no [[Definition:Odd Integer|odd integer]] $k$ satisfying $(1)$.
+It can be established by computer that $33 \, 817 \, 088$ is the smallest such [[Definition:Even Integer|even integer]] with this property.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Integer which is Sum of 2 Cubes in 4 Ways}
+Tags: Sums of Cubes, 42,549,416
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|positive integer]] which can be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Cube Number|cubes]] in $4$ different ways is:
+{{begin-eqn}}
+{{eqn | l = 42 \, 549 \, 416
+ | r = 348^3 + 74^3
+ | c =
+}}
+{{eqn | r = 282^3 + 272^3
+ | c =
+}}
+{{eqn | r = \left({-2662}\right)^3 + 2664^3
+ | c =
+}}
+{{eqn | r = \left({-475}\right)^3 + 531^3
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}<|endoftext|>
+\section{Squares whose Digits form Consecutive Integers}
+Tags: Square Numbers, Recreational Mathematics
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] whose [[Definition:Square (Algebra)|squares]] have a [[Definition:Decimal Notation|decimal representation]] consisting of the concatenation of $2$ consecutive [[Definition:Integer|integers]], either increasing or decreasing begins:
+:$91, 428, 573, 727, 846, 7810, 9079, 9901, 36 \, 365, 63 \, 636, 326 \, 734, 673 \, 267, 733 \, 674, \ldots$
+This sequence can be divided into two subsequences:
+Those where the consecutive [[Definition:Integer|integers]] are increasing:
+:$428, 573, 727, 846, 7810, 36 \, 365, 63 \, 636, 326 \, 734, 673 \, 267, \ldots$
+{{OEIS|A030467}}
+Those where the consecutive [[Definition:Integer|integers]] are decreasing:
+:$91, 9079, 9901, 733 \, 674, 999 \, 001, 88 \, 225 \, 295, \ldots$
+{{OEIS|A054216}}
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = 91^2
+ | r = 8281
+ | c =
+}}
+{{eqn | l = 428^2
+ | r = 183 \, 184
+ | c =
+}}
+{{eqn | l = 573^2
+ | r = 328 \, 329
+ | c =
+}}
+{{eqn | l = 727^2
+ | r = 528 \, 529
+ | c =
+}}
+{{eqn | l = 846^2
+ | r = 715 \, 716
+ | c =
+}}
+{{eqn | l = 7810^2
+ | r = 6099 \, 6100
+ | c =
+}}
+{{eqn | l = 9079^2
+ | r = 82 \, 428 \, 241
+ | c =
+}}
+{{eqn | l = 9901^2
+ | r = 98 \, 029 \, 801
+ | c =
+}}
+{{eqn | l = 36 \, 365^2
+ | r = 13224 \, 13225
+ | c =
+}}
+{{eqn | l = 63 \, 636^2
+ | r = 40495 \, 40496
+ | c =
+}}
+{{eqn | l = 326 \, 734^2
+ | r = 106755 \, 106756
+ | c =
+}}
+{{end-eqn}}
+They can be determined by inspection.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Squares whose Digits form Consecutive Decreasing Integers}
+Tags: Square Numbers, Recreational Mathematics
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] whose [[Definition:Square (Algebra)|squares]] have a [[Definition:Decimal Notation|decimal representation]] consisting of the concatenation of $2$ consecutive decreasing [[Definition:Integer|integers]] begins:
+:$91, 9079, 9901, 733 \, 674, 999 \, 001, 88 \, 225 \, 295, 99 \, 990 \, 001, \ldots$
+{{OEIS|A030467}}
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = 91^2
+ | r = 8281
+ | c =
+}}
+{{eqn | l = 9079^2
+ | r = 82 \, 428 \, 241
+ | c =
+}}
+{{eqn | l = 9901^2
+ | r = 98 \, 029 \, 801
+ | c =
+}}
+{{eqn | l = 733 \, 674^2
+ | r = 538 \, 277 \, 538 \, 276
+ | c =
+}}
+{{eqn | l = 999 \, 001^2
+ | r = 998 \, 002 \, 998 \, 001
+ | c =
+}}
+{{end-eqn}}
+They can be determined by inspection.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Numbers n whose Euler Phi value Divides n + 1}
+Tags: Euler Phi Function
+
+\begin{theorem}
+The following [[Definition:Integer|integers]] $n$ satisfy the equation:
+:$\exists k \in \Z: k \, \map \phi n = n + 1$
+where $\phi$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ function]]:
+:$83 \, 623 \, 935, 83 \, 623 \, 935 \times 83 \, 623 \, 937$
+\end{theorem}
+
+\begin{proof}
+From {{EulerPhiLink|83,623,935|83 \, 623 \, 935}}:
+:$\map \phi {83 \, 623 \, 935} = 41 \, 811 \, 968$
+and then:
+{{begin-eqn}}
+{{eqn | l = 2 \times 41 \, 811 \, 968
+ | r = 83 \, 623 \, 936
+ | c =
+}}
+{{eqn | r = 1 + 83 \, 623 \, 935
+ | c =
+}}
+{{end-eqn}}
+{{qed|lemma}}
+Then we have that $83 \, 623 \, 937$ is the $868 \, 421$st [[Definition:Prime Number|prime number]].
+From [[Euler Phi Function of Prime]]:
+:$\map \phi {83 \, 623 \, 937} = 83 \, 623 \, 936$
+{{begin-eqn}}
+{{eqn | l = \map \phi {83 \, 623 \, 935 \times 83 \, 623 \, 937}
+ | r = \map \phi {83 \, 623 \, 935} \times \map \phi {83 \, 623 \, 937}
+ | c = [[Euler Phi Function is Multiplicative]]
+}}
+{{eqn | r = 41 \, 811 \, 968 \times 83 \, 623 \, 936
+ | c = from above
+}}
+{{end-eqn}}
+Then we have that:
+{{begin-eqn}}
+{{eqn | l = 83 \, 623 \, 935 \times 83 \, 623 \, 937
+ | r = \paren {83 \, 623 \, 936 - 1} \times \paren {83 \, 623 \, 936 + 1}
+ | c =
+}}
+{{eqn | r = 83 \, 623 \, 936^2 - 1^2
+ | c = [[Difference of Two Squares]]
+}}
+{{end-eqn}}
+and so:
+{{begin-eqn}}
+{{eqn | l = 2 \times \map \phi {83 \, 623 \, 935 \times 83 \, 623 \, 937}
+ | r = 2 \times 41 \, 811 \, 968 \times 83 \, 623 \, 936
+ | c =
+}}
+{{eqn | r = 83 \, 623 \, 936^2
+ | c =
+}}
+{{eqn | r = 83 \, 623 \, 935 \times 83 \, 623 \, 937 + 1
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Hardy-Ramanujan Number/Examples/87,539,319}
+Tags: Hardy-Ramanujan Numbers, 87,539,319
+
+\begin{theorem}
+The $3$rd [[Definition:Hardy-Ramanujan Number|Hardy-Ramanujan number]] $\operatorname {Ta} \left({3}\right)$ is $87 \, 539 \, 319$:
+{{begin-eqn}}
+{{eqn | l = 87 \, 539 \, 319
+ | r = 167^3 + 436^3
+ | c =
+}}
+{{eqn | r = 228^3 + 423^3
+ | c =
+}}
+{{eqn | r = 255^3 + 414^3
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}<|endoftext|>
+\section{Polynomial is Linear Combination of Monomials}
+Tags: Polynomial Theory, Monomials
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+Let $R \sqbrk X$ be a [[Definition:Polynomial Ring|polynomial ring]] over $R$ in the [[Definition:Variable of Polynomial Ring|variable]] $X$.
+Let $P \in R \sqbrk X$.
+Then $P$ is a [[Definition:Linear Combination|linear combination]] of the [[Definition:Monomial of Polynomial Ring|monomials]] of $R \sqbrk X$, with coefficients in $R$.
+{{explain|this needs to be made more precise}}
+\end{theorem}
+
+\begin{proof}
+Let $S \subset R \sqbrk X$ be the [[Definition:Subset|subset]] of all elements that are [[Definition:Linear Combination|linear combinations]] of [[Definition:Monomial of Polynomial Ring|monomials]].
+Let $\iota: S \to R \sqbrk X$ denote the [[Definition:Inclusion Mapping|inclusion mapping]].
+Suppose for the moment that $S$ is a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+Then by [[Universal Property of Polynomial Ring]], there exists a [[Definition:Ring Homomorphism|ring homomorphism]] $g: R \sqbrk X \to S$ with $\map g X = X$.
+By [[Inclusion Mapping on Subring is Homomorphism]], $\iota: S \to R \sqbrk X$ is a [[Definition:Ring Homomorphism|ring homomorphism]].
+By [[Composition of Ring Homomorphisms is Ring Homomorphism]], $\iota \circ g: R \sqbrk X \to R \sqbrk X$ is a [[Definition:Ring Homomorphism|ring homomorphism]].
+By construction, $\map {\paren {\iota \circ g} } X = X$.
+By [[Universal Property of Polynomial Ring]], there exists a [[Definition:Unique|unique]] [[Definition:Ring Homomorphism|ring homomorphism]] $h : R \sqbrk X \to R \sqbrk X$ with $\map h X = X$.
+We have that $\iota \circ g$ is such a [[Definition:Ring Homomorphism|ring homomorphism]].
+By [[Identity Mapping is Ring Automorphism]], the [[Definition:Identity Mapping|identity mapping]] $I$ on $R \sqbrk X$ is one too.
+By [[Definition:Unique|uniqueness]], $\iota \circ g = I$.
+By [[Identity Mapping is Surjection]] and [[Surjection if Composite is Surjection]], $\iota$ is a [[Definition:Surjection|surjection]].
+By [[Inclusion Mapping is Surjection iff Identity]], $S = R \sqbrk X$.
+It remains to show that $S$ is a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+{{finish}}
+\end{proof}<|endoftext|>
+\section{Sum over Disjoint Union of Finite Sets}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $S$ and $T$ be [[Definition:Finite Set|finite]] [[Definition:Disjoint Sets|disjoint sets]].
+Let $S \cup T$ be their [[Definition:Set Union|union]].
+Let $f: S \cup T \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Then we have the equality of [[Definition:Summation|summations]] over [[Definition:Finite Set|finite sets]]:
+:$\displaystyle \sum_{u \mathop \in S \mathop \cup T} f \left({u}\right) = \sum_{s \mathop \in S} f \left({s}\right) + \sum_{t \mathop \in T} f \left({t}\right)$
+\end{theorem}
+
+\begin{proof}
+Note that by [[Union of Finite Sets is Finite]], the [[Definition:Set Union|union]] $S \cup T$ is [[Definition:Finite Set|finite]].
+Let $m$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $S$ and $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $T$.
+Let $\N_{< m}$ denote an [[Definition:Initial Segment of Natural Numbers|initial segment of the natural numbers]].
+Let $\sigma: \N_{< m} \to S$ and $\tau: \N_{< n} \to T$ be [[Definition:Bijection|bijections]].
+Let $\alpha: \N_{< n} \to \left[{m \,.\,.\, m + n - 1}\right]$ be the [[Definition:Mapping|mapping]] defined as:
+:$\alpha \left({k}\right) = k + m$
+By [[Translation of Integer Interval is Bijection]], $\alpha$ is a [[Definition:Bijection|bijection]].
+By [[Composite of Bijections is Bijection]] and [[Disjoint Union of Bijections is Bijection]], the [[Definition:Union of Mappings|union]]:
+:$\sigma \cup (\tau\circ \alpha) : \N_{< m} \cup \left[{m \,.\,.\, m + n - 1}\right] \to S \cup T$ is a [[Definition:Bijection|bijection]].
+By [[Union of Integer Intervals]], $\N_{< m} \cup \left[{m \,.\,.\, m + n - 1}\right] = \N_{< m + n}$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{u \mathop \in S \mathop \cup T} f \left({u}\right)
+ | r = \sum_{i \mathop = 0}^{m + n - 1} f \circ \left({\sigma \cup \left({\tau \circ \alpha}\right)}\right) \left({i}\right)
+ | c = {{Defof|Summation}}
+}}
+{{eqn | r = \sum_{i \mathop = 0}^{m - 1} f \circ \left({\sigma \cup \left({\tau \circ \alpha}\right)}\right) \left({i}\right) + \sum_{i \mathop = m}^{m + n - 1} f \circ \left({\sigma \cup \left({\tau \circ \alpha}\right)}\right) \left({i}\right)
+ | c = [[Indexed Summation over Adjacent Intervals]]
+}}
+{{eqn | r = \sum_{i \mathop = 0}^{m - 1} \left({f \circ \sigma}\right) \left({i}\right) + \sum_{i \mathop = m}^{m + n - 1} \left({g \circ \tau \circ \alpha}\right) \left({i}\right)
+ | c = [[Restriction of Union of Mappings to Component of Domain]]
+}}
+{{eqn | r = \sum_{i \mathop = 0}^{m - 1} \left({f \circ \sigma}\right) \left({i}\right) + \sum_{i \mathop = 0}^{n - 1} \left({g \circ \tau}\right) \left({i}\right)
+ | c = Definition of $\alpha$, [[Indexed Summation over Translated Interval]]
+}}
+{{eqn | r = \displaystyle \sum_{s \mathop \in S} f \left({s}\right) + \sum_{t \mathop \in T} g \left({t}\right)
+ | c = {{Defof|Summation}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Finite Summation does not Change under Permutation}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $S$ be a [[Definition:Finite Set|finite set]].
+Let $f : S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Let $\sigma : S\to S$ be a [[Definition:Permutation|permutation]].
+Then we have the equality of [[Definition:Summation|summations]] over [[Definition:Finite Set|finite sets]]:
+:$\displaystyle \sum_{s \mathop \in S} \map f s = \sum_{s \mathop \in S} \map f {\map \sigma s}$
+\end{theorem}
+
+\begin{proof}
+This is a special case of [[Change of Variables in Summation over Finite Set]].
+{{qed}}
+[[Category:Summations]]
+1posxiyi6afj4e2emhfo71ku7qmtc43
+\end{proof}<|endoftext|>
+\section{Summation over Finite Set is Well-Defined}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $S$ be a [[Definition:Finite Set|finite set]].
+Let $f: S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Let $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $S$.
+let $\N_{
+\section{Change of Variables in Summation over Finite Set}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $S$ and $T$ be [[Definition:Finite Set|finite sets]].
+Let $f: S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Let $g: T \to S$ be a [[Definition:Bijection|bijection]].
+Then we have an equality of [[Definition:Summation|summations]] over [[Definition:Finite Set|finite sets]]:
+:$\displaystyle \sum_{s \mathop \in S} f \left({s}\right) = \sum_{t \mathop \in T} f \left({g \left({t}\right)}\right)$
+\end{theorem}
+
+\begin{proof}
+Let $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $S$ and $T$.
+Let $\N_{
+\section{Indexed Summation does not Change under Permutation}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $a$ and $b$ be [[Definition:Integer|integers]].
+Let $\closedint a b$ be the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
+Let $f: \closedint a b \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Let $\sigma: \closedint a b \to \closedint a b$ be a [[Definition:Permutation|permutation]].
+Then we have an equality of [[Definition:Indexed Summation|indexed summations]]:
+:$\displaystyle \sum_{i \mathop = a}^b \map f i = \sum_{i \mathop = a}^b \map f {\map \sigma i}$
+\end{theorem}
+
+\begin{proof}
+Let $a > b$.
+Then by definition of [[Definition:Indexed Summation|indexed summation]], both sides are $0$.
+Let $a \le b$.
+=== Reduction to $a=1$ ===
+By [[Indexed Summation over Translated Interval]]:
+:$\displaystyle \sum_{i \mathop = a}^b \map f i = \sum_{j \mathop = 1}^{b - a + 1} \map f {j - a + 1}$
+:$\displaystyle \sum_{i \mathop = a}^b \map f {\map \sigma i} = \sum_{j \mathop = 1}^{b - a + 1} \map f {\map \sigma {j - a + 1} }$
+Let $n = b - a + 1$.
+Because $a \le b$, we have $n \ge 1$.
+By [[Translation of Integer Interval is Bijection]], the [[Definition:Mapping|mapping]] $T : \closedint 1 n \to \closedint a b$ defined by:
+:$\map T i = i + a - 1$
+is a [[Definition:Bijection|bijection]].
+We have:
+:$\displaystyle \sum_{i \mathop = a}^b \map f i = \sum_{j \mathop = 1}^n \map f {\map T j}$
+:$\displaystyle \sum_{i \mathop = a}^b \map f {\map \sigma i} = \sum_{j \mathop = 1}^n \map f {\map \sigma {\map T j} }$
+By [[Inverse of Bijection is Bijection]] and [[Composite of Bijections is Bijection]], the [[Definition:Composition of Mappings|composition]] $T^{-1} \circ \sigma \circ T$ is a [[Definition:Permutation|permutation]] of $\closedint 1 n$.
+Suppose we have settled the case $a = 1$.
+Then:
+:$\displaystyle \sum_{j \mathop = 1}^n \map f {\map T j} = \sum_{j \mathop = 1}^n \map {\paren {f \circ T} \circ \paren {T^{-1} \circ \sigma \circ T} } j$
+By [[Composition of Mappings is Associative]], this equals:
+:$\displaystyle \sum_{j \mathop = 1}^n \paren {f \circ \sigma \circ T} j$
+Thus:
+:$\displaystyle \sum_{i \mathop = a}^b \map f i = \sum_{i \mathop = a}^b \map f {\map \sigma i}$
+{{qed|lemma}}
+It remains to investigate the case $a = 1$.
+=== The case $a = 1$ ===
+It remains to show that
+:$\displaystyle \sum_{i \mathop = 1}^n \map f i = \sum_{i \mathop = 1}^n \map f {\map \sigma i}$
+for any [[Definition:Permutation|permutation]] $\sigma$ of $\closedint 1 n$.
+Let $S_n$ be the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]], that is, the [[Definition:Symmetric Group|group of permutations]] of $\closedint 1 n$.
+Let $U \subset S_n$ be the [[Definition:Subset|subset]] of all [[Definition:Permutation|permutations]] $\sigma$ satisfying
+:$\displaystyle \sum_{i \mathop = 1}^n \map f i = \sum_{i \mathop = 1}^n \map f {\map \sigma i}$
+for all [[Definition:mapping|mappings]] $f: \closedint 1 n \to \mathbb A$.
+{{qed|lemma}}
+==== Summation-preserving permutations form subgroup ====
+We prove that $U$ is a [[Definition:Subgroup|subgroup]] of $S_n$, using [[Two-Step Subgroup Test]].
+We have to show that $U$ is [[Definition:Non-Empty Set|non-empty]].
+By [[Identity Mapping is Permutation]] the [[Definition:Identity Mapping|identity mapping]] $I$ on $\closedint 1 n$ is a [[Definition:Permutation|permutation]].
+By [[Identity Mapping is Right Identity]], $f \circ I = f$.
+Thus:
+:$\displaystyle \sum_{i \mathop = 1}^n \map f i = \sum_{i \mathop = 1}^n \map f {\map I i}$
+Thus $I \in U$.
+Let $\sigma, \tau \in U$.
+We have to show that $\sigma \circ \tau$ is in $U$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = 1}^n \map {f \circ \paren {\sigma \circ \tau} } i
+ | r = \sum_{i \mathop = 1}^n \map {\paren {\paren {f \circ \sigma} \circ \tau} } i
+ | c = [[Composition of Mappings is Associative]]
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n \map {\paren {f \circ \sigma} } i
+ | c = $\tau \in U$
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n \map f i
+ | c = $\sigma \in U$
+}}
+{{end-eqn}}
+Thus $\sigma \circ \tau \in U$.
+We show that $\sigma^{-1} \in U$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = 1}^n \map {\paren {f \circ \sigma^{-1} } } i
+ | r = \sum_{i \mathop = 1}^n \map {\paren {\paren {f \circ \sigma^{-1} } \circ \sigma} } i
+ | c = $\sigma \in U$
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n \map {\paren {f \circ \paren {\sigma^{-1} \circ \sigma} } } i
+ | c = [[Composition of Mappings is Associative]]
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n \map {\paren {f \circ I} } i
+ | c = [[Identity Mapping is Identity Element in Group of Permutations]]
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n \map f i
+ | c = $I \in U$
+}}
+{{end-eqn}}
+Thus $\sigma^{-1} \in U$.
+Thus $U$ is a [[Definition:Subgroup|subgroup]] of $S_n$.
+{{qed|lemma}}
+==== Last adjacent transposition preserves summations ====
+If $n = 1$, then by [[First Symmetric Group is Trivial Group]] and [[Subgroups of Trivial Group]], $U = S_n$.
+Let $n \ge 2$.
+By [[Symmetric Group on n Letters is Generated by Standard Cycle and Adjacent Transposition]], it suffices to show that $U$ contains an [[Definition:Adjacent Transposition|adjacent transposition]] and the [[Definition:Standard Cycle|standard cycle]].
+Let $\sigma \in S_n$ be the [[Definition:Adjacent Transposition|adjacent transposition]] $\begin {bmatrix} n - 1 & n \end {bmatrix}$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = 1}^n \map f {\map \sigma i}
+ | r = \sum_{i \mathop = 1}^{n - 1} \map f {\map \sigma i} + \map f {\map \sigma n}
+ | c = {{Defof|Indexed Summation}}
+}}
+{{eqn | r = \sum_{i \mathop = 1}^{n - 2} \map f {\map \sigma i} + \map f {\map \sigma {n - 1} } + \map f {\map \sigma n}
+ | c = {{Defof|Indexed Summation}}
+}}
+{{eqn | r = \sum_{i \mathop = 1}^{n - 2} \map f i + \map f n + \map f {n - 1}
+ | c = {{Defof|Adjacent Transposition}}
+}}
+{{eqn | r = \sum_{i \mathop = 1}^{n - 2} \map f i + \map f {n - 1} + \map f n
+ | c = [[Commutative Law of Addition]]
+}}
+{{eqn | r = \sum_{i \mathop = 1}^{n - 1} \map f i + \map f n
+ | c = {{Defof|Indexed Summation}}
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n \map f i
+ | c = {{Defof|Indexed Summation}}
+}}
+{{end-eqn}}
+Thus $\begin {bmatrix} n - 1 & n \end {bmatrix} \in U$.
+{{qed|lemma}}
+==== Standard cycle preserves summations ====
+Let $\sigma$ be the [[Definition:Standard Cycle|standard cycle]] $\begin {bmatrix} 1 & \ldots & n \end {bmatrix}$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = 1}^n \map f {\map \sigma i}
+ | r = \sum_{i \mathop = 1}^{n - 1} \map f {\map \sigma i} + \map f {\map \sigma n}
+ | c = {{Defof|Indexed Summation}}, $n \ge 1$
+}}
+{{eqn | r = \sum_{i \mathop = 1}^{n - 1} \map f {i + 1} + \map f 1
+ | c = {{Defof|Standard Cycle}}
+}}
+{{eqn | r = \sum_{i \mathop = 2}^n \map f i + \map f 1
+ | c = [[Indexed Summation over Translated Interval]]
+}}
+{{eqn | r = \map f 1 + \sum_{i \mathop = 2}^n \map f i
+ | c = [[Commutative Law of Addition]]
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n \map f i
+ | c = [[Indexed Summation without First Term]]
+}}
+{{end-eqn}}
+Thus $\begin {bmatrix} 1 & \ldots & n \end {bmatrix} \in U$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Indexed Summation over Translated Interval}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $a$ and $b$ be [[Definition:Integer|integers]].
+Let $\closedint a b$ be the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
+Let $f: \closedint a b \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Let $c\in\Z$ be an [[Definition:Integer|integer]].
+Then we have an equality of [[Definition:Indexed Summation|indexed summations]]:
+:$\displaystyle \sum_{i \mathop = a}^b f(i) = \sum_{i \mathop = a + c}^{b + c} \map f {i - c}$
+\end{theorem}
+
+\begin{proof}
+The proof goes by [[Principle of Mathematical Induction|induction]] on $b$.
+=== Basis for the Induction ===
+Let $b < a$.
+Then:
+:$b + c < a + c$
+Thus both [[Definition:Indexed Summation|indexed summations]] are [[Definition:Zero of Standard Number System|zero]].
+This is our [[Definition:Basis for the Induction|basis for the induction]].
+=== Induction Step ===
+Let $b \ge a$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = a + c}^{b + c} \map f {i - c}
+ | r = \sum_{i \mathop = a + c}^{b + c - 1} \map f {i - c} + \map f {b + c - c}
+ | c = {{Defof|Indexed Summation}}
+}}
+{{eqn | r = \sum_{i \mathop = a}^{b - 1} \map f i + \map f b
+ | c = [[Definition:Induction Hypothesis|Induction Hypothesis]]
+}}
+{{eqn | r = \sum_{i \mathop = a}^b \map f i
+ | c = {{Defof|Indexed Summation}}
+}}
+{{end-eqn}}
+By the [[Principle of Mathematical Induction]], the proof is complete.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Indexed Summation over Adjacent Intervals}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N,\Z,\Q,\R,\C$.
+Let $a,b,c$ be [[Definition:Integer|integers]].
+Let $\left[{a \,.\,.\, c}\right]$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $c$.
+Let $b \in \left[{a-1 \,.\,.\, c}\right]$.
+Let $f : \left[{a \,.\,.\, c}\right] \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Then we have an equality of [[Definition:Indexed Summation|indexed summations]]:
+:$\displaystyle \sum_{i \mathop = a}^c f(i) = \sum_{i \mathop = a}^b f(i) + \sum_{i \mathop = b+1}^c f(i)$
+\end{theorem}
+
+\begin{proof}
+The proof goes by [[Principle of Mathematical Induction|induction]] on $b$.
+=== Basis for the Induction ===
+Let $b = a-1$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = a}^b f(i) + \sum_{i \mathop = b+1}^c f(i)
+ | r = 0 + \sum_{i \mathop = b+1}^c f(i)
+ | c = {{Defof|Indexed Summation}}, $b=a-1$
+}}
+{{eqn | l =
+ | r = \sum_{i \mathop = a}^c f(i)
+ | c = [[Identity Element of Addition on Numbers]]
+}}
+{{end-eqn}}
+This is our [[Principle of Mathematical Induction#Basis for the Induction|basis for the induction]].
+=== Induction Step ===
+Let $a \leq b \leq c$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = a}^b f(i) + \sum_{i \mathop = b+1}^c f(i)
+ | r = \sum_{i \mathop = a}^{b-1} f(i) + f(b) + \sum_{i \mathop = b+1}^c f(i)
+ | c = {{Defof|Indexed Summation}}
+}}
+{{eqn | r = \sum_{i \mathop = a}^{b-1} f(i) + \sum_{i \mathop = b}^c f(i)
+ | c = [[Indexed Summation without First Term]]
+}}
+{{end-eqn}}
+By the [[Principle of Mathematical Induction]], the proof is complete.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Indexed Summation over Interval of Length Two}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N,\Z,\Q,\R,\C$.
+Let $a\in\Z$ be an [[Definition:Integer|integer]].
+Let $f : \{a, a+1\} \to \mathbb A$ be a [[Definition:Real-Valued Function|real-valued function]].
+Then the [[Definition:Indexed Summation|indexed summation]]:
+:$\displaystyle \sum_{i \mathop = a}^{a+1} f(i) = f(a) + f(a+1)$.
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = a}^{a+1} f(i)
+ | r = \sum_{i \mathop = a}^a f(i) + f(a+1)
+ | c = {{Defof|Indexed Summation}}
+}}
+{{eqn | l =
+ | r = f(a) + f(a+1)
+ | c = [[Indexed Summation over Interval of Length One]]
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Summations]]
+1fmazkc5z3mw4wvpz1inhwswolxgv6k
+\end{proof}<|endoftext|>
+\section{Indexed Summation over Interval of Length One}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N,\Z,\Q,\R,\C$.
+Let $a\in\Z$ be an [[Definition:Integer|integer]].
+Let $f : \{a\} \to \mathbb A$ be a [[Definition:Mapping|mapping]] on the [[Definition:Singleton|singleton]] $\{a \}$.
+Then the [[Definition:Indexed Summation|indexed summation]]:
+:$\displaystyle \sum_{i \mathop = a}^{a} f(i) = f(a)$
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = a}^{a} f(i)
+ | r = \sum_{i \mathop = a}^{a-1} f(i) + f(a)
+ | c = {{Defof|Indexed Summation}}
+}}
+{{eqn | l =
+ | r = 0 + f(a)
+ | c = {{Defof|Indexed Summation}}, $a-1 < a$
+}}
+{{eqn | l =
+ | r = f(a)
+ | c = [[Identity Element of Addition on Numbers]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Change of Variables in Indexed Summation}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $a, b, c, d$ be [[Definition:Integer|integers]].
+Let $\closedint a b$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
+Let $f: \closedint a b \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Let $g: \closedint c d \to \closedint a b$ be a [[Definition:Bijection|bijection]].
+Then we have an equality of [[Definition:Indexed Summation|indexed summations]]:
+:$\displaystyle \sum_{i \mathop = a}^b \map f i = \sum_{i \mathop = c}^d \map f {\map g i}$
+\end{theorem}
+
+\begin{proof}
+Because $g : \closedint c d \to \closedint a b$ is a [[Definition:Bijection|bijection]], these [[Definition:Set|sets]] are [[Definition:Set Equivalence|equivalent]].
+By [[Cardinality of Integer Interval]], $\closedint a b$ has [[Definition:Cardinality of Finite Set|cardinality]] $b - a + 1$.
+Thus:
+:$b - a + 1 = d - c + 1$
+Thus
+:$c - a = d - b$
+By [[Indexed Summation over Translated Interval]]:
+:$\displaystyle \sum_{i \mathop = c}^d \map f {\map g i} = \sum_{i \mathop = a}^b \map f {\map g {i + c - a} }$
+By [[Translation of Integer Interval is Bijection]], the [[Definition:Mapping|mapping]] $T : \closedint a b \to \closedint c d$ defined as:
+:$\map T k = k + c - a$
+is a [[Definition:Bijection|bijection]].
+By [[Composite of Bijections is Bijection]], $g \circ T$ is a [[Definition:Permutation|permutation]] of $\closedint a b$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = c}^d \map f {\map g i}
+ | r = \sum_{i \mathop = a}^b \map f {\map g {i + c - a} }
+ | c = [[Indexed Summation over Translated Interval]]
+}}
+{{eqn | r = \sum_{i \mathop = a}^b \map f {\map g {\map T i} }
+ | c = Definition of $T$
+}}
+{{eqn | r = \sum_{i \mathop = a}^b \map f i
+ | c = [[Indexed Summation does not Change under Permutation]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Translation of Integer Interval is Bijection}
+Tags: Set Theory
+
+\begin{theorem}
+Let $a, b, c \in \Z$ be [[Definition:Integer|integers]].
+Let $\closedint a b$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
+Then the [[Definition:Mapping|mapping]] $T: \closedint a b \to \closedint {a + c} {b + c}$ defined as:
+:$\map T k = k + c$
+is a [[Definition:Bijection|bijection]].
+\end{theorem}
+
+\begin{proof}
+Note that if $k \in \closedint a b$, then indeed $k + c \in \closedint {a + c} {b + c}$.
+=== Injectivity ===
+Let $k, l \in \closedint a b$ with $k + c = l + c$.
+By [[Integer Addition is Cancellable]], $k = l$.
+Thus $T$ is [[Definition:Injection|injective]].
+{{qed|lemma}}
+=== Surjectivity ===
+Let $m \in \closedint {a + c} {b + c}$.
+Then $m - c \in \closedint a b$.
+Then $\map T {m - c} = m - c + c = m$.
+Thus $T$ is [[Definition:Surjection|surjective]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Indexed Summation without First Term}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N,\Z,\Q,\R,\C$.
+Let $a$ and $b$ be [[Definition:Integer|integers]] with $a\leq b$.
+Let $\left[{a \,.\,.\, b}\right]$ be the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
+Let $f : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Then we have an equality of [[Definition:Indexed Summation|indexed summations]]:
+:$\displaystyle \sum_{i \mathop = a}^b f(i) = f(a) + \sum_{i \mathop = a+1}^b f(\sigma(i))$
+\end{theorem}
+
+\begin{proof}
+The proof goes by [[Principle of Mathematical Induction|induction]] on $b$.
+=== Basis for the Induction ===
+Let $b=a$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = a}^a f(i)
+ | r = f(a)
+ | c = [[Indexed Summation over Interval of Length One]]
+}}
+{{eqn | r = f(a) + 0
+ | c = [[Identity Element of Addition on Numbers]]
+}}
+{{eqn | r = f(a) + \sum_{i \mathop = a+1}^{a} f(i)
+ | c = {{Defof|Indexed Summation}}, $a+1 > a$
+}}
+{{end-eqn}}
+This is our [[Principle of Mathematical Induction#Basis for the Induction|basis for the induction]].
+=== Induction Step ===
+Let $b \geq a+1$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = a}^b f(i)
+ | r = \sum_{i \mathop = a}^{b-1} f(i) + f(b)
+ | c = {{Defof|Indexed Summation}}, $b \geq a$
+}}
+{{eqn | r = f(a) + \sum_{i \mathop = a+1}^{b-1} f(i) + f(b)
+ | c = [[Principle of Mathematical Induction#Induction Hypothesis|induction hypothesis]]
+}}
+{{eqn | r = f(a) + \sum_{i \mathop = a+1}^{b} f(i)
+ | c = {{Defof|Indexed Summation}}, $b \geq a+1$
+}}
+{{end-eqn}}
+By the [[Principle of Mathematical Induction]], the proof is complete.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Summation over Interval equals Indexed Summation}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $a, b \in \Z$ be [[Definition:Integer|integers]].
+Let $\left[{a \,.\,.\, b}\right]$ be the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
+Let $f: \left[{a \,.\,.\, b}\right] \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Then the [[Definition:Summation|summation]] over the [[Definition:Finite Set|finite set]] $\left[{a \,.\,.\, b}\right]$ equals the [[Definition:Indexed Summation|indexed summation]] from $a$ to $b$:
+:$\displaystyle\sum_{k \mathop \in \left[{a \,.\,.\, b}\right]} f \left({k}\right) = \sum_{k \mathop = a}^b f \left({k}\right)$
+\end{theorem}
+
+\begin{proof}
+By [[Cardinality of Integer Interval]], $\left[{a \,.\,.\, b}\right]$ has [[Definition:Cardinality of Finite Set|cardinality]] $b - a + 1$.
+By [[Translation of Integer Interval is Bijection]], the [[Definition:Mapping|mapping]] $T : \left[{0 \,.\,.\, b - a}\right] \to \left[{a \,.\,.\, b}\right]$ defined as:
+:$T \left({k}\right) = k + a$
+is a [[Definition:Bijection|bijection]].
+By definition of [[Definition:Summation|summation]]:
+:$\displaystyle \sum_{k \mathop \in \left[{a \,.\,.\, b}\right]} f \left({k}\right) = \sum_{k \mathop = 0}^{b - a} f \left({k + a}\right)$
+By [[Indexed Summation over Translated Interval]]:
+:$\displaystyle \sum_{k \mathop = 0}^{b - a} f \left({k + a}\right) = \sum_{k \mathop = a}^b f \left({k}\right)$
+{{qed}}
+[[Category:Summations]]
+1cv3z7q8i4xg9ek6l2wzjw6t8jhuwcb
+\end{proof}<|endoftext|>
+\section{Hardy-Ramanujan Number/Examples/1729}
+Tags: Hardy-Ramanujan Numbers
+
+\begin{theorem}
+The $2$nd [[Definition:Hardy-Ramanujan Number|Hardy-Ramanujan number]] $\map {\operatorname {Ta}} 2$ is $1729$:
+{{begin-eqn}}
+{{eqn | l = 1729
+ | r = 12^3 + 1^3
+ | c =
+}}
+{{eqn | r = 10^3 + 9^3
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|that there are none smaller}}
+[[Category:Hardy-Ramanujan Numbers]]
+3zxwhv63uu98drg4wtczji4auaqcnbk
+\end{proof}<|endoftext|>
+\section{Cardinality of Integer Interval}
+Tags: Set Theory
+
+\begin{theorem}
+Let $a, b \in \Z$ be [[Definition:Integer|integers]].
+Let $\left[{a \,.\,.\, b}\right]$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
+Then $\left[{a \,.\,.\, b}\right]$ is [[Definition:Finite Set|finite]] and its [[Definition:Cardinality of Finite Set|cardinality]] equals:
+:$\begin{cases}
+b - a + 1 & : b \ge a - 1 \\
+0 & : b \le a - 1
+\end{cases}$
+\end{theorem}
+
+\begin{proof}
+Let $b < a$.
+Then $\left[{a \,.\,.\, b}\right]$ is [[Definition:Empty Set|empty]].
+By [[Empty Set is Finite]], $\left[{a \,.\,.\, b}\right]$ is [[Definition:Finite Set|finite]].
+By [[Cardinality of Empty Set]], $\left[{a \,.\,.\, b}\right]$ has [[Definition:Cardinality of Finite Set|cardinality]] $0$.
+Let $b \ge a$.
+By [[Translation of Integer Interval is Bijection]], there exists a [[Definition:Bijection|bijection]] between $\left[{a \,.\,.\, b}\right]$ and $\left[{0 \,.\,.\, b - a}\right]$.
+Thus $\left[{a \,.\,.\, b}\right]$ is [[Definition:Finite Set|finite]] of [[Definition:Cardinality of Finite Set|cardinality]] $b - a + 1$.
+{{qed}}
+[[Category:Set Theory]]
+tr6bchkgtxm9a40pp9dzcux54f7jo80
+\end{proof}<|endoftext|>
+\section{Hardy-Ramanujan Number/Examples/6,963,472,309,248}
+Tags: Hardy-Ramanujan Numbers, 6,963,472,309,248
+
+\begin{theorem}
+The $4$th [[Definition:Hardy-Ramanujan Number|Hardy-Ramanujan number]] $\operatorname {Ta} \left({4}\right)$ is $6 \, 963 \, 472 \, 309 \, 248$:
+{{begin-eqn}}
+{{eqn | l = 6 \, 963 \, 472 \, 309 \, 248
+ | r = 2421^3 + 19 \, 083^3
+ | c =
+}}
+{{eqn | r = 5436^3 + 18 \, 948^3
+ | c =
+}}
+{{eqn | r = 10 \, 200^3 + 18 \, 072^3
+ | c =
+}}
+{{eqn | r = 13 \, 322^3 + 16 \, 630^3
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}<|endoftext|>
+\section{Indexed Summation of Sum of Mappings}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $a, b$ be [[Definition:Integer|integers]].
+Let $\closedint a b$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
+Let $f, g: \closedint a b \to \mathbb A$ be [[Definition:Mapping|mappings]].
+Let $h = f + g$ be their [[Definition:Pointwise Addition|pointwise sum]].
+Then we have the equality of [[Definition:Indexed Summation|indexed summations]]:
+:$\displaystyle \sum_{i \mathop = a}^b \map h i = \sum_{i \mathop = a}^b \map f i + \sum_{i \mathop = a}^b \map g i$
+\end{theorem}
+
+\begin{proof}
+The proof proceeds by [[Principle of Mathematical Induction|induction]] on $b$.
+For all $b \in \Z_{\ge 0}$, let $\map P b$ be the [[Definition:Proposition|proposition]]:
+:$\displaystyle \sum_{i \mathop = a}^b \map h i = \sum_{i \mathop = a}^b \map f i + \sum_{i \mathop = a}^b \map g i$
+=== Basis for the Induction ===
+Let $b < a$.
+Then all [[Definition:Indexed Summation|indexed summations]] are [[Definition:Zero (Number)|zero]].
+Because $0 = 0 + 0$, the result follows.
+=== Basis for the Induction ===
+Let $b = a$.
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = a}^b \map h i
+ | r = \sum_{i \mathop = a}^a \map h i
+ | c =
+}}
+{{eqn | r = \map h a
+ | c =
+}}
+{{eqn | r = \map f a + \map g a
+ | c = {{Defof|Pointwise Addition}}
+}}
+{{eqn | r = \sum_{i \mathop = a}^a \map f i + \sum_{i \mathop = a}^a \map g i
+ | c =
+}}
+{{eqn | r = \sum_{i \mathop = a}^b \map f i + \sum_{i \mathop = a}^b \map g i
+ | c = as $a = b$
+}}
+{{end-eqn}}
+Thus $\map P a$ is seen to hold.
+This is our [[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 $k \ge a$, then it logically follows that $\map P {k + 1}$ is true.
+So this is the [[Definition:Induction Hypothesis|induction hypothesis]]:
+:$\displaystyle \sum_{i \mathop = a}^k \map h i = \sum_{i \mathop = a}^k \map f i + \sum_{i \mathop = a}^k \map g i$
+from which it is to be shown that:
+:$\displaystyle \sum_{i \mathop = a}^{k + 1} \map h i = \sum_{i \mathop = a}^{k + 1} \map f i + \sum_{i \mathop = a}^{k + 1} \map g i$
+=== Induction Step ===
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = a}^{k + 1} \map h i
+ | r = \sum_{i \mathop = a}^k \map h i + \map h {k + 1}
+ | c = {{Defof|Indexed Summation}}
+}}
+{{eqn | r = \sum_{i \mathop = a}^k \map h i + \map f {k + 1} + \map g {k + 1}
+ | c = {{Defof|Pointwise Addition}}
+}}
+{{eqn | r = \sum_{i \mathop = a}^k \map f i + \sum_{i \mathop = a}^k \map g i + \map f {k + 1} + \map g {k + 1}
+ | c = [[Indexed Summation of Sum of Mappings#Induction Hypothesis|Induction Hypothesis]]
+}}
+{{eqn | r = \sum_{i \mathop = a}^k \map f i + \map f {k + 1} + \sum_{i \mathop = a}^k \map g i + \map g {k + 1}
+ | c = [[Commutative Law of Addition]]
+}}
+{{eqn | r = \sum_{i \mathop = a}^{k + 1} \map f i + \sum_{i \mathop = a}^{k + 1} \map g i
+ | c = {{Defof|Indexed Summation}}
+}}
+{{end-eqn}}
+So $\map P k \implies \map P {k + 1}$ and the result follows by the [[Principle of Mathematical Induction]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Summation of Sum of Mappings on Finite Set}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $S$ be a [[Definition:Finite Set|finite set]].
+Let $f, g: S \to \mathbb A$ be [[Definition:Mapping|mappings]].
+Let $h = f + g$ be their [[Definition:Sum of Mappings|sum]].
+Then we have the equality of [[Definition:Summation|summations]] on [[Definition:Finite Set|finite sets]]:
+:$\displaystyle \sum_{s \mathop \in S} h \left({s}\right) = \sum_{s \mathop \in S} f \left({s}\right) + \sum_{s \mathop \in S} g \left({s}\right)$
+\end{theorem}
+
+\begin{proof}
+Let $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $S$.
+Let $\sigma: \N_{< n} \to S$ be a [[Definition:Bijection|bijection]], where $\N_{< n}$ is an [[Definition:Initial Segment of Natural Numbers|initial segment of the natural numbers]].
+By definition of [[Definition:Summation|summation]], we have to prove the following equality of [[Definition:Indexed Summation|indexed summations]]:
+:$\displaystyle \sum_{i \mathop = 0}^{n - 1} h \left({\sigma \left({i}\right)}\right) = \sum_{i \mathop = 0}^{n - 1} f \left({\sigma \left({i}\right)}\right) + \sum_{i \mathop = 0}^{n - 1} g \left({\sigma \left({i}\right)}\right)$
+By [[Sum of Mappings Composed with Mapping]], $h \circ \sigma = f \circ \sigma + g \circ \sigma$.
+The above equality now follows from [[Indexed Summation of Sum of Mappings]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Indexed Summation of Multiple of Mapping}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N,\Z,\Q,\R,\C$.
+Let $a, b$ be [[Definition:Integer|integers]].
+Let $\left[{a \,.\,.\, b}\right]$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
+Let $f: \left[{a \,.\,.\, b}\right] \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Let $\lambda \in \mathbb A$.
+Let $g = \lambda \cdot f$ be the [[Definition:Product of Mapping with Scalar|product]] of $f$ with $\lambda$.
+Then we have the equality of [[Definition:Indexed Summation|indexed summations]]:
+:$\displaystyle \sum_{i \mathop = a}^b g \left({i}\right) = \lambda \cdot \sum_{i \mathop = a}^b f \left({i}\right)$
+\end{theorem}
+
+\begin{proof}
+The proof goes by [[Principle of Mathematical Induction|induction]] on $b$.
+=== Basis for the Induction ===
+Let $b < a$.
+Then all [[Definition:Indexed Summation|indexed summations]] are [[Definition:Zero of Standard Number System|zero]].
+Because $0 = \lambda \cdot 0$, the result follows.
+This is our [[Principle of Mathematical Induction#Basis for the Induction|basis for the induction]].
+=== Induction Step ===
+Let $b \geq a$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = a}^b g \left({i}\right)
+ | r = \sum_{i \mathop = a}^{b - 1} g \left({i}\right) + g \left({b}\right)
+ | c = {{Defof|Indexed Summation}}
+}}
+{{eqn | r = \sum_{i \mathop = a}^{b - 1} g \left({i}\right) + \lambda \cdot f \left({b}\right)
+ | c = {{Defof|Product of Mapping with Scalar}}
+}}
+{{eqn | r = \lambda \cdot \sum_{i \mathop = a}^{b - 1} f \left({i}\right) + \lambda \cdot f \left({b}\right)
+ | c = [[Principle of Mathematical Induction#Induction Hypothesis|Induction Hypothesis]]
+}}
+{{eqn | r = \lambda \cdot \left({\sum_{i \mathop = a}^{b - 1} f \left({i}\right) + f \left({b}\right)}\right)
+ | c = [[Multiplication of Numbers Distributes over Addition]]
+}}
+{{eqn | r = \lambda \cdot \sum_{i \mathop = a}^b f \left({i}\right)
+ | c = {{Defof|Indexed Summation}}
+}}
+{{end-eqn}}
+By the [[Principle of Mathematical Induction]], the proof is complete.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Summation of Multiple of Mapping on Finite Set}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $S$ be a [[Definition:Finite Set|finite set]].
+Let $f: S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Let $\lambda \in \mathbb A$.
+Let $g = \lambda \cdot f$ be the [[Definition:Product of Mapping with Scalar|product]] of $f$ with $\lambda$.
+Then we have the equality of [[Definition:Summation|summations]] on [[Definition:Finite Set|finite sets]]:
+:$\displaystyle \sum_{s \mathop \in S} g \left({s}\right) = \lambda \cdot \sum_{s \mathop \in S} f \left({s}\right)$
+\end{theorem}
+
+\begin{proof}
+Let $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $S$.
+Let $\sigma : \N_{< n} \to S$ be a [[Definition:Bijection|bijection]], where $\N_{< n}$ is an [[Definition:Initial Segment of Natural Numbers|initial segment of the natural numbers]].
+By definition of [[Definition:Summation|summation]], we have to prove the following equality of [[Definition:Indexed Summation|indexed summations]]:
+:$\displaystyle \sum_{i \mathop = 0}^{n - 1} g \left({\sigma \left({i}\right)}\right) = \lambda \cdot \sum_{i \mathop = 0}^{n - 1} f \left({\sigma \left({i}\right)}\right)$
+By [[Multiple of Mapping Composed with Mapping]], $g \circ \sigma = \lambda \cdot \left({f \circ \sigma}\right)$.
+The above equality now follows from [[Indexed Summation of Multiple of Mapping]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Linear Combination of Indexed Summations}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N,\Z,\Q,\R,\C$.
+Let $a,b$ be [[Definition:Integer|integers]].
+Let $\left[{a \,.\,.\, b}\right]$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
+Let $f, g : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be [[Definition:Mapping|mappings]].
+Let $\lambda, \mu \in \mathbb A$.
+Let $\lambda \cdot f + \mu \cdot g$ be the [[Definition:Sum of Mappings|sum]] of the [[Definition:Product of Mapping with Scalar|product]] of $f$ with $\lambda$ and the [[Definition:Product of Mapping with Scalar|product]] of $g$ with $\mu$.
+Then we have the equality of [[Definition:Indexed Summation|indexed summations]]:
+:$\displaystyle \sum_{i \mathop = a}^b \left( \lambda \cdot f(i) + \mu \cdot g(i) \right) = \lambda \cdot \sum_{i \mathop = a}^b f(i) + \mu \cdot \sum_{i \mathop = a}^b g(i)$
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = a}^b \left( \lambda \cdot f(i) + \mu \cdot g(i) \right)
+ | r = \sum_{i \mathop = a}^b \left( \lambda \cdot f(i) \right) + \sum_{i \mathop = a}^b \left( \mu \cdot g(i) \right)
+ | c = [[Indexed Summation of Sum of Mappings]]
+}}
+{{eqn | r = \lambda \cdot \sum_{i \mathop = a}^b f(i) + \mu \cdot \sum_{i \mathop = a}^b g(i)
+ | c = [[Indexed Summation of Multiple of Mapping]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Triangle Inequality for Indexed Summations}
+Tags: Summations, Triangle Inequality
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N,\Z,\Q,\R,\C$.
+Let $a,b$ be [[Definition:Integer|integers]].
+Let $\left[{a \,.\,.\, b}\right]$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
+Let $f : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Let $|\cdot|$ denote the [[Definition:Standard Absolute Value|standard absolute value]].
+Let $\vert f \vert$ be the [[Definition:Absolute Value of Mapping|absolute value]] of $f$.
+Then we have the [[Definition:Inequality|inequality]] of [[Definition:Indexed Summation|indexed summations]]:
+:$\displaystyle \left\vert \sum_{i \mathop = a}^b f(i) \right\vert \leq \sum_{i \mathop = a}^b \vert f(i) \vert$
+\end{theorem}
+
+\begin{proof}
+The proof goes by [[Principle of Mathematical Induction|induction]] on $b$.
+=== Basis for the Induction ===
+Let $b < a$.
+Then all [[Definition:Indexed Summation|indexed summations]] are [[Definition:Zero of Standard Number System|zero]].
+Because $|0| \leq |0|$ by definition of the [[Definition:Standard Absolute Value|standard absolute value]], the result follows.
+This is our [[Principle of Mathematical Induction#Basis for the Induction|basis for the induction]].
+=== Induction Step ===
+Let $b \geq a$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \left\vert \sum_{i \mathop = a}^{b} f(i) \right\vert
+ | r = \left\vert \sum_{i \mathop = a}^{b-1} f(i) + f(b) \right\vert
+ | c = {{Defof|Indexed Summation}}
+}}
+{{eqn | r = \left\vert \sum_{i \mathop = a}^{b-1} f(i) \right\vert + \left\vert f(b) \right\vert
+ | c = [[Triangle Inequality]]
+ | o = \leq
+}}
+{{eqn | r = \sum_{i \mathop = a}^{b-1} \left\vert f(i) \right\vert + \left\vert f(b) \right\vert
+ | c = [[Principle of Mathematical Induction#Induction Hypothesis|Induction Hypothesis]], {{defof|Relation Compatible with Operation}}
+ | o = \leq
+}}
+{{eqn | r = \sum_{i \mathop = a}^{b} \left\vert f(i) \right\vert
+ | c = {{Defof|Indexed Summation}}
+}}
+{{end-eqn}}
+By the [[Principle of Mathematical Induction]], the proof is complete.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Triangle Inequality for Summation over Finite Set}
+Tags: Summations, Triangle Inequality
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $S$ be a [[Definition:Finite Set|finite set]].
+Let $f : S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Let $\left\vert{\, \cdot\,}\right\vert$ denote the [[Definition:Standard Absolute Value|standard absolute value]].
+Let $\left\vert{f}\right\vert$ be the [[Definition:Absolute Value of Mapping|absoute value]] of $f$.
+Then we have the [[Definition:Inequality|inequality]] of [[Definition:Summation|summations]] on [[Definition:Finite Set|finite sets]]:
+:$\displaystyle \left\vert \sum_{s \mathop \in S} f(s) \right\vert \leq \sum_{s \mathop \in S} \vert f(s) \vert$
+\end{theorem}
+
+\begin{proof}
+Let $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $S$.
+Let $\sigma: \N_{< n} \to S$ be a [[Definition:Bijection|bijection]], where $\N_{< n}$ is an [[Definition:Initial Segment of Natural Numbers|initial segment of the natural numbers]].
+By definition of [[Definition:Summation|summation]], we have to prove the following [[Definition:Inequality|inequality]] of [[Definition:Indexed Summation|indexed summations]]:
+:$\displaystyle \left\vert{\sum_{i \mathop = 0}^{n - 1} f \left({\sigma \left({i}\right)}\right)}\right\vert \le \sum_{i \mathop = 0}^{n-1} \left({\left\vert{f}\right\vert \circ \sigma}\right) \left({i}\right) $
+By [[Absolute Value of Mapping Composed with Mapping]]:
+: $\left\vert{f}\right\vert \circ \sigma = \left\vert{f \circ \sigma}\right\vert$
+The above equality now follows from [[Triangle Inequality for Indexed Summations]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Exchange of Order of Indexed Summations}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $a, b, c, d \in \Z$ be [[Definition:Integer|integers]].
+Let $\closedint a b$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
+\end{theorem}<|endoftext|>
+\section{Exchange of Order of Indexed Summations/Rectangular Domain}
+Tags: Summations
+
+\begin{theorem}
+Let $D = \closedint a b \times \closedint c d$ be the [[Definition:Cartesian Product|cartesian product]].
+Let $f: D \to \mathbb A$ be a [[Definition:Mapping|mapping]]
+Then we have an equality of [[Definition:Indexed Summation|indexed summations]]:
+:$\displaystyle \sum_{i \mathop = a}^b \sum_{j \mathop = c}^d \map f {i, j} = \sum_{j \mathop = c}^d \sum_{i \mathop = a}^b \map f {i, j}$
+\end{theorem}
+
+\begin{proof}
+The proof proceeds by [[Principle of Mathematical Induction|induction]] on $d$.
+=== Basis for the Induction ===
+Let $d < c$.
+Then the [[Definition:Indexed Summation|indexed summation]] in the {{RHS}} is [[Definition:Zero (Number)|zero]].
+By [[Indexed Summation of Zero]], so is the {{LHS}}.
+This is our [[Definition:Basis for the Induction|basis for the induction]].
+=== Induction Step ===
+Let $d \ge c$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{i \mathop = a}^b \sum_{j \mathop = c}^d \map f {i, j}
+ | r = \sum_{i \mathop = a}^b \paren {\sum_{j \mathop = c}^{d - 1} \map f {i, j} + \map f {i, d} }
+ | c = {{Defof|Indexed Summation}}
+}}
+{{eqn | r = \sum_{i \mathop = a}^b \sum_{j \mathop = c}^{d - 1} \map f {i, j} + \sum_{i \mathop = a}^b \map f {i, d}
+ | c = [[Indexed Summation of Sum of Mappings]]
+}}
+{{eqn | r = \sum_{j \mathop = c}^{d - 1} \sum_{i \mathop = a}^b \map f {i, j} + \sum_{i \mathop = a}^b \map f {i, d}
+ | c = [[Principle of Mathematical Induction#Induction Hypothesis|Induction Hypothesis]]
+}}
+{{eqn | r = \sum_{j \mathop = c}^d \sum_{i \mathop = a}^b \map f {i, j}
+ | c = {{Defof|Indexed Summation}}
+}}
+{{end-eqn}}
+By the [[Principle of Mathematical Induction]], the proof is complete.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Exchange of Order of Summations over Finite Sets/Cartesian Product}
+Tags: Summations, Exchange of Order of Summations over Finite Sets
+
+\begin{theorem}
+Let $f: S \times T \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Then we have an equality of [[Definition:Summation|summations]] over [[Definition:Finite Set|finite sets]]:
+:$\displaystyle \sum_{s \mathop \in S} \sum_{t \mathop \in T} \map f {s, t} = \sum_{t \mathop \in T} \sum_{s \mathop \in S} \map f {s, t}$
+\end{theorem}<|endoftext|>
+\section{Exchange of Order of Summations over Finite Sets}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $S, T$ be [[Definition:Finite Set|finite sets]].
+Let $S \times T$ be their [[Definition:Cartesian Product|cartesian product]].
+\end{theorem}<|endoftext|>
+\section{Sum over Complement of Finite Set}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $S$ be a [[Definition:Finite Set|finite set]].
+Let $f: S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Let $T \subseteq S$ be a [[Definition:Subset|subset]].
+Let $S \setminus T$ be its [[Definition:Relative Complement|relative complement]].
+Then we have the equality of [[Definition:Summation|summations]] over [[Definition:Finite Set|finite sets]]:
+:$\displaystyle \sum_{s \mathop \in S \setminus T} f \left({s}\right) = \sum_{s \mathop \in S} f \left({s}\right) - \sum_{t \mathop \in T} f \left({t}\right)$
+\end{theorem}
+
+\begin{proof}
+Note that by [[Subset of Finite Set is Finite]], $T$ is indeed [[Definition:Finite Set|finite]].
+By [[Set is Disjoint Union of Subset and Relative Complement]], $S$ is the [[Definition:Disjoint Union|disjoint union]] of $S \setminus T$ and $T$.
+The result now follows from [[Sum over Disjoint Union of Finite Sets]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Mapping Defines Additive Function of Subalgebra of Power Set}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $S$ be a [[Definition:Finite Set|finite set]].
+Let $f: S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Let $B$ be an [[Definition:Algebra of Sets|algebra of sets]] over $S$.
+Define $\Sigma: B \to \mathbb A$ using [[Definition:Summation|summation]] as:
+:$\Sigma \left({T}\right) = \displaystyle \sum_{t \mathop \in T} f \left({t}\right)$
+for $T\subseteq S$.
+Then $\Sigma$ is an [[Definition:Additive Function (Measure Theory)|additive function]] on $B$.
+\end{theorem}
+
+\begin{proof}
+Note that by [[Subset of Finite Set is Finite]], $B$ consists of [[Definition:Finite Set|finite sets]].
+The result now follows from [[Sum over Disjoint Union of Finite Sets]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum over Union of Finite Sets}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $S$ and $T$ be [[Definition:Finite Set|finite sets]].
+Let $f: S \cup T \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Then we have the equality of [[Definition:Summation|summations]] over [[Definition:Finite Set|finite sets]]:
+:$\displaystyle \sum_{u \mathop \in S \mathop \cup T} \map f u = \sum_{s \mathop \in S} \map f s + \sum_{t \mathop \in T} \map f t - \sum_{v \mathop \in S \mathop \cap T} \map f v$
+\end{theorem}
+
+\begin{proof}
+Follows from:
+:[[Mapping Defines Additive Function of Subalgebra of Power Set]]
+:[[Power Set is Algebra of Sets]]
+:[[Inclusion-Exclusion Principle]]
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Summation over Finite Set Equals Summation over Support}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
+Let $S$ be a [[Definition:Finite Set|finite set]].
+Let $f: S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
+Let $\operatorname{Supp} \left({f}\right)$ be its [[Definition:Support of Mapping to Algebraic Structure|support]].
+Then we have an equality of [[Definition:Summation|summations]] over [[Definition:Finite Set|finite sets]]:
+:$\displaystyle \sum_{s \mathop \in S} f \left({s}\right) = \sum_{s \mathop \in \operatorname{Supp} \left({f}\right)} f \left({s}\right)$
+\end{theorem}
+
+\begin{proof}
+Note that by [[Subset of Finite Set is Finite]], $\operatorname{Supp} \left({f}\right)$ is indeed [[Definition:Finite Set|finite]].
+The result now follows from:
+* [[Sum over Complement of Finite Set]]
+* [[Sum of Zero over Finite Set]]
+* [[Identity Element of Addition on Numbers]]
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Summation of Zero}
+Tags: Summations
+
+\begin{theorem}
+Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N,\Z,\Q,\R,\C$.
+\end{theorem}<|endoftext|>
+\section{Summation of Zero/Indexed Summation}
+Tags: Summations
+
+\begin{theorem}
+Let $a,b$ be [[Definition:Integer|integers]].
+Let $\left[{a \,.\,.\, b}\right]$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
+Let $0 : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be the [[Definition:Zero Mapping|zero mapping]].
+Then the [[Definition:Indexed Summation|indexed summation]] of $0$ from $a$ to $b$ equals [[Definition:Zero of Standard Number System|zero]]:
+:$\displaystyle \sum_{i \mathop = a}^b 0(i) = 0$
+\end{theorem}
+
+\begin{proof}
+At least three proofs are possible:
+* by induction, using [[Identity Element of Addition on Numbers]]
+* using [[Indexed Summation of Multiple of Mapping]]
+* using [[Indexed Summation of Sum of Mappings]]
+{{ProofWanted}}
+[[Category:Summations]]
+3drcgu3jy45li9v3a1ivxrq7qiai7y4
+\end{proof}<|endoftext|>
+\section{Summation of Zero/Finite Set}
+Tags: Summations
+
+\begin{theorem}
+Let $S$ be a [[Definition:Finite Set|finite set]].
+Let $0 : S \to \mathbb A$ be the [[Definition:Zero Mapping|zero mapping]].
+{{explain|Presumably the above is a [[Definition:Constant Mapping|constant mapping]] on $0$ -- needs to be made explicit.}}
+Then the [[Definition:Summation|summation]] of $0$ over $S$ equals [[Definition:Zero of Standard Number System|zero]]:
+:$\displaystyle \sum_{s \mathop \in S} 0 \left({s}\right) = 0$
+\end{theorem}
+
+\begin{proof}
+At least three proofs are possible:
+* Using the definition of [[Definition:Summation|summation]] and [[Indexed Summation of Zero]]
+* using [[Indexed Summation of Sum of Mappings]]
+* using [[Summation of Multiple of Mapping on Finite Set]]
+{{ProofWanted}}
+[[Category:Summations]]
+oqey1cchdp8n9xxz38hcng5j4dierj5
+\end{proof}<|endoftext|>
+\section{Summation of Zero/Set}
+Tags: Summations
+
+\begin{theorem}
+Let $S$ be a [[Definition:Set|set]].
+Let $0: S \to \mathbb A$ be the [[Definition:Zero Mapping|zero mapping]].
+Then the [[Definition:Summation over Set with Finite Support|summation with finite support]] of $0$ over $S$ equals [[Definition:Zero of Standard Number System|zero]]:
+:$\displaystyle \sum_{s \mathop \in S} \map 0 s = 0$
+\end{theorem}
+
+\begin{proof}
+By [[Support of Zero Mapping]], the [[Definition:Support of Mapping to Algebraic Structure|support]] of $0$ is [[Definition:Empty Set|empty]].
+By [[Empty Set is Finite]], the [[Definition:Support of Mapping to Algebraic Structure|support]] of $0$ is indeed [[Definition:Finite Set|finite]].
+By [[Summation over Empty Set]], $\displaystyle \sum_{s \mathop \in S} \map 0 s = \sum_{s \mathop \in \O} \map 0 s = 0$
+{{qed}}
+[[Category:Summations]]
+0vnmuzgyu2aw8ll79sxek547d0ijrwv
+\end{proof}<|endoftext|>
+\section{Exchange of Order of Summations over Finite Sets/Subset of Cartesian Product}
+Tags: Summations
+
+\begin{theorem}
+Let $D\subset S \times T$ be a [[Definition:Subset|subset]].
+Let $\pi_1 : D \to S$ and $\pi_2 : D \to T$ be the [[Definition:Restriction of Mapping|restrictions]] of the [[Definition:Projection from Cartesian Product|projections]] of $S\times T$.
+Then we have an equality of [[Definition:Summation|summations]] over [[Definition:Finite Set|finite sets]]:
+:$\displaystyle \sum_{s \mathop \in S} \sum_{t \mathop \in \pi_2 \left({\pi_1^{-1} \left({s}\right)}\right)} f \left({s, t}\right) = \sum_{t \mathop \in T} \sum_{s \mathop \in \pi_1 \left({\pi_2^{-1} \left({t}\right)}\right)} f \left({s, t}\right)$
+\end{theorem}
+
+\begin{proof}
+Define an [[Definition:Extension of Mapping|extension]] $\overline f$ of $f$ to $S \times T$ by:
+:$\overline f \left({s, t}\right) = \begin{cases}
+f \left({s, t}\right) & : \left({s, t}\right) \in D \\
+0 & : \left({s, t}\right) \notin D
+\end{cases}$
+Then for all $s \in S$, by:
+: [[Preimage of Disjoint Union is Disjoin Union]]
+: [[Sum over Disjoint Union of Finite Sets]]
+: [[Summation over Finite Set of Zero]]:
+:$\displaystyle \sum_{t \mathop \in \pi_2 \left({\pi_1^{-1} \left({s}\right)}\right)} f \left({s, t}\right) = \sum_{t \mathop \in T} \overline f \left({s, t}\right)$
+Thus:
+:$\displaystyle \sum_{s \mathop \in S} \sum_{t \mathop \in \pi_2 \left({\pi_1^{-1} \left({s}\right)}\right)} f \left({s, t}\right) = \sum_{s \mathop \in S} \sum_{t \mathop \in T} \overline f \left({s, t}\right)$
+Similarly:
+:$\displaystyle \sum_{t \mathop \in T} \sum_{s \mathop \in \pi_1 \left({\pi_2^{-1} \left({t}\right)}\right)} f \left({s, t}\right) = \sum_{t \mathop \in T} \sum_{s \mathop \in S} \overline f \left({s, t}\right)$
+By [[Exchange of Order of Summation over Cartesian Product of Finite Sets]], the result follows.
+{{qed}}
+[[Category:Summations]]
+1dvrkgudjt280sscptdkqnz44ldqhi3
+\end{proof}<|endoftext|>
+\section{Universal Property of Polynomial Ring}
+Tags: Polynomial Theory, Universal Properties
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+The different definitions of a [[Definition:Polynomial Ring|polynomial ring]] $(R(x), \iota, x)$ on $R$ satisfy the [[Definition:Universal Property of Polynomial Ring|universal property of a polynomial ring]]:
+\end{theorem}<|endoftext|>
+\section{Pandigital Properties of 123,456,789}
+Tags: 123,456,789, Pandigital Integers
+
+\begin{theorem}
+$123 \, 456 \, 789$ has the following properties:
+It is [[Definition:Pandigital Number|pandigital]], and remains so when multiplied by $2$, $4$, $5$, $7$ and $8$:
+{{begin-eqn}}
+{{eqn | l = 123 \, 456 \, 789 \times 1
+ | r = 123 \, 456 \, 789
+ | c =
+}}
+{{eqn | l = 123 \, 456 \, 789 \times 2
+ | r = 246 \, 913 \, 578
+ | c =
+}}
+{{eqn | l = 123 \, 456 \, 789 \times 3
+ | r = 370 \, 370 \, 367
+ | c =
+}}
+{{eqn | l = 123 \, 456 \, 789 \times 4
+ | r = 493 \, 827 \, 156
+ | c =
+}}
+{{eqn | l = 123 \, 456 \, 789 \times 5
+ | r = 617 \, 283 \, 945
+ | c =
+}}
+{{eqn | l = 123 \, 456 \, 789 \times 6
+ | r = 740 \, 740 \, 734
+ | c =
+}}
+{{eqn | l = 123 \, 456 \, 789 \times 7
+ | r = 864 \, 197 \, 523
+ | c =
+}}
+{{eqn | l = 123 \, 456 \, 789 \times 8
+ | r = 987 \, 654 \, 312
+ | c =
+}}
+{{eqn | l = 123 \, 456 \, 789 \times 9
+ | r = 1 \, 111 \, 111 \, 101
+ | c =
+}}
+{{end-eqn}}
+{{expand|Add some mathematical analysis explaining this phenomenon}}
+\end{theorem}<|endoftext|>
+\section{Pandigital Integers remaining Pandigital on Multiplication}
+Tags: Pandigital Integers
+
+\begin{theorem}
+Certain [[Definition:Pandigital Integer|pandigital integers]] remain [[Definition:Pandigital Integer|pandigital]] when [[Definition:Integer Multiplication|multiplying]] them by certain single-[[Definition:Digit|digit]] [[Definition:Integer|integers]]:
+{{begin-eqn}}
+{{eqn | l = 1 \, 098 \, 765 \, 432 \times 1
+ | r = 1 \, 098 \, 765 \, 432
+ | c = which is [[Definition:Pandigital Integer|pandigital]]
+}}
+{{eqn | l = 1 \, 098 \, 765 \, 432 \times 2
+ | r = 2 \, 197 \, 530 \, 864
+ | c = which is [[Definition:Pandigital Integer|pandigital]]
+}}
+{{eqn | l = 1 \, 098 \, 765 \, 432 \times 3
+ | r = 3 \, 296 \, 296 \, 296
+ | c =
+}}
+{{eqn | l = 1 \, 098 \, 765 \, 432 \times 4
+ | r = 4 \, 395 \, 061 \, 728
+ | c = which is [[Definition:Pandigital Integer|pandigital]]
+}}
+{{eqn | l = 1 \, 098 \, 765 \, 432 \times 5
+ | r = 5 \, 493 \, 827 \, 160
+ | c = which is [[Definition:Pandigital Integer|pandigital]]
+}}
+{{eqn | l = 1 \, 098 \, 765 \, 432 \times 6
+ | r = 6 \, 592 \, 592 \, 592
+ | c =
+}}
+{{eqn | l = 1 \, 098 \, 765 \, 432 \times 7
+ | r = 7 \, 691 \, 358 \, 024
+ | c = which is [[Definition:Pandigital Integer|pandigital]]
+}}
+{{eqn | l = 1 \, 098 \, 765 \, 432 \times 8
+ | r = 8 \, 790 \, 123 \, 456
+ | c = which is [[Definition:Pandigital Integer|pandigital]]
+}}
+{{eqn | l = 1 \, 098 \, 765 \, 432 \times 9
+ | r = 9 \, 888 \, 888 \, 888
+ | c =
+}}
+{{end-eqn}}
+The sequence:
+:$1039675824, 1053826974, 1068253974, 1068379524, 1073968254, 1075396824, 1098765432, 1204756839, 1234567890, 1357802469$
+contains all [[Definition:Pandigital Integer|pandigital integers]] with at least $4$ nontrivial [[Definition:Pandigital Integer|pandigital]] multiples, of which:
+:$1098765432, 1234567890$
+has $5$.
+{{OEIS|A167476}}
+{{expand|Add some mathematical analysis explaining this phenomenon.}}
+\end{theorem}<|endoftext|>
+\section{Algebra Defined by Ring Homomorphism on Ring with Unity is Unitary}
+Tags: Algebras
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
+Let $\struct {S, +, *}$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $f: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
+Let $\struct {S_R, *}$ be the [[Definition:Algebra Defined by Ring Homomorphism|algebra defined by the ring homomorphism]] $f$.
+Then $\struct {S_R, *}$ is a [[Definition:Unitary Algebra|unitary algebra]].
+\end{theorem}
+
+\begin{proof}
+By definition, the [[Definition:Multiplication of Algebra|multiplication]] of $\struct {S_R, *}$ is the [[Definition:Ring Product|ring product]] of $S$.
+Thus it follows immediately from the fact that $S$ is a [[Definition:Ring with Unity|ring with unity]], that $\struct {S_R, *}$ is a [[Definition:Unitary Algebra|unitary algebra]].
+{{qed}}
+[[Category:Algebras]]
+cpt3ip7bivyes7cyh5urvhe82e5sjko
+\end{proof}<|endoftext|>
+\section{Algebra Defined by Ring Homomorphism is Algebra}
+Tags: Algebras
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
+Let $\struct {S, +, *}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
+Let $f : R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
+Let the [[Definition:Image of Set under Mapping|image]] of $f$ be a [[Definition:Subset|subset]] of the [[Definition:Center of Ring|center]] of $S$.
+Let $\struct {S_R, *}$ be the [[Definition:Algebra Defined by Ring Homomorphism|algebra defined by the ring homomorphism]] $f$.
+Then $\struct {S_R, *}$ is an [[Definition:Algebra over Ring|algebra over]] $R$.
+\end{theorem}<|endoftext|>
+\section{Algebra Defined by Ring Homomorphism is Associative}
+Tags: Algebras
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
+Let $\struct {S, +, *}$ be a [[Definition:Ring with Unity|ring with unity]].
+Let $f: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
+Let the [[Definition:Image of Set under Mapping|image]] of $f$ be a [[Definition:Subset|subset]] of the [[Definition:Center of Ring|center]] of $S$.
+Let $\struct {S_R, *}$ be the [[Definition:Algebra Defined by Ring Homomorphism|algebra defined by the ring homomorphism]] $f$.
+Then $\struct {S_R, *}$ is an [[Definition:Associative Algebra|associative algebra]].
+\end{theorem}
+
+\begin{proof}
+By definition, the [[Definition:Multiplication of Algebra|multiplication]] of $\struct {S_R, *}$ is the [[Definition:Ring Product|ring product]] of $S$.
+Thus it follows immediately from the fact that $S$ is a [[Definition:Ring (Abstract Algebra)|ring]], that $\struct {S_R, *}$ is an [[Definition:Associative Algebra|associative algebra]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Algebra Defined by Ring Homomorphism on Commutative Ring is Commutative}
+Tags: Algebras
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
+Let $\left({S, +, *}\right)$ be a [[Definition:Commutative Ring|commutative ring]].
+Let $f: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
+Let $\left({S_R, *}\right)$ be the [[Definition:Algebra Defined by Ring Homomorphism|algebra defined by the ring homomorphism]] $f$.
+Then $\left({S_R, *}\right)$ is an [[Definition:Commutative Algebra|commutative algebra]].
+\end{theorem}
+
+\begin{proof}
+Note that by [[Center of Commutative Ring]], the [[Definition:Image of Set under Mapping|image]] of $f$ is indeed a [[Definition:Subset|subset]] of the [[Definition:Center of Ring|center]] of $S$.
+By definition, the [[Definition:Multiplication of Algebra|multiplication]] of $(S_R, *)$ is the [[Definition:Ring Product|ring product]] of $S$.
+Thus it follows immediately from the fact that $S$ is a [[Definition:Ring (Abstract Algebra)|ring]], that $(S_R, *)$ is an [[Definition:Commutative Algebra|commutative algebra]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Integer which is Sum of 3 Sixth Powers in 2 Ways}
+Tags: Sixth Powers, 160,426,514
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|positive integer]] which can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Sixth Power|sixth powers]] in $2$ different ways is:
+{{begin-eqn}}
+{{eqn | l = 160 \, 426 \, 514
+ | r = 3^6 + 19^6 + 22^6
+ | c =
+}}
+{{eqn | r = 10^6 + 15^6 + 23^6
+ | c =
+}}
+{{end-eqn}}
+Also note that:
+{{begin-eqn}}
+{{eqn | l = 854
+ | r = 3^2 + 19^2 + 22^2
+ | c =
+}}
+{{eqn | r = 10^2 + 15^2 + 23^2
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+We have that:
+{{begin-eqn}}
+{{eqn | l = 160 \, 426 \, 514
+ | r = 729 + 47 \, 045 \, 881 + 113 \, 379 \, 904
+ | c =
+}}
+{{eqn | r = 3^6 + 19^6 + 22^6
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 160 \, 426 \, 514
+ | r = 1 \, 000 \, 000 + 11 \, 390 \, 625 + 148 \, 035 \, 889
+ | c =
+}}
+{{eqn | r = 10^6 + 15^6 + 23^6
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown there are no smaller}}
+\end{proof}<|endoftext|>
+\section{Infinite Number of Integers which are Sum of 3 Sixth Powers in 2 Ways}
+Tags: Sixth Powers, 160,426,514
+
+\begin{theorem}
+There exist an [[Definition:Infinite Set|infinite number]] of [[Definition:Positive Integer|positive integers]] which can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Sixth Power|sixth powers]] in $2$ different ways.
+\end{theorem}
+
+\begin{proof}
+There are many parametric solutions to $x^6 + y^6 + z^6 = u^6 + v^6 + w^6$. One is given by:
+{{begin-eqn}}
+{{eqn | l = x
+ | r = 2 m^4 + 4 m^3 n - 5 m^2 n^2 - 12 m n^3 - 9 n^4
+}}
+{{eqn | l = y
+ | r = 3 m^4 + 9 m^3 n + 18 m^2 n^2 + 21 m n^3 + 9 n^4
+}}
+{{eqn | l = z
+ | r = -m^4 - 10 m^3 n - 17 m^2 n^2 - 12 m n^3
+}}
+{{eqn | l = u
+ | r = m^4 - 3 m^3 n - 14 m^2 n^2 - 15 m n^3 - 9 n^4
+}}
+{{eqn | l = v
+ | r = 3 m^4 + 8 m^3 n + 9 m^2 n^2
+}}
+{{eqn | l = w
+ | r = 2 m^4 + 12 m^3 n + 19 m^2 n^2 + 18 m n^3 + 9 n^4
+}}
+{{end-eqn}}
+This set of solutions also satisfy:
+{{begin-eqn}}
+{{eqn | l = x^2 + y^2 + z^2
+ | r = u^2 + v^2 + w^2
+}}
+{{eqn | l = 3 x + y + z
+ | r = 3 u + v + w
+}}
+{{end-eqn}}
+{{finish|For people who enjoy degree $24$ polynomials in $2$ unknowns}}
+\end{proof}<|endoftext|>
+\section{Consecutive Primes of form 4n+1}
+Tags: Prime Numbers
+
+\begin{theorem}
+The [[Definition:Integer Sequence|sequence]] of $16$ consecutive [[Definition:Prime Number|prime numbers]] beginning from $207 \, 622 \, 273$ are all of the form $4 n + 1$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | n = 1
+ | l = 207 \, 622 \, 273
+ | r = 4 \times 51 \, 905 \, 568 + 1
+ | c = and is the $11 \, 477 \, 482$nd [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 2
+ | l = 207 \, 622 \, 297
+ | r = 4 \times 51 \, 905 \, 574 + 1
+ | c = and is the $11 \, 477 \, 483$rd [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 3
+ | l = 207 \, 622 \, 301
+ | r = 4 \times 51 \, 905 \, 575 + 1
+ | c = and is the $11 \, 477 \, 484$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 4
+ | l = 207 \, 622 \, 313
+ | r = 4 \times 51 \, 905 \, 578 + 1
+ | c = and is the $11 \, 477 \, 485$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 5
+ | l = 207 \, 622 \, 321
+ | r = 4 \times 51 \, 905 \, 580 + 1
+ | c = and is the $11 \, 477 \, 486$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 6
+ | l = 207 \, 622 \, 381
+ | r = 4 \times 51 \, 905 \, 595 + 1
+ | c = and is the $11 \, 477 \, 487$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 7
+ | l = 207 \, 622 \, 409
+ | r = 4 \times 51 \, 905 \, 602 + 1
+ | c = and is the $11 \, 477 \, 488$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 8
+ | l = 207 \, 622 \, 417
+ | r = 4 \times 51 \, 905 \, 604 + 1
+ | c = and is the $11 \, 477 \, 489$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 9
+ | l = 207 \, 622 \, 421
+ | r = 4 \times 51 \, 905 \, 605 + 1
+ | c = and is the $11 \, 477 \, 490$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 10
+ | l = 207 \, 622 \, 489
+ | r = 4 \times 51 \, 905 \, 622 + 1
+ | c = and is the $11 \, 477 \, 491$st [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 11
+ | l = 207 \, 622 \, 501
+ | r = 4 \times 51 \, 905 \, 625 + 1
+ | c = and is the $11 \, 477 \, 492$nd [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 12
+ | l = 207 \, 622 \, 517
+ | r = 4 \times 51 \, 905 \, 629 + 1
+ | c = and is the $11 \, 477 \, 493$rd [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 13
+ | l = 207 \, 622 \, 537
+ | r = 4 \times 51 \, 905 \, 634 + 1
+ | c = and is the $11 \, 477 \, 494$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 14
+ | l = 207 \, 622 \, 549
+ | r = 4 \times 51 \, 905 \, 637 + 1
+ | c = and is the $11 \, 477 \, 495$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 15
+ | l = 207 \, 622 \, 553
+ | r = 4 \times 51 \, 905 \, 638 + 1
+ | c = and is the $11 \, 477 \, 496$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 16
+ | l = 207 \, 622 \, 561
+ | r = 4 \times 51 \, 905 \, 640 + 1
+ | c = and is the $11 \, 477 \, 497$th [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+Note that the $11 \, 477 \, 481$st [[Definition:Prime Number|prime]]:
+:$207 \, 622 \, 271 = 4 \times 51 \, 905 \, 568 - 1$
+and the $11 \, 477 \, 498$th [[Definition:Prime Number|prime]]:
+:$207 \, 622 \, 567 = 4 \times 51 \, 905 \, 642 - 1$
+and so are not of the form $4 n + 1$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integer which is Sum of 3 Fourth Powers in 2 Ways and Products of Those Roots}
+Tags: Fourth Powers, 256,103,393
+
+\begin{theorem}
+The [[Definition:Positive Integer|positive integer]] $256 \, 103 \, 393$ can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Fourth Power|fourth powers]] in $2$ different ways:
+{{begin-eqn}}
+{{eqn | l = 256 \, 103 \, 393
+ | r = 22^4 + 93^4 + 116^4
+ | c =
+}}
+{{eqn | r = 29^4 + 66^4 + 124^4
+ | c =
+}}
+{{end-eqn}}
+Also note that:
+{{begin-eqn}}
+{{eqn | l = 237 \, 336
+ | r = 22 \times 93 \times 116
+ | c =
+}}
+{{eqn | r = 29 \times 66 \times 124
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+We have that:
+{{begin-eqn}}
+{{eqn | l = 256 \, 103 \, 393
+ | r = 234 \, 256 + 74 \, 805 \, 201 + 181 \, 063 \, 936
+ | c =
+}}
+{{eqn | r = 22^4 + 93^4 + 116^4
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 256 \, 103 \, 393
+ | r = 707 \, 281 + 18 \, 974 \, 736 + 236 \, 421 \, 376
+ | c =
+}}
+{{eqn | r = 29^4 + 66^4 + 124^4
+ | c =
+}}
+{{end-eqn}}
+Then we have:
+{{begin-eqn}}
+{{eqn | l = 22 \times 93 \times 116
+ | r = \left({2 \times 11}\right) \times \left({3 \times 31}\right) \times \left({2^2 \times 29}\right)
+ | c =
+}}
+{{eqn | r = 2^3 \times 3 \times 11 \times 29 \times 31
+ | c =
+}}
+{{eqn | r = 237 \, 336
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 29 \times 66 \times 124
+ | r = 29 \times \left({2 \times 3 \times 11}\right) \times \left({2^2 \times 31}\right)
+ | c =
+}}
+{{eqn | r = 2^3 \times 3 \times 11 \times 29 \times 31
+ | c =
+}}
+{{eqn | r = 237 \, 336
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Groups of Order 8}
+Tags: Order of Groups, Groups of Order 8
+
+\begin{theorem}
+Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $8$.
+Then $G$ is [[Definition:Group Isomorphism|isomorphic]] to one of the following:
+:$\Z_8$
+:$\Z_4 \oplus \Z_2$
+:$\Z_2 \oplus \Z_2 \oplus \Z_2$
+:$D_4$
+:$\Dic 2$
+where:
+:$\Z_n$ is the [[Definition:Cyclic Group|cyclic group]] of order $n$
+:$D_4$ is the [[Definition:Dihedral Group|dihedral group]] of order $8$
+:$\Dic 2$ is the [[Definition:Dicyclic Group|dicyclic group]] of order $8$.
+\end{theorem}
+
+\begin{proof}
+The [[Definition:Abelian Group|abelian]] cases are handled by the [[Abelian Group Factored by Prime/Corollary|corollary to Abelian Group Factored by Prime]].
+{{qed|lemma}}
+Let $G$ be non-[[Definition:Abelian Group|abelian]].
+By [[Lagrange's Theorem (Group Theory)|Lagrange's theorem]] the [[Definition:Order of Group Element|order]] of non-[[Definition:Identity Element|identity]] elements in $G$ is either $2$, $4$ or $8$.
+{{AimForCont}} that there exists an [[Definition:Order of Group Element|order $8$ element]].
+Then $G$ is [[Definition:Generated Subgroup|generated]] by this [[Definition:Element|element]].
+So $G$ is by definition [[Definition:Cyclic Group|cyclic]].
+But [[Cyclic Group is Abelian]], contradicting the assumption that $G$ is non-[[Definition:Abelian Group|abelian]].
+So there is no [[Definition:Order of Group Element|order $8$ element]].
+By [[Non-Abelian Order 8 Group has Order 4 Element]], there exists at least one [[Definition:Order of Group Element|order $4$ element]] in $G$.
+Let it be denoted by $a$.
+Let $A$ denote the [[Definition:Generated Subgroup|subgroup generated]] by $a$.
+By [[Lagrange's Theorem (Group Theory)|Lagrange's theorem]] there are two [[Definition:Coset|cosets]] in $G$: $A$ and $G \setminus A$.
+Let $b \in G \setminus A$.
+Then $\set {a, b}$ is a [[Definition:Generator of Group|generator]] of $G$.
+Now we consider how $a$ and $b$ interact with each other.
+Consider the element $x = b a b^{-1}$.
+By [[Subgroup of Index 2 is Normal]], $b A b^{-1} = A$.
+So $x \in A$.
+By [[Order of Conjugate Element equals Order of Element]], the only possible choices are $x = a$ or $x = a^3$.
+If $x = a$, then $a$ and $b$ [[Definition:Commuting Elements|commute]].
+Since $\set {a, b}$ [[Definition:Generator of Group|generates]] $G$, this makes $G$ an [[Definition:Abelian Group|abelian group]], which is a [[Definition:Contradiction|contradiction]].
+So $b a b^{-1} = a^3$.
+It suffices to consider the [[Definition:Order of Group Element|order]] of $b$:
+:If $\order b = 2$, then $G \cong D_4$.
+:If $\order b = 4$, then $G \cong \Dic 2$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Triangular Numbers which are Product of 3 Consecutive Integers}
+Tags: Triangular Numbers
+
+\begin{theorem}
+The $6$ [[Definition:Triangular Number|triangular numbers]] which can be expressed as the [[Definition:Integer Multiplication|product]] of $3$ consecutive [[Definition:Integer|integers]] are:
+:$6, 120, 210, 990, 185 \, 836, 258 \, 474 \, 216$
+{{OEIS|A001219}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = T_3
+ | r = \frac {3 \left({3 + 1}\right)} 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{eqn | r = 6
+ | c =
+}}
+{{eqn | r = 1 \times 2 \times 3
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = T_{15}
+ | r = \frac {15 \left({15 + 1}\right)} 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{eqn | r = 120
+ | c =
+}}
+{{eqn | r = 4 \times 5 \times 6
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = T_{20}
+ | r = \frac {20 \left({20 + 1}\right)} 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{eqn | r = 210
+ | c =
+}}
+{{eqn | r = 5 \times 6 \times 7
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = T_{44}
+ | r = \frac {44 \left({44 + 1}\right)} 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{eqn | r = 990
+ | c =
+}}
+{{eqn | r = 9 \times 10 \times 11
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = T_{608}
+ | r = \frac {608 \left({608 + 1}\right)} 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{eqn | r = 185 \, 136
+ | c =
+}}
+{{eqn | r = 56 \times 57 \times 58
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = T_{22 \, 736}
+ | r = \frac {22 \, 736 \left({22 \, 736 + 1}\right)} 2
+ | c = [[Closed Form for Triangular Numbers]]
+}}
+{{eqn | r = 258 \, 474 \, 216
+ | c =
+}}
+{{eqn | r = 2^3 \times 3 \times 7^2 \times 11 \times 13 \times 29 \times 53
+ | c =
+}}
+{{eqn | r = \left({2^2 \times 3 \times 53}\right) \times \left({7^2 \times 13}\right) \times \left({2 \times 11 \times 29}\right)
+ | c =
+}}
+{{eqn | r = 636 \times 637 \times 638
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|It remains to be shown these are the only ones}}
+\end{proof}<|endoftext|>
+\section{First Harmonic Number to exceed 20}
+Tags: Harmonic Numbers, 272,400,600
+
+\begin{theorem}
+The first [[Definition:Harmonic Number|harmonic number]] that is greater than $20$ is $H_{272 \, 400 \, 600}$.
+That is, the number of [[Definition:Term of Sequence|terms]] of the [[Definition:Harmonic Series|harmonic series]] required for its [[Definition:Partial Sum|partial sum]] to exceed $20$ is $272 \, 400 \, 600$.
+\end{theorem}
+
+\begin{proof}
+We have:
+:$H_{272 \, 400 \, 599} = \displaystyle \sum_{k \mathop = 1}^{272 \, 400 \, 599} \frac 1 k \approx 19 \cdotp 99999 \, 99979$
+and:
+:$H_{272 \, 400 \, 600} = \displaystyle \sum_{k \mathop = 1}^{272 \, 400 \, 600} \frac 1 k \approx 20 \cdotp 00000 \, 00016$
+\end{proof}<|endoftext|>
+\section{First Harmonic Number to exceed 10}
+Tags: Harmonic Numbers, 12,367
+
+\begin{theorem}
+The first [[Definition:Harmonic Number|harmonic number]] that is greater than $10$ is $H_{12 \, 367}$.
+That is, the number of [[Definition:Term of Sequence|terms]] of the [[Definition:Harmonic Series|harmonic series]] required for its [[Definition:Partial Sum|partial sum]] to exceed $10$ is $12 \, 367$.
+\end{theorem}
+
+\begin{proof}
+We have:
+:$H_{12 \, 366} = \displaystyle \sum_{k \mathop = 1}^{12 \, 366} \frac 1 k \approx 9 \cdotp 99996 \, 214$
+and:
+:$H_{12 \, 367} = \displaystyle \sum_{k \mathop = 1}^{12 \, 367} \frac 1 k \approx 10 \cdotp 00004 \, 30083$
+\end{proof}<|endoftext|>
+\section{First Harmonic Number to exceed 100}
+Tags: Harmonic Numbers
+
+\begin{theorem}
+The first [[Definition:Harmonic Number|harmonic number]] that is greater than $100$ is $H_n$ where $n \approx 1.5 \times 10^{43}$.
+That is, it takes approximately $1.5 \times 10^{43}$ [[Definition:Term of Sequence|terms]] of the [[Definition:Harmonic Series|harmonic series]] required for its [[Definition:Partial Sum|partial sum]] to exceed $100$.
+\end{theorem}<|endoftext|>
+\section{Number of Magic Squares of Order 5}
+Tags: Magic Squares
+
+\begin{theorem}
+Up to rotations and reflections, there are $275 \, 305 \, 224$ [[Definition:Distinct|distinct]] [[Definition:Magic Square|magic squares]] of [[Definition:Order of Magic Square|order $5$]].
+\end{theorem}<|endoftext|>
+\section{Canonical Homomorphism to Polynomial Ring is Ring Monomorphism}
+Tags: Polynomial Theory
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+Let $(R[X], \iota, X)$ be a [[Definition:Polynomial Ring in one Indeterminate|polynomial ring]] over $R$ in one [[Definition:Indeterminate of Polynomial Ring|indeterminate]] $X$.
+Then the [[Definition:Embedding into Polynomial Ring|canonical homomorphism]] $\iota : R \to R[X]$ is a [[Definition:Ring Monomorphism|ring monomorphism]].
+\end{theorem}
+
+\begin{proof}
+Let $\operatorname{id} : R \to R$ be the [[Definition:Identity Mapping|identity mapping]].
+Let $1$ be the [[Definition:Unity of Ring|unity]] of $R$.
+By [[Identity Mapping is Ring Automorphism]], $\operatorname{id}$ is a [[Definition:Ring Homomorphism|ring homomorphism]].
+By [[Universal Property of Polynomial Ring]], there exists a [[Definition:Ring Homomorphism|ring homomorphism]] $h : R[X] \to R$ with $h\circ \iota = \operatorname{id}$.
+By [[Identity Mapping is Injection]], $\operatorname{id}$ is an [[Definition:Injection|injection]].
+By [[Injection if Composite is Injection]], $\iota$ is an [[Definition:Injection|injection]].
+Thus $\iota$ is a [[Definition:Ring Monomorphism|ring monomorphism]].
+{{qed}}
+[[Category:Polynomial Theory]]
+9v6ubd9bbba5c4eqfgu96fv9lmn3i0o
+\end{proof}<|endoftext|>
+\section{Universal Property of Field of Rational Fractions}
+Tags: Polynomial Theory, Universal Properties
+
+\begin{theorem}
+Let $R$ be an [[Definition:Integral Domain|integral domain]].
+Let $(R(x), \iota, x)$ be the [[Definition:Field of Rational Fractions|field of rational fractions]] over $R$.
+Let $(K, f, a)$ be an [[Definition:Ordered Triple|ordered triple]], where:
+:$K$ is a [[Definition:Field (Abstract Algebra)|field]]
+:$f : R \to K$ is a [[Definition:Unital Ring Homomorphism|unital ring homomorphism]]
+:$a$ is an [[Definition:Element|element]] of $K$.
+Then there exists a [[Definition:Unique|unique]] [[Definition:Unital Ring Homomorphism|unital ring homomorphism]] $\bar f : R(x) \to K$ such that $\bar f\circ\iota = f$ and $\bar f(x) = a$.
+:$\xymatrix{
+R \ar[d]^\iota \ar[r]^{\forall f} & K\\
+R(x) \ar[ru]_{\exists ! \bar f} }$
+\end{theorem}
+
+\begin{proof}
+Use [[Universal Property of Polynomial Ring]] and [[Universal Poperty of Field of Fractions]].
+{{ProofWanted}}
+[[Category:Polynomial Theory]]
+[[Category:Universal Properties]]
+6grwged78wrdv6kuqtrvbjrflq9g02z
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Unital Subalgebra}
+Tags: Algebras
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
+Let $\struct {A_R, *}$ be an [[Definition:Unital Algebra|unital algebra]] over $R$ whose [[Definition:Unit of Algebra|unit]] is $1_A$.
+Let $\struct {B_R, *}$ be a [[Definition:Subalgebra|subalgebra]] of $A_R$.
+{{TFAE|def = Unital Subalgebra}}
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Algebras]]
+0cfyiqjazgztxulfzqobfd020aim4vs
+\end{proof}<|endoftext|>
+\section{Polydivisible Number/Examples/381,654,729}
+Tags: Polydivisible Numbers, Pandigital Integers, 381,654,729
+
+\begin{theorem}
+The [[Definition:Integer|integer]] $381 \, 654 \, 729$ is the only [[Definition:Polydivisible Number|polydivisible number]] which is [[Definition:Pandigital Integer|pandigital]] in the sense of excluding [[Definition:Zero Digit|zero]].
+\end{theorem}
+
+\begin{proof}
+First it is demonstrated that indeed $381 \, 654 \, 729$ has this property:
+{{begin-eqn}}
+{{eqn | l = 3
+ | r = 1 \times 3
+}}
+{{eqn | l = 38
+ | r = 2 \times 19
+}}
+{{eqn | l = 381
+ | r = 3 \times 127
+}}
+{{eqn | l = 3816
+ | r = 4 \times 954
+}}
+{{eqn | l = 38 \, 165
+ | r = 5 \times 7633
+}}
+{{eqn | l = 381 \, 654
+ | r = 6 \times 63 \, 609
+}}
+{{eqn | l = 3 \, 816 \, 547
+ | r = 7 \times 545 \, 221
+}}
+{{eqn | l = 38 \, 165 \, 472
+ | r = 8 \times 4 \, 770 \, 684
+}}
+{{eqn | l = 381 \, 654 \, 729
+ | r = 9 \times 42 \, 406 \, 081
+}}
+{{end-eqn}}
+{{qed|lemma}}
+It remains to be demonstrated that it is the only such number.
+Let $\sqbrk {abcdefghi}$ be a [[Definition:Polydivisible Number|polydivisible number]] which is [[Definition:Pandigital Integer|pandigital]].
+By [[Divisibility by 5]]:
+:$e = 5$
+By [[Divisibility by 2]]:
+:$b, d, f, h$ are [[Definition:Even Integer|even]].
+Hence:
+:$a, c, g, i$ are [[Definition:Odd Integer|odd]].
+By [[Divisibility by 3]]:
+:$a + b + c$, $d + 5 + f$, $g + h + i$ are [[Definition:Divisor of Integer|divisible]] by $3$.
+By [[Divisibility by 8]]:
+:$\sqbrk {fgh}$ is [[Definition:Divisor of Integer|divisible]] by $8$.
+Because $f$ is [[Definition:Even Integer|even]]:
+:$\sqbrk {f00}$ is [[Definition:Divisor of Integer|divisible]] by $8$.
+We must therefore have:
+:$\sqbrk {gh}$ [[Definition:Divisor of Integer|divisible]] by $8$.
+Thus:
+:$\sqbrk {gh}$ must be one of $16, 32, 72, 96$.
+Because $i$ is [[Definition:Odd Integer|odd]] and $g + h + i$ are [[Definition:Divisor of Integer|divisible]] by $3$:
+:$\sqbrk {ghi}$ must be one of $321, 327, 723, 729, 963$.
+By [[Divisibility by 4]]:
+:$\sqbrk {cd}$ is [[Definition:Divisor of Integer|divisible]] by $4$.
+Because $c$ is [[Definition:Odd Integer|odd]]:
+:$d$ must be $2$ or $6$.
+Suppose $h = 6$.
+Then $\sqbrk {ghi} = 963$ and $d = 2$.
+Since $d + 5 + f$ is divisible by $3$ and $f$ is [[Definition:Even Integer|even]]:
+:$f = 8$
+The remaining [[Definition:Even Integer|even number]] $b$ must be $4$.
+So:
+:$\sqbrk {abcdefghi} = \sqbrk {a4c258963}$
+Thus the only remaining options for $a$ and $c$ are $1$ and $7$.
+However, we find that neither $1472589$ nor $7412689$ is [[Definition:Divisor of Integer|divisible]] by $7$.
+Therefore we must have:
+:$h = 2$
+It follow therefore that:
+:$d = 6$
+Because $d + 5 + f$ is [[Definition:Divisor of Integer|divisible]] by $3$, and $f$ is [[Definition:Even Integer|even]]:
+:$f = 4$
+The remaining [[Definition:Even Integer|even number]] $b$ must be $8$.
+So:
+:$\sqbrk {abcdefghi} = \sqbrk {a8c654g2i}$
+Since $a + 8 + c$ is divisible by $3$:
+:$\sqbrk {a8c}$ must be one of $183, 189, 381, 387, 783, 789, 981, 987$.
+Suppose $g = 3$.
+Then:
+:$\sqbrk {a8c}$ must be one of $189, 789, 981, 987$.
+Inspection of the possible numbers reveals that none of $1896543$, $7896543$, $9816543$, $9876543$ is [[Definition:Divisor of Integer|divisible]] by $7$.
+Suppose $g = 7$.
+Then:
+:$\sqbrk {a8c}$ must be one of $183, 189, 381, 981$.
+Among $1836547$, $1896547$, $3816547$, $9816547$, only $3816547$ is [[Definition:Divisor of Integer|divisible]] by $7$.
+The results follow.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Pandigital Product of Pandigital Pairs in 3 Ways}
+Tags: Pandigital Sets
+
+\begin{theorem}
+The [[Definition:Pandigital Integer|pandigital integer]] $0 \, 429 \, 315 \, 678$ can be expressed as the [[Definition:Integer Multiplication|product]] of a [[Definition:Pandigital Set|pandigital]] [[Definition:Doubleton|doubleton]] in $3$ different ways:
+{{begin-eqn}}
+{{eqn | l = 0 \, 429 \, 315 \, 678
+ | r = 04 \, 926 \times 87 \, 153
+}}
+{{eqn | r = 07 \, 923 \times 54 \, 186}}
+{{eqn | r = 15 \, 846 \times 27 \, 093}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+We have that:
+:$0 \, 429 \, 315 \, 678 = 2 \times 3^2 \times 11 \times 19 \times 139 \times 821$
+Then:
+{{begin-eqn}}
+{{eqn | l = 04 \, 926 \times 87 \, 153
+ | r = \paren {2 \times 3 \times 821} \times \paren {3 \times 11 \times 19 \times 139}
+}}
+{{eqn | l = 07 \, 923 \times 54 \, 186
+ | r = \paren {3 \times 19 \times 139} \times \paren {2 \times 3 \times 11 \times 821}
+}}
+{{eqn | l = 15 \, 846 \times 27 \, 093
+ | r = \paren {2 \times 3 \times 19 \times 139} \times \paren {3 \times 11 \times 821}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Integer which is Sum of 2 Fourth Powers in 2 Ways}
+Tags: Fourth Powers, 635,318,657
+
+\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:
+{{begin-eqn}}
+{{eqn | l = 635 \, 318 \, 657
+ | r = 59^4 + 158^4
+ | c =
+}}
+{{eqn | r = 133^4 + 134^4
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}<|endoftext|>
+\section{Largest Pandigital Square less Zero}
+Tags: Square Numbers, Pandigital Integers, 923,187,456
+
+\begin{theorem}
+The largest [[Definition:Pandigital Integer|pandigital]] [[Definition:Square Number|square]] (in the sense where [[Definition:Pandigital Integer|pandigital]] excludes the [[Definition:Zero Digit|zero]]) is $923 \, 187 \, 456$:
+:$923 \, 187 \, 456 = 30 \, 384^2$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Needs to be demonstrated that there are none higher. Could be done by checking all the squares from $30 \, 385^2$ up to $31 \, 426$ but that's too boring for now.}}
+\end{proof}<|endoftext|>
+\section{Pandigital Properties of 987,654,321}
+Tags: 987,654,321, Pandigital Integers
+
+\begin{theorem}
+$987 \, 654 \, 321$ has the following properties:
+It is [[Definition:Pandigital Number|pandigital]], and remains so when multiplied by $1$, $2$, $4$, $5$, $7$ and $8$:
+{{begin-eqn}}
+{{eqn | l = 987 \, 654 \, 321 \times 1
+ | r = 987 \, 654 \, 321
+ | c =
+}}
+{{eqn | l = 987 \, 654 \, 321 \times 2
+ | r = 1 \, 975 \, 308 \, 642
+ | c =
+}}
+{{eqn | l = 987 \, 654 \, 321 \times 3
+ | r = 2 \, 962 \, 962 \, 963
+ | c =
+}}
+{{eqn | l = 987 \, 654 \, 321 \times 4
+ | r = 3 \, 950 \, 617 \, 284
+ | c =
+}}
+{{eqn | l = 987 \, 654 \, 321 \times 5
+ | r = 4 \, 938 \, 271 \, 605
+ | c =
+}}
+{{eqn | l = 987 \, 654 \, 321 \times 6
+ | r = 5 \, 925 \, 925 \, 925
+ | c =
+}}
+{{eqn | l = 987 \, 654 \, 321 \times 7
+ | r = 6 \, 975 \, 308 \, 642
+ | c =
+}}
+{{eqn | l = 987 \, 654 \, 321 \times 8
+ | r = 7 \, 901 \, 234 \, 568
+ | c =
+}}
+{{eqn | l = 987 \, 654 \, 321 \times 9
+ | r = 8 \, 888 \, 888 \, 889
+ | c =
+}}
+{{end-eqn}}
+{{expand|Add some mathematical analysis explaining this phenomenon}}
+Also:
+:$987 \, 654 \, 321 - 123 \, 456 \, 789 = 864 \, 197 \, 532$
+which is also [[Definition:Pandigital Number|pandigital]].
+\end{theorem}<|endoftext|>
+\section{Largest 9-Digit Prime Number}
+Tags: Specific Numbers, 999,999,937
+
+\begin{theorem}
+The largest [[Definition:Prime Number|prime number]] with $9$ [[Definition:Digit|digits]] is $999 \, 999 \, 937$.
+\end{theorem}
+
+\begin{proof}
+Consider the numbers $\sqbrk {999 \, 999 \, 9ab}$.
+Since $999 \, 999 \, 000$ is [[Definition:Divisor|divisible]] by $2, 3, 5, 7, 11, 13$,
+if $\sqbrk {9ab}$ is [[Definition:Divisor|divisible]] by these [[Definition:Prime Number|primes]], so is $\sqbrk {999 \, 999 \, 9ab}$.
+After this elimination the only $\sqbrk {ab} > 37$ that remains are:
+:$41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97$
+We have:
+{{begin-eqn}}
+{{eqn | l = 999 \, 999 \, 941
+ | r = 113 \times 149 \times 59 \, 393
+}}
+{{eqn | l = 999 \, 999 \, 943
+ | r = 5 \, 623 \times 177 \, 841
+}}
+{{eqn | l = 999 \, 999 \, 947
+ | r = 163 \times 6 \, 134 \, 969
+}}
+{{eqn | l = 999 \, 999 \, 953
+ | r = 29 \times 31 \times 773 \times 1 \, 439
+}}
+{{eqn | l = 999 \, 999 \, 961
+ | r = 3 \, 673 \times 272 \, 257
+}}
+{{eqn | l = 999 \, 999 \, 967
+ | r = 3 \, 257 \times 307 \, 031
+}}
+{{eqn | l = 999 \, 999 \, 971
+ | r = 193 \times 5 \, 181 \, 347
+}}
+{{eqn | l = 999 \, 999 \, 977
+ | r = 2 \, 971 \times 336 \, 587
+}}
+{{eqn | l = 999 \, 999 \, 983
+ | r = 337 \times 2 \, 967 \, 359
+}}
+{{eqn | l = 999 \, 999 \, 989
+ | r = 4 \, 327 \times 231 \, 107
+}}
+{{eqn | l = 999 \, 999 \, 991
+ | r = 67 \times 14 \, 925 \, 373
+}}
+{{eqn | l = 999 \, 999 \, 997
+ | r = 71 \times 2 \, 251 \times 6 \, 257
+}}
+{{end-eqn}}
+And we do have $999 \, 999 \, 937$ is [[Definition:Prime Number|prime]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Pandigital Square with Zero}
+Tags: Square Numbers, Pandigital Integers, 1,026,753,849
+
+\begin{theorem}
+The smallest [[Definition:Pandigital Integer|pandigital]] [[Definition:Square Number|square]] (in the sense where [[Definition:Pandigital Integer|pandigital]] includes the [[Definition:Zero Digit|zero]]) is $1 \, 026 \, 753 \, 849$:
+:$1 \, 026 \, 753 \, 849 = 32 \, 043^2$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Needs to be demonstrated that there are none lower. Could be done by checking all the squares from $31 \, 992^2$ up but, while feasible, that's too boring for now. CBA to write a program.}}
+\end{proof}<|endoftext|>
+\section{Sound Proof System is Consistent}
+Tags: Formal Systems
+
+\begin{theorem}
+Let $\mathcal L$ be a [[Definition:Logical Language|logical language]].
+Let $\mathscr M$ be a [[Definition:Formal Semantics|formal semantics]] for $\mathcal L$.
+Let $\mathscr P$ be a [[Definition:Proof System|proof system]] for $\mathcal L$.
+Suppose that $\mathscr P$ is [[Definition:Sound Proof System|sound]] for $\mathscr M$.
+Then $\mathscr P$ is [[Definition:Consistent Proof System|consistent]].
+\end{theorem}
+
+\begin{proof}
+By assumption, some [[Definition:Logical Formula|logical formula]] $\phi$ is not an $\mathscr M$-[[Definition:Tautology (Formal Semantics)|tautology]].
+Since $\mathscr P$ is [[Definition:Sound Proof System|sound]] for $\mathscr M$, $\phi$ is also not a $\mathscr P$-[[Definition:Theorem (Formal Systems)|theorem]].
+But then by definition $\mathscr P$ is [[Definition:Consistent Proof System|consistent]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Consistent Proof System}
+Tags: Formal Systems, Propositional Logic
+
+\begin{theorem}
+{{TFAE|def = Consistent (Logic)/Proof System/Propositional Logic|view = Consistent Proof System for Propositional Logic}}
+Let $\LL_0$ be the [[Definition:Language of Propositional Logic|language of propositional logic]].
+Let $\mathscr P$ be a [[Definition:Proof System|proof system]] for $\LL_0$.
+\end{theorem}
+
+\begin{proof}
+=== Definition 1 implies Definition 2 ===
+Suppose that $\neg \vdash_{\mathscr P} \phi$.
+Suppose additionally that there is some [[Definition:Logical Formula|logical formula]] $\psi$ such that:
+:$\vdash_{\mathscr P} \psi, \neg \psi$
+By the [[Rule of Explosion/Variant 3|Rule of Explosion]]:
+:$\psi, \neg \psi \vdash_{\mathscr P} \phi$
+By [[Provable Consequence of Theorems is Theorem]], we conclude:
+:$\vdash_{\mathscr P} \phi$
+in contradiction to our assumption.
+{{qed|lemma}}
+=== Definition 2 implies Definition 1 ===
+Suppose either $\phi$ or $\neg \phi$ is not a [[Definition:Theorem (Formal Systems)|theorem]] of $\mathscr P$.
+The implication follows trivially.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Rule of Explosion/Variant 3}
+Tags: Rule of Explosion
+
+\begin{theorem}
+:$p, \neg p \vdash q$
+\end{theorem}
+
+\begin{proof}
+{{BeginTableau|p, \neg p \vdash q|[[Definition:Hilbert Proof System/Instance 2|Instance 2 of the Hilbert-style systems]]}}
+{{Assumption|1|p}}
+{{Assumption|2|\neg p}}
+{{TableauLine
+ |n = 3
+ |f = q \implies (p \lor q)
+ |rlnk = Definition:Hilbert Proof System/Instance 2
+ |rtxt = Axiom $A2$
+}}
+{{TableauLine
+ |n = 4
+ |f = \neg p \implies (q \lor \neg p)
+ |rlnk = Definition:Hilbert Proof System/Instance 2
+ |rtxt = Rule $RST \, 1$
+ |dep = 3
+ |c = $\neg p \, / \, q$, $q \, / \, p$
+}}
+{{TableauLine
+ |n = 5
+ |f = q \lor \neg p
+ |rlnk = Definition:Hilbert Proof System/Instance 2
+ |rtxt = Rule $RST \, 3$
+ |dep = 2, 4
+}}
+{{TableauLine
+ |n = 6
+ |f = (q \lor \neg p) \implies (\neg p \lor q)
+ |rlnk = Definition:Hilbert Proof System/Instance 2
+ |rtxt = Axiom $A3$, Rule $RST \, 1$
+ |c = $\neg p \, / \, q$, $q \, / \, p$
+}}
+{{TableauLine
+ |n = 7
+ |f = \neg p \lor q
+ |rlnk = Definition:Hilbert Proof System/Instance 2
+ |rtxt = Rule $RST \, 3$
+ |dep = 5, 6
+}}
+{{TableauLine
+ |n = 8
+ |f = p \implies q
+ |rlnk = Definition:Hilbert Proof System/Instance 2
+ |rtxt = Rule $RST \, 2 \, (2)$
+}}
+{{TableauLine
+ |n = 9
+ |f = q
+ |rlnk = Definition:Hilbert Proof System/Instance 2
+ |rtxt = Rule $RST \, 3$
+ |dep = 1, 8
+}}
+{{EndTableau}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest Integer which is Sum of 3 Fifth Powers in 2 Ways}
+Tags: Fifth Powers, 1,375,298,099
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|positive integer]] which can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Fifth Power|fifth powers]] in $2$ different ways:
+The [[Definition:Positive Integer|positive integer]] $1 \, 375 \, 298 \, 099$ can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Fifth Power|fifth powers]] in $2$ different ways:
+{{begin-eqn}}
+{{eqn | l = 1 \, 375 \, 298 \, 099
+ | r = 24^5 + 28^5 + 67^5
+ | c =
+}}
+{{eqn | r = 3^5 + 54^5 + 62^5
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1 \, 375 \, 298 \, 099
+ | r = 7 \, 962 \, 624 + 17 \, 210 \, 368 + 1 \, 350 \, 125 \, 107
+ | c =
+}}
+{{eqn | r = 24^5 + 28^5 + 67^5
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 1 \, 375 \, 298 \, 099
+ | r = 243 + 459 \, 165 \, 024 + 916 \, 132 \, 832
+ | c =
+}}
+{{eqn | r = 3^5 + 54^5 + 62^5
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|that it is the smallest}}
+\end{proof}<|endoftext|>
+\section{Automorphic Numbers with 10 Digits}
+Tags: Automorphic Numbers
+
+\begin{theorem}
+The only $10$-[[Definition:Digit|digit]] [[Definition:Automorphic Number|automorphic numbers]] are:
+:$1 \, 787 \, 109 \, 376$
+:$8 \, 212 \, 890 \, 625$
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = 1 \, 787 \, 109 \, 376^2
+ | r = \enspace 3 \, 193 \, 759 \, 92 \mathbf {1 \, 787 \, 109 \, 376}
+}}
+{{eqn | l = 8 \, 212 \, 890 \, 625^2
+ | r = 67 \, 451 \, 572 \, 41 \mathbf {8 \, 212 \, 890 \, 625}
+}}
+{{end-eqn}}
+thus demonstrating they are [[Definition:Automorphic Number|automorphic]].
+By [[Automorphic Numbers in Base 10]], there are no others.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Left-Truncated Automorphic Number is Automorphic}
+Tags: Automorphic Numbers, Left-Truncated Automorphic Number is Automorphic
+
+\begin{theorem}
+Let $n$ be an [[Definition:Automorphic Number|automorphic number]], expressed in some conventional [[Definition:Number Base|number base]].
+Let any number of [[Definition:Digit|digits]] be removed from the left-hand end of $n$.
+Then what remains is also an [[Definition:Automorphic Number|automorphic number]].
+\end{theorem}
+
+\begin{proof}
+Let $n$ be an [[Definition:Automorphic Number|automorphic number]] of $d$ [[Definition:Digit|digits]], expressed in [[Definition:Number Base|base]] $b$.
+By {{Defof|Automorphic Number}}, we have:
+:$n^2 \equiv n \pmod {b^d}$
+Let some [[Definition:Digit|digits]] be removed from the left-hand end of $n$, so that only $d'$ digits remain.
+This only makes sense when $d' < d$.
+Define this new number as $n'$.
+Then we have:
+:$n \equiv n' \pmod {b^{d'} }$
+Thus we have:
+{{begin-eqn}}
+{{eqn | l = n^2
+ | o = \equiv
+ | r = n
+ | rr = \pmod {b^{d'} }
+ | c = [[Congruence by Divisor of Modulus]]: from $n^2 \equiv n \pmod {b^d}$ and $b^{d'} \divides b^d$
+}}
+{{eqn | l = n^2
+ | o = \equiv
+ | r = \paren {n'}^2
+ | rr = \pmod {b^{d'} }
+ | c = [[Congruence of Powers]]: from $n \equiv n' \pmod {b^{d'} }$
+}}
+{{eqn | ll = \leadsto
+ | l = \paren {n'}^2
+ | o = \equiv
+ | r = n'
+ | rr = \pmod {b^{d'} }
+}}
+{{end-eqn}}
+Hence $n'$ is an [[Definition:Automorphic Number|automorphic number]] of $d'$ [[Definition:Digit|digits]] in [[Definition:Number Base|base]] $b$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Square whose Sigma is Cubic}
+Tags: Sigma Function, Square Numbers, Cube Numbers, 1,857,437,604, Numbers whose Sigma is Cubic
+
+\begin{theorem}
+The number $1 \, 857 \, 437 \, 604$ is a [[Definition:Square Number|square number]] whose [[Definition:Sigma Function|$\sigma$ value]] is a [[Definition:Cube Number|cube]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1 \, 857 \, 437 \, 604
+ | r = 43 \, 098^2
+ | c =
+}}
+{{eqn | l = \map \sigma {1 \, 857 \, 437 \, 604}
+ | r = 5 \, 168 \, 743 \, 489
+ | c = {{SigmaLink|1,857,437,604|1 \, 857 \, 437 \, 604}}
+}}
+{{eqn | r = 1729^3
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Largest Right-Truncatable Primes allowing 1}
+Tags: Right-Truncatable Primes
+
+\begin{theorem}
+Let $1$ be temporarily considered to be a [[Definition:Prime Number|prime number]].
+Under that consideration, the largest [[Definition:Right-Truncatable Prime|right-truncatable]] [[Definition:Prime Number|prime numbers]] are:
+:$1 \, 979 \, 339 \, 333$
+:$1 \, 979 \, 339 \, 339$
+\end{theorem}
+
+\begin{proof}
+We have that:
+{{begin-eqn}}
+{{eqn | o =
+ | r = 1 \, 979 \, 339 \, 333
+ | c = is [[Definition:Prime Number|prime]]
+}}
+{{eqn | o =
+ | r = 1 \, 979 \, 339 \, 339
+ | c = is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+For both, the truncation process is the same:
+{{begin-eqn}}
+{{eqn | o =
+ | r = 197 \, 933 \, 933
+ | c = is the $10 \, 970 \, 817$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | o =
+ | r = 19 \, 793 \, 393
+ | c = is the $1 \, 252 \, 285$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | o =
+ | r = 1 \, 979 \, 339
+ | c = is the $147 \, 488$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | o =
+ | r = 197 \, 933
+ | c = is the $17 \, 815$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | o =
+ | r = 19 \, 793
+ | c = is the $2240$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | o =
+ | r = 1979
+ | c = is the $299$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | o =
+ | r = 197
+ | c = is the $45$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | o =
+ | r = 19
+ | c = is the $8$th [[Definition:Prime Number|prime]]
+}}
+{{eqn | o =
+ | r = 1
+ | c = has been defined temporarily to be [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Completely Multiplicative Function is Multiplicative}
+Tags: Number Theory, Completely Multiplicative Functions, Multiplicative Functions
+
+\begin{theorem}
+Let $f: \Z \to \Z$ be a [[Definition:Function|function]] on the [[Definition:Integer|integers]] $\Z$.
+Let $f$ be [[Definition:Completely Multiplicative Function|completely multiplicative]].
+Then $f$ is [[Definition:Multiplicative Arithmetic Function|multiplicative]].
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Completely Multiplicative Function|complete multiplicativity]]:
+:$\forall m, n \in \Z: \map f {m n} = \map f m \map f n$
+Hence by [[True Statement is implied by Every Statement]]:
+:$\forall m, n \in \Z: m \perp n \implies \map f {m n} = \map f m \map f n$
+So $f$ is [[Definition:Multiplicative Arithmetic Function|multiplicative]].
+{{qed}}
+[[Category:Number Theory]]
+[[Category:Completely Multiplicative Functions]]
+[[Category:Multiplicative Functions]]
+st2el7aotybal8kzslr6xuxckfq4fqy
+\end{proof}<|endoftext|>
+\section{Pandigital Numbers Divisible by All Integers up to 18}
+Tags: Pandigital Integers
+
+\begin{theorem}
+The following [[Definition:Pandigital Integer|pandigital integers]] are [[Definition:Divisor of Integer|divisible]] by all the [[Definition:Positive Integer|positive integers]] up to $18$:
+:$2 \, 438 \, 195 \, 760$
+:$3 \, 785 \, 942 \, 160$
+:$4 \, 753 \, 869 \, 120$
+:$4 \, 876 \, 391 \, 520$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 2 \, 438 \, 195 \, 760
+ | r = 2 \times 1 \, 219 \, 097 \, 880
+}}
+{{eqn | r = 3 \times 812 \, 731 \, 920}}
+{{eqn | r = 4 \times 609 \, 548 \, 940}}
+{{eqn | r = 5 \times 487 \, 639 \, 152}}
+{{eqn | r = 6 \times 406 \, 365 \, 960}}
+{{eqn | r = 7 \times 348 \, 313 \, 680}}
+{{eqn | r = 8 \times 304 \, 774 \, 470}}
+{{eqn | r = 9 \times 270 \, 910 \, 640}}
+{{eqn | r = 10 \times 243 \, 819 \, 576}}
+{{eqn | r = 11 \times 221 \, 654 \, 160}}
+{{eqn | r = 12 \times 203 \, 182 \, 980}}
+{{eqn | r = 13 \times 187 \, 553 \, 520}}
+{{eqn | r = 14 \times 174 \, 156 \, 840}}
+{{eqn | r = 15 \times 162 \, 546 \, 384}}
+{{eqn | r = 16 \times 152 \, 387 \, 235}}
+{{eqn | r = 17 \times 143 \, 423 \, 280}}
+{{eqn | r = 18 \times 135 \, 553 \, 320}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 3 \, 785 \, 942 \, 160
+ | r = 2 \times 1 \, 892 \, 971 \, 080
+}}
+{{eqn | r = 3 \times 1 \, 261 \, 980 \, 720}}
+{{eqn | r = 4 \times 946 \, 485 \, 540}}
+{{eqn | r = 5 \times 757 \, 188 \, 432}}
+{{eqn | r = 6 \times 630 \, 990 \, 360}}
+{{eqn | r = 7 \times 540 \, 848 \, 880}}
+{{eqn | r = 8 \times 473 \, 242 \, 770}}
+{{eqn | r = 9 \times 420 \, 660 \, 240}}
+{{eqn | r = 10 \times 378 \, 594 \, 216}}
+{{eqn | r = 11 \times 344 \, 176 \, 560}}
+{{eqn | r = 12 \times 315 \, 495 \, 180}}
+{{eqn | r = 13 \times 291 \, 226 \, 320}}
+{{eqn | r = 14 \times 270 \, 424 \, 440}}
+{{eqn | r = 15 \times 252 \, 396 \, 144}}
+{{eqn | r = 16 \times 236 \, 621 \, 385}}
+{{eqn | r = 17 \times 222 \, 702 \, 480}}
+{{eqn | r = 18 \times 210 \, 330 \, 120}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 4 \, 753 \, 869 \, 120
+ | r = 2 \times 2 \, 376 \, 934 \, 560
+}}
+{{eqn | r = 3 \times 1 \, 584 \, 623 \, 040}}
+{{eqn | r = 4 \times 1 \, 188 \, 467 \, 280}}
+{{eqn | r = 5 \times 950 \, 773 \, 824}}
+{{eqn | r = 6 \times 792 \, 311 \, 520}}
+{{eqn | r = 7 \times 679 \, 124 \, 160}}
+{{eqn | r = 8 \times 594 \, 233 \, 640}}
+{{eqn | r = 9 \times 528 \, 207 \, 680}}
+{{eqn | r = 10 \times 475 \, 386 \, 912}}
+{{eqn | r = 11 \times 432 \, 169 \, 920}}
+{{eqn | r = 12 \times 396 \, 155 \, 760}}
+{{eqn | r = 13 \times 365 \, 682 \, 240}}
+{{eqn | r = 14 \times 339 \, 562 \, 080}}
+{{eqn | r = 15 \times 316 \, 924 \, 608}}
+{{eqn | r = 16 \times 297 \, 116 \, 820}}
+{{eqn | r = 17 \times 279 \, 639 \, 360}}
+{{eqn | r = 18 \times 264 \, 103 \, 840}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = 4 \, 876 \, 391 \, 520
+ | r = 2 \times 2 \, 438 \, 195 \, 760
+ | c = see above
+}}
+{{eqn | r = 3 \times 1 \, 625 \, 463 \, 840}}
+{{eqn | r = 4 \times 1 \, 210 \, 097 \, 880}}
+{{eqn | r = 5 \times 975 \, 278 \, 304}}
+{{eqn | r = 6 \times 812 \, 731 \, 920}}
+{{eqn | r = 7 \times 696 \, 627 \, 360}}
+{{eqn | r = 8 \times 609 \, 548 \, 940}}
+{{eqn | r = 9 \times 541 \, 821 \, 280}}
+{{eqn | r = 10 \times 487 \, 639 \, 152}}
+{{eqn | r = 11 \times 443 \, 308 \, 320}}
+{{eqn | r = 12 \times 406 \, 365 \, 960}}
+{{eqn | r = 13 \times 375 \, 107 \, 040}}
+{{eqn | r = 14 \times 348 \, 313 \, 680}}
+{{eqn | r = 15 \times 325 \, 092 \, 768}}
+{{eqn | r = 16 \times 304 \, 774 \, 470}}
+{{eqn | r = 17 \times 286 \, 846 \, 560}}
+{{eqn | r = 18 \times 270 \, 910 \, 640}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Smallest n for which 2^n-3 is Divisible by n}
+Tags: Number Theory
+
+\begin{theorem}
+The smallest [[Definition:Positive Integer|positive integer]] $n$ such that $2^n - 3$ is [[Definition:Divisor of Integer|divisible]] by $n$ is $4 \, 700 \, 063 \, 497$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Brute force by the Lehmers, probably}}
+\end{proof}<|endoftext|>
+\section{Smallest Odd Abundant Number not Divisible by 3}
+Tags: Abundant Numbers, 5,391,411,025
+
+\begin{theorem}
+The smallest [[Definition:Odd Integer|odd]] [[Definition:Abundant Number|abundant number]] not [[Definition:Divisor of Integer|divisible]] by $3$ is $5 \, 391 \, 411 \, 025$.
+\end{theorem}
+
+\begin{proof}
+We have:
+:$5 \, 391 \, 411 \, 025 = 5^2 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29$
+showing it is not [[Definition:Divisor of Integer|divisible]] by $3$.
+Then from {{SigmaLink|5,391,411,025|5 \, 391 \, 411 \, 025}} we have:
+:$\map \sigma {5 \, 391 \, 411 \, 025} = 10 \, 799 \, 308 \, 800 = 2 \times 5 \, 391 \, 411 \, 025 + 16 \, 486 \, 750$
+demonstrating that $5 \, 391 \, 411 \, 025$ is [[Definition:Abundant Number|abundant]].
+{{ProofWanted|It is still to be shown it is the smallest such.}}
+\end{proof}<|endoftext|>
+\section{Closure of Irreducible Subspace is Irreducible}
+Tags: Irreducible Spaces
+
+\begin{theorem}
+Let $X$ be a [[Definition:Topological Space|topological space]].
+Let $Y \subset X$ be an [[Definition:Irreducible Space|irreducible]] [[Definition:Topological Subspace|subspace]].
+Then its [[Definition:Closure (Topology)|closure]] $\overline Y$ is also [[Definition:Irreducible Space|irreducible]].
+\end{theorem}
+
+\begin{proof}
+By the definition of an [[Definition:Irreducible Space|irreducible]] subset, $Y\subset X$ is irreducible if and only if any nonempty [[Definition:Open Set|open]] subset in $Y$ intersects.
+Since the open subsets in $\bar{Y}$ is the same as the open subsets in $Y$, any two of them still trivially intersects in $\bar{Y}$, showing that $\bar{Y}$ is also irreducible.
+More generally, we can also show that if $\bar{Y}$ is irreducible for a subset $Y\subset X$, then $Y$ is also irreducible in $X$.
+We prove it by contradiction.
+Assume $Y$ is not irreducible, then there exist two closed proper subsets $Y_1$, $Y_2$ such that $Y=Y_1\cup Y_2$, then $\bar{Y}=\bar{Y_1}\cup \bar{Y_2}$, which contradicts with the assumption.
+\end{proof}<|endoftext|>
+\section{Point is Contained in Irreducible Component}
+Tags: Irreducible Spaces
+
+\begin{theorem}
+Let $X$ be a [[Definition:Topological Space|topological space]].
+Let $x\in X$ be a point.
+Then $x$ is [[Definition:Set Inclusion|contained]] in some [[Definition:Irreducible Component|irreducible component]] of $X$.
+\end{theorem}
+
+\begin{proof}
+Because [[Trivial Topological Space is Irreducible]], this is a special case of [[Irreducible Subspace is Contained in Irreducible Component]].
+{{qed}}
+[[Category:Irreducible Spaces]]
+0c7d4q4hjsg3o0hxvk280dbnknp8qme
+\end{proof}<|endoftext|>
+\section{Irreducible Subspace is Contained in Irreducible Component}
+Tags: Irreducible Spaces
+
+\begin{theorem}
+Let $X$ be a [[Definition:Topological Space|topological space]].
+Let $Y\subset X$ be an [[Definition:Irreducible Space|irreducible]] [[Definition:Topological Subspace|subspace]].
+Then there exists an [[Definition:Irreducible Component|irreducible component]] of $X$ [[Definition:Set Inclusion|containing]] $Y$.
+\end{theorem}
+
+\begin{proof}
+By definition, an [[Definition:Irreducible Component|irreducible component]] of $X$ is an [[Definition:Irreducible Space|irreducible]] [[Definition:Topological Subspace|subspace]] that is [[Definition:Maximal Element|maximal]] among the [[Definition:Irreducible Space|irreducible]] [[Definition:Topological Subspace|subspaces]], [[Definition:Inclusion Ordering|ordered by inclusion]].
+{{ProofWanted}}
+{{AoC|Zorn's Lemma}}
+\end{proof}<|endoftext|>
+\section{Irreducible Component is Closed}
+Tags: Irreducible Spaces
+
+\begin{theorem}
+Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
+Let $Y$ be an [[Definition:Irreducible Component|irreducible component]] of $T$.
+Then $Y$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
+\end{theorem}
+
+\begin{proof}
+By [[Closure of Irreducible Subspace is Irreducible]], the [[Definition:Topological Closure|closure]] $Y^-$ of $Y$ is [[Definition:Irreducible Space|irreducible]].
+By [[Set is Subset of its Topological Closure]], $Y \subseteq Y^-$.
+Because $Y$ is an [[Definition:Irreducible Component|irreducible component]], we must have $Y = Y^-$.
+By [[Set is Closed iff Equals Topological Closure]], $Y$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
+{{qed}}
+[[Category:Irreducible Spaces]]
+o1xyk5a7bvm1x768k3roqodm8ipyohl
+\end{proof}<|endoftext|>
+\section{Trivial Topological Space is Irreducible}
+Tags: Irreducible Spaces, Trivial Topological Space
+
+\begin{theorem}
+Let $X$ be a [[Definition:Trivial Topological Space|trivial topological space]].
+Then $X$ is [[Definition:Irreducible Space|irreducible]].
+\end{theorem}
+
+\begin{proof}
+Follows from:
+:[[Trivial Topological Space is Indiscrete]]
+:[[Indiscrete Space is Irreducible]]
+{{qed}}
+[[Category:Irreducible Spaces]]
+[[Category:Trivial Topological Space]]
+j50l4l5dfsvui33i4zypvnnzjtiwbcx
+\end{proof}<|endoftext|>
+\section{Trivial Topological Space is Indiscrete}
+Tags: Trivial Topological Space
+
+\begin{theorem}
+Let $X$ be a [[Definition:Trivial Topological Space|trivial topological space]].
+Then $X$ is [[Definition:Indiscrete Space|indiscrete]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Trivial Topological Space]]
+219q1co9xraqdc1zgy242fsjy30zqp2
+\end{proof}<|endoftext|>
+\section{Irreducible Components of Hausdorff Space are Points}
+Tags: Irreducible Spaces
+
+\begin{theorem}
+Let $X$ be a [[Definition:Hausdorff Space|Hausdorff space]].
+Then the [[Definition:Irreducible Component|irreducible components]] of $X$ are the [[Definition:Singleton|singleton sets]].
+\end{theorem}
+
+\begin{proof}
+By [[Subspace of Hausdorff Space is Hausdorff]], the [[Definition:Irreducible Component|irreducible components]] of $X$ are also [[Definition:Hausdorff Space|Hausdorff]].
+By [[Irreducible Hausdorff Space is Singleton]], they can only be [[Definition:Singleton|singletons]].
+By [[Trivial Topological Space is Irreducible]], every [[Definition:Singleton|singleton]] of $X$ is indeed [[Definition:Irreducible Space|irreducible]].
+{{qed}}
+[[Category:Irreducible Spaces]]
+8ylvofgxxdjnjln8zk4c1fjg1geuhjw
+\end{proof}<|endoftext|>
+\section{Pandigital Integer Formed by Digits in Alphabetical Order}
+Tags: Pandigital Integers, 8,549,176,320
+
+\begin{theorem}
+The number $8 \, 549 \, 176 \, 320$ is the [[Definition:Pandigital Integer|pandigital integer]] formed from the [[Definition:Digit|digits]] from $0$ to $9$ arranged in alphabetical order.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | o =
+ | r = 8
+ | c = eight
+}}
+{{eqn | o =
+ | r = 5
+ | c = five
+}}
+{{eqn | o =
+ | r = 4
+ | c = four
+}}
+{{eqn | o =
+ | r = 9
+ | c = nine
+}}
+{{eqn | o =
+ | r = 1
+ | c = one
+}}
+{{eqn | o =
+ | r = 7
+ | c = seven
+}}
+{{eqn | o =
+ | r = 6
+ | c = six
+}}
+{{eqn | o =
+ | r = 3
+ | c = three
+}}
+{{eqn | o =
+ | r = 2
+ | c = two
+}}
+{{eqn | o =
+ | r = 0
+ | c = zero
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Largest Pandigital Square including Zero}
+Tags: Square Numbers, Pandigital Integers, 9,814,072,356
+
+\begin{theorem}
+The largest [[Definition:Pandigital Integer|pandigital]] [[Definition:Square Number|square]] (in the sense where [[Definition:Pandigital Integer|pandigital]] includes the [[Definition:Zero Digit|zero]]) is $9 \, 814 \, 072 \, 356$:
+:$9 \, 814 \, 072 \, 356 = 99 \, 066^2$
+\end{theorem}
+
+\begin{proof}
+We check all the [[Definition:Square Number|square]] of numbers from $99 \, 067$ up to $\floor {9876543210} = 99 \, 380$, with the following constraints:
+Since all these [[Definition:Square Number|squares]] has $9$ as its leftmost [[Definition:Digit|digit]], the number cannot end with $3$ or $7$.
+The number cannot end with $0$ since its [[Definition:Square Number|square]] will end in $00$.
+A [[Definition:Pandigital Integer|pandigital]] number is [[Definition:Divisor of Integer|divisible]] by $9$, so our number must be [[Definition:Divisor of Integer|divisible]] by $3$.
+These constraints leaves us with the around $60$ candidates:
+:$99069, 99072, 99075, 99078, 99081, 99084, 99096, 99099, 99102, 99105, 99108, 99111, 99114, 99126, 99129, 99132, \dots, 99378$
+{{finish}}
+\end{proof}<|endoftext|>
+\section{Pandigital Properties of 9,876,543,210}
+Tags: 9,876,543,210, Pandigital Integers
+
+\begin{theorem}
+$9 \, 876 \, 543 \, 210$ has the following properties:
+:$9 \, 876 \, 543 \, 210 - 0 \, 123 \, 456 \, 789 = 9 \, 753 \, 086 \, 421$
+all three terms of which are [[Definition:Pandigital Integer|pandigital]].
+{{expand|Add more}}
+\end{theorem}<|endoftext|>
+\section{Palindromic Cube with Non-Palindromic Root}
+Tags: Cube Numbers, Palindromic Numbers, 10,662,526,601
+
+\begin{theorem}
+The only known [[Definition:Palindromic Integer|palindromic]] [[Definition:Cube Number|cube]] with a [[Definition:Cube Root|root]] that is not itself [[Definition:Palindromic Integer|palindromic]] is $10 \, 662 \, 526 \, 601$.
+\end{theorem}
+
+\begin{proof}
+We have that:
+:$10 \, 662 \, 526 \, 601 = 2201^3$
+There are no others whose [[Definition:Cube Root|cube root]] is below $10^{15}$.
+\end{proof}<|endoftext|>
+\section{Open Set of Irreducible Space is Irreducible}
+Tags: Irreducible Spaces
+
+\begin{theorem}
+Let $T = \struct {S, \tau}$ be an [[Definition:Irreducible Space|irreducible topological space]].
+Let $U$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Set (Topology)|open set]] of $T$.
+Then $U$ is [[Definition:Irreducible Space|irreducible]] in its [[Definition:Subspace Topology|induced subspace topology]].
+\end{theorem}
+
+\begin{proof}
+Let $T = \struct {S, \tau}$ be an [[Definition:Irreducible Space|irreducible topological space]].
+Let $U$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Set (Topology)|open set]] of $T$.
+{{AimForCont}} $U$ is not [[Definition:Irreducible Space|irreducible]] in $T$.
+Then $U = V_1 \cup V_2$ for some [[Definition:Closed Set (Topology)|closed sets]] $V_1$ and $V_2$ of $\struct {U, \tau_U}$.
+By definition of [[Definition:Subspace Topology|subspace topology]]:
+:$V_1 = U \cap W_1$
+and:
+:$V_2 = U \cap W_2$
+for some [[Definition:Closed Set (Topology)|closed sets]] $W_1$ and $W_2$ of $T$.
+Because $W_1 = S \implies V_1 = U \cap W_1 = U$, it follows that:
+:$W_1 \ne S$
+This is a [[Definition:Contradiction|contradiction]], because $V_1$ is a [[Definition:Proper Subset|proper subset]] of $U$.
+Now $U \ne \O$ implies that $S \setminus U$ is a [[Definition:Proper Subset|proper subset]] of $S$ which is a [[Definition:Closed Set (Topology)|closed set]] of $T$.
+Let $W_3 := W_2 \cup \paren {S \setminus U}$.
+Then $W_3$ is a [[Definition:Closed Set (Topology)|closed set]] of $T$.
+Also, $W_3 \ne S$, because otherwise $V_2 = U \cap W_2 = U \cap W_3 = U$.
+Thus $S = W_1 \cup W_3$ shows that $T$ is not [[Definition:Irreducible Space|irreducible]].
+The result follows by [[Proof by Contradiction]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Union of Open Irreducible Non-Disjoint Subspaces is Irreducible}
+Tags: Irreducible Spaces
+
+\begin{theorem}
+Let $T = \left({S, \tau}\right)$ be an [[Definition:Irreducible Space|irreducible toplogical space]].
+Let $U$ and $V$ be [[Definition:Open Set (Topology)|open]] [[Definition:Irreducible Space|irreducible]] [[Definition:Topological Subspace|subspaces]].
+Let their [[Definition:Set Intersection|intersection]] $U \cap V$ be [[Definition:Non-Empty Set|non-empty]].
+Then their [[Definition:Set Union|union]] $U \cup V$ is [[Definition:Irreducible Space|irreducible]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Irreducible Spaces]]
+c2et37ojcyg591nbskw2bej1966454m
+\end{proof}<|endoftext|>
+\section{Closed Set of Ultraconnected Space is Ultraconnected}
+Tags: Ultraconnected Spaces
+
+\begin{theorem}
+Let $T = \struct {S, \tau}$ be an [[Definition:Ultraconnected Space|ultraconnected topological space]].
+Let $F \subset S$ be a [[Definition:Closed Set (Topology)|closed set]] in $T$.
+Then $F$ is [[Definition:Ultraconnected Space|ultraconnected]].
+\end{theorem}
+
+\begin{proof}
+Let $A, B$ be two [[Definition:Non-Empty Set|non-empty]] [[Definition:Closed Set (Topology)|closed sets]] in $\struct {F, \tau}$.
+By [[Closed Set in Topological Subspace/Corollary|Closed Set in Topological Subspace]], $A, B$ are [[Definition:Closed Set (Topology)|closed]] in $T$ as well.
+By {{Defof|Ultraconnected Space}}, $A$ and $B$ are not [[Definition:Disjoint Sets|disjoint]].
+Since $A$, $B$ are arbitrary, no two [[Definition:Non-Empty Set|non-empty]] [[Definition:Closed Set (Topology)|closed sets]] of $\struct {F, \tau}$ are [[Definition:Disjoint Sets|disjoint]].
+Hence the result from {{Defof|Ultraconnected Space}}.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Noetherian Space is Compact}
+Tags: Noetherian Spaces
+
+\begin{theorem}
+Let $X$ be a [[Definition:Noetherian Topological Space|noetherian topological space]].
+Then $X$ is [[Definition:Compact Space|compact]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Noetherian Spaces]]
+7bbes81c9tle64y3qikhqsxyy9hk3la
+\end{proof}<|endoftext|>
+\section{Subspace of Noetherian Space is Noetherian}
+Tags: Noetherian Spaces
+
+\begin{theorem}
+Let $X$ be a [[Definition:Noetherian Topological Space|noetherian topological space]].
+Let $Y\subseteq X$ be a [[Definition:Topological Subspace|subspace]].
+Then $Y$ is [[Definition:Noetherian Topological Space|noetherian]].
+\end{theorem}<|endoftext|>
+\section{Zero Locus of Larger Set is Smaller}
+Tags: Zariski Topology, Algebraic Geometry
+
+\begin{theorem}
+Let $k$ be a [[Definition:Field (Abstract Algebra)|field]].
+Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
+Let $A = k \sqbrk {X_1, \ldots, X_n}$ be the [[Definition:Ring of Polynomials|ring of polynomials]] in $n$ [[Definition:Variable of Polynomial Ring|variables]] over $k$.
+Let $I, J \subseteq A$ be [[Definition:Subset|subsets]], and $\map V I$ and $\map V J$ their [[Definition:Zero Locus of Set of Polynomials|zero loci]].
+Let $I \subseteq J$.
+Then $\map V I \supseteq \map V J$.
+\end{theorem}
+
+\begin{proof}
+Assume $p \in \map V J$.
+Then:
+{{begin-eqn}}
+{{eqn | l = p
+ | o = \in
+ | r = \map V J
+}}
+{{eqn | ll= \leadsto
+ | lo= \forall x \in J:
+ | l = \map p x
+ | r = 0
+}}
+{{eqn | ll= \leadsto
+ | lo= \forall x \in I:
+ | l = \map p x
+ | r = 0
+ | c = $I \subseteq J$ by assumption
+}}
+{{eqn | ll= \leadsto
+ | l = p
+ | o = \in
+ | r = \map V I
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Zariski Topology]]
+[[Category:Algebraic Geometry]]
+3q31apmcqb9tcrnpx8ij71hun45jp2g
+\end{proof}<|endoftext|>
+\section{Smallest Multiply Perfect Number of Order 5}
+Tags: Multiply Perfect Numbers, 14,182,439,040
+
+\begin{theorem}
+The number $14 \, 182 \, 439 \, 040$ is [[Definition:Multiply Perfect Number|multiply perfect]] of [[Definition:Order of Multiply Perfect Number|order]] $5$:
+:$\sigma \left({14 \, 182 \, 439 \, 040}\right) = 70 \, 912 \, 195 \, 200 = 5 \times 14 \, 182 \, 439 \, 040$
+It is the smallest [[Definition:Positive Integer|positive integer]] to be so.
+\end{theorem}
+
+\begin{proof}
+From {{SigmaLink|14,182,439,040|14 \, 182 \, 439 \, 040}}:
+:$\sigma \left({14 \, 182 \, 439 \, 040}\right) = 70 \, 912 \, 195 \, 200$
+{{ProofWanted|That it is the smallest one remains to be proved.}}
+\end{proof}<|endoftext|>
+\section{Smallest Fourth Power expressible as Sum of 4 Fourth Powers}
+Tags: Fourth Powers, 15,527,402,881
+
+\begin{theorem}
+$15 \, 527 \, 402 \, 881$ is the smallest [[Definition:Fourth Power|fourth power]] which can be expressed as the [[Definition:Integer Addition|sum]] of $4$ [[Definition:Fourth Power|fourth powers]]:
+:$15 \, 527 \, 402 \, 881 = 353^4 = 30^4 + 120^4 + 272^4 + 315^4$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | o =
+ | r = 30^4 + 120^4 + 272^4 + 315^4
+ | c =
+}}
+{{eqn | r = 810 \, 000 + 207 \, 360 \, 000 + 5 \, 473 \, 632 \, 256 + 9 \, 845 \, 600 \, 625
+ | c =
+}}
+{{eqn | r = 15 \, 527 \, 402 \, 881
+ | c =
+}}
+{{eqn | r = 353^4
+ | c =
+}}
+{{end-eqn}}
+{{ProofWanted|That it is the smallest remains to be proved.}}
+\end{proof}<|endoftext|>
+\section{Largest Known Lead by 4n+1 in Prime Number Race}
+Tags: Prime Number Races
+
+\begin{theorem}
+In the [[Definition:Prime Number Race|prime number race]] between [[Definition:Prime Number|prime numbers]] of the form $4 n - 1$ and $4 n + 1$, the highest known stretch of [[Definition:Integer|integers]] where $4 n + 1$ is not less than $4 n - 1$ is between $18 \, 465 \, 126 \, 293$ and $19 \, 033 \, 524 \, 538$.
+\end{theorem}
+
+\begin{proof}
+First note that:
+{{begin-eqn}}
+{{eqn | l = 18 \, 465 \, 126 \, 257
+ | r = 4 \times 4 \, 616 \, 281 \, 564 + 1
+ | c =
+}}
+{{eqn | l = 18 \, 465 \, 126 \, 293
+ | r = 4 \times 4 \, 616 \, 281 \, 573 + 1
+ | c =
+}}
+{{end-eqn}}
+where it can be seen that [[Definition:Prime Number|prime numbers]] of the form $4 n + 1$ are locally increasing, and:
+{{begin-eqn}}
+{{eqn | l = 19 \, 033 \, 524 \, 533
+ | r = 4 \times 4 \, 758 \, 381 \, 133 + 1
+ | c =
+}}
+{{eqn | l = 19 \, 033 \, 524 \, 539
+ | r = 4 \times 4 \, 758 \, 381 \, 135 - 1
+ | c =
+}}
+{{end-eqn}}
+where it can be seen at that point [[Definition:Prime Number|prime numbers]] of the form $4 n - 1$ may now be in front, but this is inconclusive.
+{{ProofWanted|I'm not actually sure this is correct. If $4 n - 1$ is now in front for the first time since $18 \, 465 \, 126 \, 293$, I would expect the previous prime also to be of the same form -- but it is not.}}
+\end{proof}<|endoftext|>
+\section{Number whose Square is in 2 Identical Halves}
+Tags: Square Numbers, 36,363,636,364
+
+\begin{theorem}
+The number $36 \, 363 \, 636 \, 364$ has the property that its [[Definition:Square (Algebra)|square]] can be split into two identical halves:
+:$36 \, 363 \, 636 \, 364 = 1 \, 322 \, 314 \, 049 \, 613 \, 223 \, 140 \, 496$
+\end{theorem}
+
+\begin{proof}
+It just does.
+\end{proof}<|endoftext|>
+\section{Fifth Power expressible as Sum of 4 Fifth Powers}
+Tags: Fifth Powers, 61,917,364,224
+
+\begin{theorem}
+$61 \, 917 \, 364 \, 224$ can be expressed as the [[Definition:Integer Addition|sum]] of $4$ [[Definition:Fifth Power|fifth powers]]:
+:$61 \, 917 \, 364 \, 224 = 144^5 = 27^5 + 84^5 + 110^5 + 133^5$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | o =
+ | r = 27^5 + 84^5 + 110^5 + 133^5
+ | c =
+}}
+{{eqn | r = 14 \, 348 \, 907 + 4 \, 182 \, 119 \, 424 + 16 \, 105 \, 100 \, 000 + 41 \, 615 \, 795 \, 893
+ | c =
+}}
+{{eqn | r = 61 \, 917 \, 364 \, 224
+ | c =
+}}
+{{eqn | r = 144^5
+ | c =
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Common Sum of 3 Distinct Amicable Pairs}
+Tags: Amicable Pairs
+
+\begin{theorem}
+The [[Definition:Integer|integer]] $64 \, 795 \, 852 \, 800$ is the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Distinct|distinct]] [[Definition:Amicable Pair|amicable pairs]]:
+:$29 \, 912 \, 035 \, 725$ and $34 \, 883 \, 817 \, 075$
+:$31 \, 695 \, 652 \, 275$ and $33 \, 100 \, 200 \, 525$
+:$32 \, 129 \, 958 \, 525$ and $32 \, 665 \, 894 \, 275$
+all of them odd.
+\end{theorem}
+
+\begin{proof}
+We have that:
+From [[Odd Amicable Pair/Examples/29,912,035,725-34,883,817,075|$29 \, 912 \, 035 \, 725$ and $34 \, 883 \, 817 \, 075$ are amicable]]:
+{{:Odd Amicable Pair/Examples/29,912,035,725-34,883,817,075}}
+From [[Odd Amicable Pair/Examples/31,695,652,275-33,100,200,525|$31 \, 695 \, 652 \, 275$ and $33 \, 100 \, 200 \, 525$ are amicable]]:
+{{:Odd Amicable Pair/Examples/31,695,652,275-33,100,200,525}}
+From [[Odd Amicable Pair/Examples/32,129,958,525-32,665,894,275|$32 \, 129 \, 958 \, 525$ and $32 \, 665 \, 894 \, 275$ are amicable]]:
+{{:Odd Amicable Pair/Examples/32,129,958,525-32,665,894,275}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Multiply Perfect Number of Order 6}
+Tags: Multiply Perfect Numbers
+
+\begin{theorem}
+The number defined as:
+:$n = 2^{36} \times 3^8 \times 5^5 \times 7^7 \times 11 \times 13^2 \times 19 \times 31^2$
+::$\times \ 43 \times 61 \times 83 \times 223 \times 331 \times 379 \times 601 \times 757 \times 1201$
+::$\times \ 7019 \times 112 \, 303 \times 898 \, 423 \times 616 \, 318 \, 177$
+is [[Definition:Multiply Perfect Number|multiply perfect]] of [[Definition:Order of Multiply Perfect Number|order]] $6$.
+\end{theorem}
+
+\begin{proof}
+From [[Sigma Function is Multiplicative]], we may take each [[Definition:Prime Factor|prime factor]] separately and form $\sigma \left({n}\right)$ 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 {11}
+ | r = 12
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^2 \times 3
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {19}
+ | r = 20
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^2 \times 5
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {43}
+ | r = 44
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^2 \times 11
+ | 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 {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 {223}
+ | r = 224
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^5 \times 7
+ | 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 {379}
+ | r = 380
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^2 \times 5 \times 19
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {601}
+ | r = 602
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 7 \times 43
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {757}
+ | r = 758
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 379
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {1201}
+ | r = 1202
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 601
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {7019}
+ | r = 7020
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^2 \times 3^3 \times 5 \times 13
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {112 \, 303}
+ | r = 112 \, 304
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^4 \times 7019
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {898 \, 423}
+ | r = 898 \, 424
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2^3 \times 112 \, 303
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {616 \, 318 \, 177}
+ | r = 616 \, 318 \, 178
+ | c = [[Sigma Function of Prime Number]]
+}}
+{{eqn | r = 2 \times 7^3 \times 898 \, 423
+ | 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^{36} }
+ | r = 2 \times 2^{36} - 1
+ | c = [[Sigma Function of Power of 2]]
+}}
+{{eqn | r = 137 \, 438 \, 953 \, 471
+ | c =
+}}
+{{eqn | r = 223 \times 616 \, 318 \, 177
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {3^8}
+ | r = \dfrac {3^9 - 1} {3 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {19 \, 683 - 1} 2
+ | c =
+}}
+{{eqn | r = 9841
+ | c =
+}}
+{{eqn | r = 13 \times 757
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {5^5}
+ | r = \dfrac {5^6 - 1} {5 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {15 \, 625 - 1} 4
+ | c =
+}}
+{{eqn | r = 3906
+ | c =
+}}
+{{eqn | r = 2 \times 3^2 \times 7 \times 31
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {7^7}
+ | r = \dfrac {7^8 - 1} {7 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {5 \, 764 \, 801 - 1} 6
+ | c =
+}}
+{{eqn | r = 960 \, 800
+ | c =
+}}
+{{eqn | r = 2^5 \times 5^2 \times 1201
+ | c =
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \map \sigma {13^2}
+ | r = \dfrac {13^3 - 1} {13 - 1}
+ | c = [[Sigma Function of Power of Prime]]
+}}
+{{eqn | r = \dfrac {2197 - 1} {12}
+ | c =
+}}
+{{eqn | r = 183
+ | c =
+}}
+{{eqn | r = 3 \times 61
+ | 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}}
+Gathering up the [[Definition:Prime Factor|prime factors]], we have:
+:$\map \sigma n = 2^{37} \times 3^9 \times 5^5 \times 7^7 \times 11 \times 13^2 \times 19 \times 31^2$
+::$\times \ 43 \times 61 \times 83 \times 223 \times 331 \times 379 \times 601 \times 757 \times 1201$
+::$\times \ 7019 \times 112 \, 303 \times 898 \, 423 \times 616 \, 318 \, 177$
+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 $2$ and $3$.
+It follows that $\map \sigma n = 2 \times 3 \times n = 6 n$.
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Probability of Receiving Complete Suit as Hand at Bridge}
+Tags: Bridge (Game)
+
+\begin{theorem}
+The [[Definition:Probability|probability]] of being dealt a complete [[Definition:Suit of Cards|suit]] in a deal at [[Definition:Bridge (Game)|Bridge]] is $1$ in $158 \, 753 \, 389 \, 900$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Straightforward but boring exercise in combinatorics}}
+\end{proof}<|endoftext|>
+\section{Polynomial Ring is Generated by Indeterminate over Ground Ring}
+Tags: Polynomial Theory
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+Let $R \sqbrk X$ be a [[Definition:Polynomial Ring|polynomial ring]] over $R$.
+Let $\iota: R \to R \sqbrk X$ be the [[Definition:Embedding into Polynomial Ring|embedding]].
+Then $R \sqbrk X$ is [[Definition:Generator of Ring Extension|generated]] by $X$ over $R$.
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Use [[Polynomial is Linear Combination of Monomials]]}}
+[[Category:Polynomial Theory]]
+2x9lzdnsljh8sd5unxy7zgtw235pu2z
+\end{proof}<|endoftext|>
+\section{Smallest Cunningham Chain of the First Kind of Length 12}
+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]] $12$ is:
+:$554 \, 688 \, 278 \, 429$, $1 \, 109 \, 376 \, 556 \, 859$, $2 \, 218 \, 753 \, 113 \, 719$, $4 \, 437 \, 506 \, 227 \, 439$,
+::$8 \, 875 \, 012 \, 454 \, 879$, $17 \, 750 \, 024 \, 909 \, 759$, $35 \, 500 \, 049 \, 819 \, 519$, $71 \, 000 \, 099 \, 639 \, 039$,
+::$142 \, 000 \, 199 \, 278 \, 079$, $284 \, 000 \, 398 \, 556 \, 159$, $568 \, 000 \, 797 \, 112 \, 319$, $1 \, 136 \, 001 \, 594 \, 224 \, 639$
+\end{theorem}
+
+\begin{proof}
+Let $C$ denote the [[Definition:Sequence|sequence]] in question.
+We have that $554 \, 688 \, 278 \, 429$ is [[Definition:Prime Number|prime]].
+First note that:
+:$\dfrac {554 \, 688 \, 278 \, 429 - 1} 2 = 277 \, 344 \, 139 \, 214 = 2 \times 138 \, 672 \, 069 \, 607$
+and so is not [[Definition:Prime Number|prime]].
+Thus $554 \, 688 \, 278 \, 429$ is not a [[Definition:Safe Prime|safe prime]], and thus fulfils the requirement for $C$ to be a [[Definition:Cunningham Chain of the First Kind|Cunningham chain of the first kind]].
+Then:
+{{begin-eqn}}
+{{eqn | n = 1
+ | l = 2 \times 554 \, 688 \, 278 \, 429 + 1
+ | r = 1 \, 109 \, 376 \, 556 \, 859
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 2
+ | l = 2 \times 1 \, 109 \, 376 \, 556 \, 859 + 1
+ | r = 2 \, 218 \, 753 \, 113 \, 719
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 3
+ | l = 2 \times 2 \, 218 \, 753 \, 113 \, 719 + 1
+ | r = 4 \, 437 \, 506 \, 227 \, 439
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 4
+ | l = 2 \times 4 \, 437 \, 506 \, 227 \, 439 + 1
+ | r = 8 \, 875 \, 012 \, 454 \, 879
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 5
+ | l = 2 \times 8 \, 875 \, 012 \, 454 \, 879 + 1
+ | r = 17 \, 750 \, 024 \, 909 \, 759
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 6
+ | l = 2 \times 17 \, 750 \, 024 \, 909 \, 759 + 1
+ | r = 35 \, 500 \, 049 \, 819 \, 519
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 7
+ | l = 2 \times 35 \, 500 \, 049 \, 819 \, 519 + 1
+ | r = 71 \, 000 \, 099 \, 639 \, 039
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 8
+ | l = 2 \times 71 \, 000 \, 099 \, 639 \, 039 + 1
+ | r = 142 \, 000 \, 199 \, 278 \, 079
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 9
+ | l = 2 \times 142 \, 000 \, 199 \, 278 \, 079 + 1
+ | r = 284 \, 000 \, 398 \, 556 \, 159
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 10
+ | l = 2 \times 284 \, 000 \, 398 \, 556 \, 159 + 1
+ | r = 568 \, 000 \, 797 \, 112 \, 319
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 11
+ | l = 2 \times 568 \, 000 \, 797 \, 112 \, 319 + 1
+ | r = 1 \, 136 \, 001 \, 594 \, 224 \, 639
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 12
+ | l = 2 \times 1 \, 136 \, 001 \, 594 \, 224 \, 639 + 1
+ | r = 2 \, 272 \, 003 \, 188 \, 449 \, 279
+ | c = which is $19 \times 119 \, 579 \, 115 \, 181 \, 541$ [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+Establishing that this is indeed the smallest such [[Definition:Cunningham Chain of the First Kind|Cunningham chain of the first kind]] of [[Definition:Length of Sequence|length]] $12$ can be done by a computer search.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Set of Integers Bounded Above by Real Number has Greatest Element}
+Tags: Set of Integers Bounded Above has Greatest Element
+
+\begin{theorem}
+Let $\Z$ be the [[Definition:Integer|set of integers]].
+Let $\le$ be the [[Definition:Usual Ordering|usual ordering on the real numbers]] $\R$.
+Let $\O \subset S \subseteq \Z$ such that $S$ is [[Definition:Bounded Above Set|bounded above]] in $\struct {\R, \le}$.
+Then $S$ has a [[Definition:Greatest Element|greatest element]].
+\end{theorem}
+
+\begin{proof}
+Let $S$ be [[Definition:Bounded Above Set|bounded above]] by $x \in \R$.
+By the [[Archimedean Principle]], there exists an [[Definition:Integer|integer]] $n \ge x$.
+Then $S$ is [[Definition:Bounded Above Set|bounded above]] by $n$.
+By [[Set of Integers Bounded Above by Integer has Greatest Element]], $S$ has a [[Definition:Greatest Element|greatest element]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Set of Integers Bounded Below by Real Number has Smallest Element}
+Tags: Set of Integers Bounded Below has Smallest Element
+
+\begin{theorem}
+Let $\Z$ be the [[Definition:Integer|set of integers]].
+Let $\le$ be the [[Definition:Usual Ordering|usual ordering on the real numbers]] $\R$.
+Let $\O \subset S \subseteq \Z$ such that $S$ is [[Definition:Bounded Below Set|bounded below]] in $\struct {\R, \le}$.
+Then $S$ has a [[Definition:Smallest Element|smallest element]].
+\end{theorem}
+
+\begin{proof}
+Let $S$ be [[Definition:Bounded Below Set|bounded below]] by $x \in \R$.
+By the [[Archimedean Principle]], there exists an [[Definition:Integer|integer]] $n \le x$.
+Then $S$ is [[Definition:Bounded Below Set|bounded below]] by $n$.
+By [[Set of Integers Bounded Below by Integer has Smallest Element]], $S$ has a [[Definition:Smallest Element|smallest element]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Real Number lies between Unique Pair of Consecutive Integers}
+Tags: Integers, Floor Function
+
+\begin{theorem}
+Let $x$ be a [[Definition:Real Number|real number]].
+\end{theorem}
+
+\begin{proof}
+=== Existence ===
+By [[Set of Integers Bounded Above by Real Number has Greatest Element]], the [[Definition:Set|set]]:
+:$S = \left\{{m \in \Z: m \le x}\right\}$
+has a [[Definition:Greatest Element|greatest element]], say $n$.
+Because $n+1>n$, $n+1\notin S$.
+Thus $n+1> x$.
+Thus $n\leq x < n+1$.
+{{qed|lemma}}
+=== Uniqueness ===
+Let $n\in\Z$ be such that:
+:$n \leq x < n + 1$
+We show that $n$ is a [[Definition:Greatest Element|greatest element]] of the [[Definition:Set|set]]:
+:$S = \left\{{m \in \Z: m \le x}\right\}$
+so that the uniqueness follows from [[Greatest Element is Unique]].
+Because $n\leq x$, we have $n \in S$.
+Let $m \in S$.
+Because $m \leq x < n+1$, $n+1 > m$.
+By [[Weak Inequality of Integers iff Strict Inequality with Integer plus One]]:
+:$n \geq m$.
+Because $m$ was arbitrary, $n$ is a [[Definition:Greatest Element|greatest element]] of $S$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Supremum of Set of Integers is Integer}
+Tags: Integers
+
+\begin{theorem}
+Let $S \subset \Z$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Integer|set of integers]].
+Let $S$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]] in the [[Definition:Real Number|set of real numbers]].
+Then its [[Definition:Supremum of Subset of Real Numbers|supremum]] $\sup S$ is an [[Definition:Integer|integer]].
+\end{theorem}
+
+\begin{proof}
+By [[Supremum of Set of Integers equals Greatest Element]], $S$ has a [[Definition:Greatest Element|greatest element]] $n \in \Z$, that is equals to the [[Definition:Supremum of Subset of Real Numbers|supremum]] of $S$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Definition:Constructed Semantics/Instance 1/Rule of Idempotence}
+Tags: Formal Semantics
+
+\begin{theorem}
+The [[Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication|Rule of Idempotence]]:
+:$(p \lor p) \implies p$
+is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 1|Instance 1]] of [[Definition:Constructed Semantics|constructed semantics]].
+\end{theorem}
+
+\begin{proof}
+By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
+:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
+the [[Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication|Rule of Idempotence]] can be written as:
+: $\neg \left({p \lor p}\right) \lor p$
+This evaluates as follows:
+:$\begin{array}{|cccc|c|c|} \hline
+\neg & (p & \lor & p) & \lor & p \\
+\hline
+2 & 2 & 1 & 1 & 2 & 1 \\
+1 & 2 & 2 & 2 & 2 & 2 \\
+\hline
+\end{array}$
+{{qed}}
+[[Category:Formal Semantics]]
+r07evkbzlhssv5d3yrb3yass8oq8v9l
+\end{proof}<|endoftext|>
+\section{Definition:Constructed Semantics/Instance 1/Rule of Addition}
+Tags: Formal Semantics
+
+\begin{theorem}
+The [[Rule of Addition/Sequent Form/Formulation 2/Form 2|Rule of Addition]]:
+:$q \implies (q \lor p)$
+is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 1|Instance 1]] of [[Definition:Constructed Semantics|constructed semantics]].
+\end{theorem}
+
+\begin{proof}
+By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
+:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
+the [[Rule of Addition/Sequent Form/Formulation 2/Form 2|Rule of Addition]] can be written as:
+: $\neg q \lor \left({p \lor q}\right)$
+This evaluates as follows:
+:$\begin{array}{|cc|c|ccc|} \hline
+\neg & q & \lor & (p & \lor & q) \\
+\hline
+2 & 1 & 2 & 1 & 1 & 1 \\
+1 & 2 & 2 & 1 & 2 & 2 \\
+2 & 1 & 2 & 2 & 2 & 1 \\
+1 & 2 & 2 & 2 & 2 & 2 \\
+\hline
+\end{array}$
+{{qed}}
+[[Category:Formal Semantics]]
+rk497e2li883lwe4k6qna1aco309ty8
+\end{proof}<|endoftext|>
+\section{Definition:Constructed Semantics/Instance 1/Rule of Commutation}
+Tags: Formal Semantics
+
+\begin{theorem}
+The [[Rule of Commutation/Disjunction/Formulation 2/Forward Implication|Rule of Commutation]]:
+:$\left({p \lor q}\right) \implies \left({q \lor p}\right)$
+is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 1|Instance 1]] of [[Definition:Constructed Semantics|constructed semantics]].
+\end{theorem}
+
+\begin{proof}
+By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
+:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
+the [[Rule of Commutation/Disjunction/Formulation 2/Forward Implication|Rule of Commutation]] can be written as:
+:$\neg \left({p \lor q}\right) \lor \left({q \lor p}\right)$
+This evaluates as follows:
+:$\begin{array}{|cccc|c|ccc|} \hline
+\neg & (p & \lor & q) & \lor & (q & \lor & p) \\
+\hline
+2 & 1 & 1 & 1 & 2 & 1 & 1 & 1 \\
+1 & 1 & 2 & 2 & 2 & 2 & 2 & 1 \\
+1 & 2 & 2 & 1 & 2 & 1 & 2 & 2 \\
+1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
+\hline
+\end{array}$
+{{qed}}
+[[Category:Formal Semantics]]
+nhvy61sitluk232bj2i72se5qmw6xrs
+\end{proof}<|endoftext|>
+\section{Supremum of Set of Integers equals Greatest Element}
+Tags: Integers
+
+\begin{theorem}
+Let $S \subset \Z$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Integer|set of integers]].
+Let $S$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]] in the [[Definition:Real Number|set of real numbers]] $\R$.
+Then $S$ has a [[Definition:Greatest Element|greatest element]], and it is equal to the [[Definition:Supremum of Subset of Real Numbers|supremum]] $\sup S$.
+\end{theorem}
+
+\begin{proof}
+By [[Set of Integers Bounded Above by Real Number has Greatest Element]], $S$ has a [[Definition:Greatest Element|greatest element]], say $n \in S$.
+By [[Greatest Element is Supremum]], $n$ is the [[Definition:Supremum of Subset of Real Numbers|supremum]] of $S$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Definition:Constructed Semantics/Instance 1/Factor Principle}
+Tags: Formal Semantics
+
+\begin{theorem}
+The [[Factor Principles/Disjunction on Left/Formulation 2|Factor Principle]]:
+:$\left({p \implies q}\right) \implies \left({\left({r \lor p}\right) \implies \left ({r \lor q}\right)}\right)$
+is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 1|Instance 1]] of [[Definition:Constructed Semantics|constructed semantics]].
+\end{theorem}
+
+\begin{proof}
+By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
+:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
+the [[Factor Principles/Disjunction on Left/Formulation 2|Factor Principle]] can be written as:
+:$\neg \left({\neg p \lor q}\right) \lor \left({\neg \left({r \lor p}\right) \lor \left ({r \lor q}\right)}\right)$
+This evaluates as follows:
+:$\begin{array}{|ccccc|c|cccccccc|} \hline
+\neg & (\neg & p & \lor & q) & \lor & (\neg & (r & \lor & p) & \lor & (r & \lor & q)) \\
+\hline
+1 & 2 & 1 & 2 & 1 & 2 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 1 \\
+1 & 2 & 1 & 2 & 1 & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 \\
+1 & 2 & 1 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & 2 & 1 & 2 & 2 \\
+1 & 2 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 2 \\
+2 & 1 & 2 & 1 & 1 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 1 & 1 \\
+2 & 1 & 2 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 1 \\
+1 & 1 & 2 & 2 & 2 & 2 & 1 & 1 & 2 & 2 & 2 & 1 & 2 & 2 \\
+1 & 1 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
+\hline
+\end{array}$
+{{qed}}
+[[Category:Formal Semantics]]
+g378broae9uk77f3lxwet48mpsu2f3y
+\end{proof}<|endoftext|>
+\section{Weak Inequality of Integers iff Strict Inequality with Integer plus One}
+Tags: Orderings on Integers
+
+\begin{theorem}
+Let $a, b \in \Z$ be [[Definition:Integer|integers]].
+{{TFAE}}
+:$(1): \quad a \le b$
+:$(2): \quad a < b + 1$
+where:
+:$\le$ is the [[Definition:Ordering on Integers|ordering on the integers]]
+:$<$ is the [[Definition:Strict Ordering on Integers|strict ordering on the integers]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Orderings on Integers]]
+o50pjpmex3eprkvlhweb2aux9n6498h
+\end{proof}<|endoftext|>
+\section{Set of Orbits forms Partition}
+Tags: Group Actions
+
+\begin{theorem}
+Let $G$ be a [[Definition:Group|group]].
+Let $X$ be a [[Definition:Set|set]].
+Let $G$ [[Definition:Group Action|act]] on $X$.
+Then the [[Definition:Set of Orbits|set of orbits]] of the [[Definition:Group Action|group action]] forms a [[Definition:Set Partition|partition]] of $X$.
+\end{theorem}
+
+\begin{proof}
+Follows from the [[Fundamental Theorem on Equivalence Relations]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Interior of Closure is Regular Open}
+Tags: Regular Open Sets
+
+\begin{theorem}
+Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
+Let $H \subseteq S$.
+Then $H^{- \circ}$ is [[Definition:Regular Open Set|regular open]].
+\end{theorem}
+
+\begin{proof}
+We must show that $H^{- \circ - \circ} = H^{- \circ}$.
+First we show that $H^{- \circ - \circ} \subseteq H^{- \circ}$.
+By the definition of [[Definition:Interior (Topology)/Definition 2|interior]]:
+:$H^{- \circ} \subseteq H^-$
+By [[Topological Closure of Subset is Subset of Topological Closure]]:
+:$H^{- \circ -} \subseteq H^{--} = H^-$
+By [[Interior of Subset]]:
+:$H^{- \circ - \circ } \subseteq H^{- \circ}$
+{{qed|lemma}}
+Next we show $H^{- \circ - \circ} \supseteq H^{- \circ}$.
+Again by the definition of [[Definition:Interior (Topology)/Definition 2|interior]]:
+: $H^{- \circ -} \supseteq H^{- \circ}$
+By [[Interior of Subset]]:
+: $H^{- \circ - \circ} \supseteq H^{- \circ \circ} = H^{- \circ}$
+{{qed|lemma}}
+Thus by definition of [[Definition:Set Equality/Definition 2|set equality]]:
+: $H^{- \circ - \circ} = H^{- \circ}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Jordan's Lemma}
+Tags: Contour Integration, Complex Analysis
+
+\begin{theorem}
+Consider a [[Definition:Complex-Valued Function|complex-valued]], [[Definition:Continuous Complex Function|continuous function]] $f$ defined on the contour:
+:$C_r = \left\{r e^{i \theta}: 0 \le \theta \le \pi \right\}, \ r > 0$
+If the function $f$ is of the form:
+:$f \left({z}\right) = e^{i a z} g \left({z}\right), \ a > 0, \ z \in C_r$
+Then:
+:$\displaystyle \left\vert{\int_{C_r} f \left({z}\right) \rd z}\right\vert \le \frac \pi a \max_{0 \le \theta \le \pi} \left\vert{g \left(re^{i \theta}\right)}\right\vert$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \left\vert{\int_{C_r} f\left({z}\right) \rd z}\right\vert
+ | r = \left\vert{\int_{C_r} e^{i a z} g\left({z}\right) \rd z}\right\vert
+}}
+{{eqn | r = \left\vert{i r}\right\vert \left\vert{\int_0^\pi e^{i a r e^{i \theta} } g \left({r e^{i \theta} }\right) e^{i \theta} \rd \theta}\right\vert
+ | c = {{Defof|Complex Contour Integral}}
+}}
+{{eqn | r = \left\vert{i r}\right\vert \left\vert{\int_0^\pi e^{i a r \left({i \sin \theta + \cos \theta}\right)} g \left({r e^{i \theta} }\right) e^{i \theta} \rd \theta}\right\vert
+ | c = [[Euler's Formula]]
+}}
+{{eqn | r = r \left\vert{\int_0^\pi e^{a r \left({i \cos \theta - \sin \theta}\right)} g \left({r e^{i \theta} }\right) e^{i \theta} \rd \theta}\right\vert
+ | c = $i^2 = -1$, $\left\vert{i}\right\vert = 1$
+}}
+{{eqn | o = \le
+ | r = r \int_0^\pi \left\vert{e^{i a r \cos \theta} }\right\vert \left\vert{e^{-a r \sin \theta} }\right\vert \left\vert{g \left({r e^{i \theta} }\right)}\right\vert \left\vert{e^{i \theta} }\right\vert \rd \theta
+ | c = [[Modulus of Complex Integral]]
+}}
+{{eqn | r = r \int_0^\pi e^{-a r \sin \theta} \left\vert{g \left({r e^{i\theta} }\right)}\right\vert \rd \theta
+ | c = $\left\vert{e^{i \theta} }\right\vert = 1$ for real $\theta$
+}}
+{{eqn | o = \le
+ | r = r \int_0^\pi e^{-a r \sin \theta} \max_{0 \le \theta \le \pi} \left\vert{g \left({r e^{i \theta} }\right)}\right\vert \rd \theta
+ | c = {{Defof|Supremum of Real-Valued Function}}
+}}
+{{eqn | o = \le
+ | r = r \max_{0 \le \theta \le \pi} \left\vert{g \left({r e^{i \theta} }\right)}\right\vert \int_0^\pi e^{-\left({2 a r \theta}\right) / \pi} \rd \theta
+ | c = [[Shape of Sine Function]]
+}}
+{{eqn | r = r \max_{0 \le \theta \le \pi} \left\vert{g\left({r e^{i \theta} }\right)}\right\vert \left[{- \frac{\pi e^{\left({2 a r \theta}\right) / \pi} } {2 a r} }\right]_0^\pi
+ | c = [[Primitive of Exponential of a x]], [[Fundamental Theorem of Calculus]]
+}}
+{{eqn | r = \max_{0 \le \theta \le \pi} \left\vert{g\left({r e^{i \theta} }\right)}\right\vert \frac{\pi \left({1 - e^{-2 a r} }\right)} {2 a}
+}}
+{{eqn | o = \le
+ | r = \frac \pi {2 a} \max_{0 \le \theta \le \pi} \left\vert{g\left({r e^{i \theta} }\right)}\right\vert
+ | c = $e^{-ar} < 1$ for positive $a, r$
+}}
+{{eqn | o = \le
+ | r = \frac \pi a \max_{0 \le \theta \le \pi} \left\vert{g\left( {r e^{i \theta} }\right)}\right\vert
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Lobachevsky Integral Formula}
+Tags: Integral Calculus, Definite Integrals involving Sine Function
+
+\begin{theorem}
+Let $f$ be a [[Definition:Continuous Real Function|continuous function]], [[Definition:Periodic Function|periodic]] in $\pi$.
+Then:
+:$\displaystyle \int_0^\infty \frac {\sin x} x f \left({x}\right) \rd x = \int_0^{\frac \pi 2} f \left({x}\right) \rd x$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+{{Namedfor|Nikolai Ivanovich Lobachevsky|cat = Lobachevsky}}
+[[Category:Integral Calculus]]
+[[Category:Definite Integrals involving Sine Function]]
+46bfrzsei4xhrzcxvdiel6v9e49cld1
+\end{proof}<|endoftext|>
+\section{Dirichlet Series of Convolution of Arithmetic Functions}
+Tags: Dirichlet Series
+
+\begin{theorem}
+Let $f, g: \N \to \C$ be [[Definition:Arithmetic Function|arithmetic functions]].
+Let $h = f * g$ be their [[Definition:Dirichlet Convolution|Dirichlet convolution]].
+Let $F, G, H$ be their [[Definition:Dirichlet Series|Dirichlet series]].
+Let $s$ be a [[Definition:Complex Number|complex number]] such that $\map F s$ and $\map G s$ [[Definition:Absolutely Convergent Series|converge absolutely]].
+Then $\map H s$ [[Definition:Absolutely Convergent Series|converges absolutely]] to $\map F s \times \map G s$.
+\end{theorem}<|endoftext|>
+\section{Upper Bound for Abscissa of Absolute Convergence of Product of Dirichlet Series}
+Tags: Dirichlet Series
+
+\begin{theorem}
+Let $f, g: \N \to \C$ be [[Definition:Arithmetic Function|arithmetic functions]] with [[Definition:Dirichlet Convolution|Dirichlet convolution]] $h = f * g$.
+Let $F, G, H$ be their [[Definition:Dirichlet Series|Dirichlet series]].
+Let $\sigma_f, \sigma_g, \sigma_h$ be their [[Definition:Abscissa of Absolute Convergence|abscissae of absolute convergence]].
+Then:
+:$\sigma_h \le \max \set {\sigma_f, \sigma_g}$
+\end{theorem}
+
+\begin{proof}
+Follows from [[Dirichlet Series of Convolution of Arithmetic Functions]]
+{{ProofWanted}}
+[[Category:Dirichlet Series]]
+e03r49gdkg8ry56gqsgk754xuk3dgql
+\end{proof}<|endoftext|>
+\section{Riemann Zeta Function in terms of Dirichlet Eta Function}
+Tags: Riemann Zeta Function, Dirichlet Eta Function
+
+\begin{theorem}
+Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
+Let $\eta$ be the [[Definition:Dirichlet Eta Function|Dirichlet eta function]].
+Let $s \in \C$ be a [[Definition:Complex Number|complex number]] with [[Definition:Real Part|real part]] $\sigma > 1$.
+Then $\zeta(s) = \dfrac 1 {1 - 2^{1-s}} \eta \left({s}\right)$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn |l = \zeta\left(s\right) - \eta\left(s\right)
+ |r = \sum_{n \mathop = 1}^\infty \frac 1 {n^s} - \sum_{n\mathop = 1}^\infty \frac{(-1)^{n-1} }{n^s}
+ |c = {{Defof|Riemann Zeta Function}}, {{Defof|Dirichlet Eta Function}}
+}}
+{{eqn |r = \sum_{n \mathop = 1}^\infty \left( \frac 1 {n^s} +\frac{(-1)^n}{n^s} \right)
+ |c = [[Linearity of Series]]
+}}
+{{eqn |r = 2\sum_{n=1}^\infty \frac 1 {\left(2n\right)^s}
+}}
+{{eqn |r = 2^{1-s}\sum_{n=1}^\infty \frac 1 {n^s}
+}}
+{{eqn |r = 2^{1-s}\zeta\left(s\right)
+ |c = {{Defof|Riemann Zeta Function}}
+}}
+{{end-eqn}}
+Rearranging,
+{{begin-eqn}}
+{{eqn |l=\left(2^{1-s}-1\right)\zeta\left(s\right)
+ | r = -\eta\left(s\right)
+}}
+{{eqn |ll=\leadsto
+ |l =\zeta\left(s\right)
+ |r = \frac 1 {1-2^{1-s} } \eta\left(s\right)
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Functional Equation for Riemann Zeta Function}
+Tags: Riemann Zeta Function
+
+\begin{theorem}
+Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
+Let $\map \zeta s$ have an [[Definition:Analytic Continuation|analytic continuation]] for $\map \Re s > 0$.
+Then:
+:$\displaystyle \map \Gamma {\frac s 2} \pi^{-s/2} \map \zeta s = \map \Gamma {\frac {1 - s} 2} \pi^{\frac {s - 1} 2} \map \zeta {1 - s}$
+where $\Gamma$ is the [[Definition:Gamma Function|gamma function]]
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+Using [[Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function]]:
+The functional equation:
+:$\map \xi s = \map \xi {1 - s}$
+follows upon observing that this integral is invariant under $s \mapsto 1 - s$.
+{{ProofWanted|using the formula for $\zeta\cdot\Gamma$ and contour integration}}
+\end{proof}<|endoftext|>
+\section{Integral Representation of Riemann Zeta Function in terms of Fractional Part}
+Tags: Riemann Zeta Function
+
+\begin{theorem}
+Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
+Let $s\in\C$ be a [[Definition:Complex Number|complex number]] with [[Definition:Real Part|real part]] $\sigma>1$.
+Then
+:$\displaystyle \zeta \left({s}\right) = \frac s {s - 1} - s \int_1^\infty \left\{ {x}\right\} x^{-s - 1} \rd x$
+where $\left\{{x}\right\}$ denotes the [[Definition:Fractional Part|fractional part]] of $x$.
+\end{theorem}
+
+\begin{proof}
+We have:
+{{begin-eqn}}
+{{eqn | l = \sum_{n \mathop = 1}^\infty n^{-s}
+ | r = \sum_{n \mathop = 1}^\infty n \left({ n^{-s} - \left({n + 1}\right)^{-s} }\right)
+ | c = [[Abel's Lemma: Formulation 2]]
+}}
+{{eqn | r = s \sum_{n \mathop = 1}^\infty n \int_n^{n+1} x^{-s - 1} \rd x
+ | c =
+}}
+{{eqn | r = s \int_1^\infty \left\lfloor{x}\right\rfloor x^{-s - 1} \rd x
+ | c = where $\left\lfloor{x}\right\rfloor$ denotes the [[Definition:Floor Function|floor function]] of $x$
+}}
+{{eqn | l =
+ | r = \frac s {s - 1} - s \int_1^\infty \left\{ {x}\right\} x^{-s - 1} \rd x
+ | c = where $\left\{ {x}\right\}$ denotes the [[Definition:Fractional Part|fractional part]] of $x$
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Binomial Form of Relation between Riemann Zeta Function and Dirichlet Eta Function}
+Tags: Riemann Zeta Function
+
+\begin{theorem}
+Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
+Let $s\in\C$ be a [[Definition:Complex Number|complex number]] with [[Definition:Real Part|real part]] $\Re(s)>1$.
+Then:
+$\displaystyle \zeta \left({s}\right) = \frac 1 {1 - 2^{1-s} } \sum_{n \mathop = 0}^\infty \left({\frac 1 {2^{n+1} } \sum_{k \mathop = 0}^n \left({-1}\right)^k \binom n k \left({k + 1}\right)^{-s} }\right)$
+\end{theorem}
+
+\begin{proof}
+Use [[Riemann Zeta Function in terms of Dirichlet Eta Function]] and [[Binomial Theorem]].
+{{ProofWanted}}
+[[Category:Riemann Zeta Function]]
+9ik01re4fpmavul7jfabh52lrh1ne3z
+\end{proof}<|endoftext|>
+\section{Analytic Continuations of Riemann Zeta Function to Complex Plane}
+Tags: Riemann Zeta Function
+
+\begin{theorem}
+The [[Definition:Riemann Zeta Function|Riemann zeta function]] $\zeta$ has a [[Definition:Unique|unique]] [[Definition:Analytic Continuation|analytic continuation]] to $\C\setminus\{1\}$.
+\end{theorem}
+
+\begin{proof}
+Note that by [[Riemann Zeta Function is Analytic]], $\zeta(s)$ is indeed [[Definition:Analytic Complex Function|analytic]] for $\Re(s)>1$.
+By [[Complex Plane minus Point is Connected]], $\C\setminus\{1\}$ is [[Definition:Connected Subset of Complex Plane|connected]].
+By [[Uniqueness of Analytic Continuation]], there is at most one [[Definition:Analytic Continuation|analytic continuation]] of $\zeta$ to $\C\setminus\{1\}$.
+{{finish|link to proofs of various analytic continuations}}
+* [[Functional Equation for Riemann Zeta Function]], analyticity is shown at [[Analytic Continuation of Riemann Zeta Function]]
+* The stepwise method as in [[Riemann Zeta Function in terms of Dirichlet Eta Function]]
+\end{proof}<|endoftext|>
+\section{Analytic Continuations of Riemann Zeta Function to Right Half-Plane}
+Tags: Riemann Zeta Function
+
+\begin{theorem}
+The [[Definition:Riemann Zeta Function|Riemann zeta function]] has a [[Definition:Unique|unique]] [[Definition:Analytic Continuation|analytic continuation]] to $\{s \in \C : \Re(s) > 0\}\setminus\{1\}$, the [[Definition:Complex Half-Plane|half-plane]] $\Re(s)>0$ minus the point $s=1$.
+\end{theorem}
+
+\begin{proof}
+Note that by [[Riemann Zeta Function is Analytic]], $\zeta(s)$ is indeed [[Definition:Analytic Complex Function|analytic]] for $\Re(s)>1$.
+By [[Complex Half-Plane minus Point is Connected]], $\{\sigma>0\}\setminus\{1\}$ is [[Definition:Connected Subset of Complex Plane|connected]].
+By [[Uniqueness of Analytic Continuation]], there is at most one [[Definition:Analytic Continuation|analytic continuation]] of $\zeta$ to $\{\sigma>0\}\setminus\{1\}$.
+By either:
+* [[Analytic Continuation of Riemann Zeta Function using Dirichlet Eta Function]]
+* [[Analytic Continuation of Riemann Zeta Function using Mellin Transform of Fractional Part]]
+* [[Analytic Continuation of Riemann Zeta Function using Jacobi Theta Function]]
+there exists one.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Functional Equation for Completed Riemann Zeta Function}
+Tags: Riemann Zeta Function
+
+\begin{theorem}
+Let $\xi : \C \to \C$ be the [[Definition:Completed Riemann Zeta Function|completed Riemann zeta function]].
+Let $s\in \C$ be a [[Definition:Complex Number|complex number]].
+Then:
+:$\map \xi s = \map \xi {1 - s}$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Use [[Functional Equation for Riemann Zeta Function]].}}
+[[Category:Riemann Zeta Function]]
+0wb6hf9c3s8vwlyid6qewmaur6ei6zt
+\end{proof}<|endoftext|>
+\section{Dirichlet Series of Inverse of Arithmetic Function}
+Tags: Dirichlet Series
+
+\begin{theorem}
+Let $f : \N \to\C$ be an [[Definition:Arithmetic Function|arithmetic function]].
+Let $g : \N \to \C$ be an [[Definition:Dirichlet Inverse of Arithmetic Function|Dirichlet inverse]] of $f$.
+Let $F, G$ be their [[Definition:Dirichlet Series|Dirichlet series]].
+Let $s \in \C$ such that both $F(s)$ and $G(s)$ [[Definition:Absolutely Convergent Series|converge absolutely]].
+Then $F(s) \cdot G(s) = 1$.
+\end{theorem}
+
+\begin{proof}
+Let $\varepsilon$ be the [[Definition:Identity Arithmetic Function|identity arithmetic function]].
+By [[Dirichlet Series of Identity Arithmetic Function]], $\varepsilon$ has [[Definition:Dirichlet Series|Dirichlet series]] $E(s) = 1$.
+By [[Dirichlet Series of Convolution of Arithmetic Functions]], $F(s)G(s) = 1$.
+{{qed}}
+[[Category:Dirichlet Series]]
+b8l8wup13ftuuhffklh4wu1yb17bdtn
+\end{proof}<|endoftext|>
+\section{Invertibility of Arithmetic Functions}
+Tags: Arithmetic Functions
+
+\begin{theorem}
+Let $f: \N \to \C$ be an [[Definition:Arithmetic Function|arithmetic function]].
+Then $f$ has a [[Definition:Dirichlet Inverse of Arithmetic Function|Dirichlet inverse]] {{iff}}:
+:$\map f 1 \ne 0$
+\end{theorem}
+
+\begin{proof}
+Let $*$ denote [[Definition:Dirichlet Convolution|Dirichlet convolution]].
+Let $\varepsilon$ denote the [[Definition:Identity Arithmetic Function|identity arithmetic function]].
+=== Sufficient Condition ===
+Let $f$ have a [[Definition:Dirichlet Inverse of Arithmetic Function|Dirichlet inverse]] $g$:
+:$f * g = \varepsilon$
+Then:
+{{begin-eqn}}
+{{eqn | l = 1
+ | r = \map \varepsilon 1
+ | c = {{Defof|Identity Arithmetic Function}}
+}}
+{{eqn | r = \map {\paren {f * g} } 1
+}}
+{{eqn | r = \sum_{d \mathop \divides 1} \map f d \, \map g {d^{-1} }
+ | c = {{Defof|Dirichlet Convolution}}
+}}
+{{eqn | r = \map f 1 \, \map g 1
+ | c = [[Divisors of One]]
+}}
+{{end-eqn}}
+Thus $\map f 1 \ne 0$.
+{{qed|lemma}}
+=== Necessary Condition ===
+Suppose that $\map f 1 \ne 0$.
+We want to find an [[Definition:Arithmetic Function|arithmetic function]] $g$ such that:
+:$(1): \quad \map {\paren {f * g} } 1 = 1$
+:$(2): \quad \map {\paren {f * g} } n = 0$ for all $n > 1$
+We have:
+:$\map {\paren {f * g} } 1 = \map f 1 \, \map g 1$
+So we have no choice but to define:
+:$\map g 1 = \paren {\map f 1}^{-1}$
+and condition $(1)$ is satisfied.
+Condition $(2)$ can be written as:
+{{begin-eqn}}
+{{eqn | l = 0
+ | r = \sum_{d \mathop \divides n} \map f d \, \map g {\frac n d}
+}}
+{{eqn | r = \map f 1 \, \map g n + \sum_{\substack {d \mathop \divides n \\ d \mathop > 1} } \map f d \, \map g {\frac n d}
+}}
+{{end-eqn}}
+That is:
+:$\map g n = -\dfrac 1 {\map f 1} \displaystyle \sum_{\substack {d \mathop \divides n \\ d \mathop > 1} } \map f d \, \map g {\frac n d}$
+We can [[Definition:Recursive Definition|recursively define]] $g$ by this formula, starting with $\map g 1$ as above.
+By definition, $g$ then satisfies:
+:$g * f = \varepsilon$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Consistent Set of Formulas}
+Tags: Formal Systems, Propositional Logic
+
+\begin{theorem}
+{{TFAE|def = Consistent (Logic)/Proof System/Propositional Logic|view = Consistent Proof System for Propositional Logic}}
+Let $\LL_0$ be the [[Definition:Language of Propositional Logic|language of propositional logic]].
+Let $\mathscr P$ be a [[Definition:Proof System|proof system]] for $\LL_0$.
+Let $\FF$ be a collection of [[Definition:Logical Formula|logical formulas]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+[[Category:Formal Systems]]
+[[Category:Propositional Logic]]
+9t09z7qppb0chm01ypqc1qfkngt4pzz
+\end{proof}<|endoftext|>
+\section{Analytic Continuation of Riemann Zeta Function using Mellin Transform of Fractional Part}
+Tags: Riemann Zeta Function
+
+\begin{theorem}
+Let $\zeta$ denote the [[Definition:Riemann Zeta Function|Riemann zeta function]].
+The [[Definition:Analytic Continuation|analytic continuation]] of $\zeta$ to the [[Definition:Complex Halfplane|half-plane]] $\map \Re s > 0$ is given by:
+:$\displaystyle \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x$
+where $x^{-s - 1}$ takes the principle value $e^{-\map \ln x \paren {s + 1} }$
+\end{theorem}
+
+\begin{proof}
+Let $s = \sigma + i t$.
+By [[Integral Representation of Riemann Zeta Function in terms of Fractional Part]], the above integral coincides with $\map \zeta s$ for $\sigma > 1$.
+We show that it is [[Definition:Analytic Complex Function|analytic]] for $0 \le \sigma \le 1$.
+For $n \ge 1$, let:
+{{begin-eqn}}
+{{eqn | l = \cmod {a_n}
+ | r = \cmod {\int_n^{n + 1} \fractpart x x^{-s - 1} \rd x}
+ | c =
+}}
+{{eqn | r = \cmod s \int_n^{n + 1} \cmod {\fractpart x x^{-s - 1} } \rd x
+ | o = \le
+ | c = [[Modulus of Complex Integral]]
+}}
+{{eqn | r = \cmod s \int_n^{n + 1} \fractpart x x^{-\sigma - 1} \rd x
+}}
+{{eqn | o = \le
+ | r = \cmod s \int_n^{n + 1} x^{-\sigma-1} \rd x
+ | c = as $\fractpart x \le 1$
+}}
+{{eqn | r = -\intlimits {\dfrac {\cmod s} \sigma x^{-\sigma} } n {n + 1}
+ | c =
+}}
+{{eqn | r = \dfrac {\cmod s} \sigma \paren {\frac 1 {n^\sigma} - \frac 1 {\paren {n + 1}^\sigma} }
+ | c =
+}}
+{{eqn | r = \dfrac {\cmod s} \sigma \paren {\frac {\paren {n + 1}^\sigma - n^\sigma} {n^\sigma \paren {n + 1}^\sigma} }
+ | c =
+}}
+{{end-eqn}}
+By the [[Mean Value Theorem]], for some $n \le \theta \le n + 1$:
+:$\paren {n + 1}^\sigma - n^\sigma = \sigma \theta^{\sigma - 1}$
+and for $\sigma \le 1$ we have:
+:$\sigma \theta^{\sigma - 1} \le \sigma \paren {n + 1}^{\sigma - 1}$
+Thus we have:
+{{begin-eqn}}
+{{eqn | l = \cmod {a_n}
+ | o = \le
+ | r = \dfrac {\cmod s} \sigma \paren {\frac {\paren {n + 1}^\sigma - n^\sigma} {n^\sigma \paren {n + 1}^\sigma} }
+}}
+{{eqn | o = \le
+ | r = \dfrac {\cmod s} \sigma \paren {\frac {\sigma \paren {n + 1}^{\sigma - 1} } {n^\sigma \paren {n + 1}^\sigma} }
+}}
+{{eqn | r = \cmod s \frac 1 {n^\sigma \paren {n + 1} }
+}}
+{{eqn | o = \le
+ | r = \frac {\cmod s} {n^{\sigma + 1} }
+}}
+{{end-eqn}}
+Thus on any compact subset of the half-plane $\sigma > 0$ which is bounded away from $0$, we have that the series:
+:$\displaystyle \frac s {s - 1} - \sum_{n \mathop = 1}^\infty a_n = \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x$
+converges [[Definition:Uniform Convergence|uniformly]] as:
+{{begin-eqn}}
+{{eqn | l = \cmod {\sum_{n \mathop = M}^\infty a_n}
+ | o = \le
+ | r = \sum_{n \mathop = M}^\infty \cmod {a_n}
+}}
+{{eqn | o = \le
+ | r = \sum_{n \mathop = M}^\infty \frac {\cmod s} {n^{\sigma + 1} }
+}}
+{{end-eqn}}
+Since it is a compact region, $\cmod s$ is bounded by some constant $C$.
+So:
+:$\cmod {\displaystyle \sum_{n \mathop = M}^\infty a_n} \le C \sum_{n \mathop = M}^\infty \frac 1 {n^{\sigma + 1} }$
+As Dirichlet series are uniformly convergent on compact regions bounded away from their abscissa of convergence, we may find an $M$ large enough so that the above is smaller than $\epsilon$ for all $\sigma$.
+Therefore the series is [[Definition:Uniform Convergence|uniformly]] convergent.
+Thus by [[Uniform Limit of Analytic Functions is Analytic]] the series defines an analytic function on the strip $0 < \sigma \le 1$ and $s \ne 1$.
+This is also equal to $\zeta$ on the halfplane $\sigma>1$.
+Thus the series defines the unique [[Definition:Analytic Continuation|analytic continuation]] of $\zeta$ onto the halfplane $\sigma > 0$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Analytic Continuation of Riemann Zeta Function using Dirichlet Eta Function}
+Tags: Riemann Zeta Function, Dirichlet Eta Function
+
+\begin{theorem}
+Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
+Let $\eta$ be the [[Definition:Dirichlet Eta Function|Dirichlet Eta Function]].
+Then:
+:$\dfrac 1 {1 - 2^{1 - s} } \map \eta s$
+defines an [[Definition:Analytic Continuation|analytic continuation]] of $\zeta$ to the [[Definition:Complex Halfplane|half-plane]] $\map \Re s > 0$ minus $s = 1$.
+\end{theorem}
+
+\begin{proof}
+By [[Riemann Zeta Function in terms of Dirichlet Eta Function]], it coincides with $\zeta$ for $\map \Re s > 1$.
+By [[Dirichlet Eta Function is Analytic]], it is [[Definition:Analytic Complex Function|analytic]] for $\map \Re s > 0$, except at $s = 1$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Analytic Continuation of Riemann Zeta Function using Jacobi Theta Function}
+Tags: Riemann Zeta Function
+
+\begin{theorem}
+Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
+Then
+:$\displaystyle\frac{\pi^{s/2}}{\Gamma \left({\frac s 2}\right)} \cdot\left( - \frac 1 {s \left({1 - s}\right)} + \int_1^\infty \left({x^{s / 2 - 1} + x^{- \left({s + 1}\right) / 2} }\right) \omega \left({x}\right) \ \mathrm d x \right)$
+defines an [[Definition:Analytic Continuation|analytic continuation]] of $\zeta$ to the [[Definition:Complex Halfplane|half-plane]] $\Re(s)>0$ minus $s=1$.
+\end{theorem}
+
+\begin{proof}
+By [[Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function]], it coincides with $\zeta(s)$ for $\Re(s)>1$.
+Interchanging integral and derivative, one shows that the integral is [[Definition:Analytic Complex Function|analytic]] for $\Re(s)>0$.
+{{finish}}
+\end{proof}<|endoftext|>
+\section{Complex Modulus of Real Number equals Absolute Value}
+Tags: Complex Modulus, Absolute Value Function
+
+\begin{theorem}
+Let $x \in \R$ be a [[Definition:Real Number|real number]].
+Then the [[Definition:Complex Modulus|complex modulus]] of $x$ equals the [[Definition:Absolute Value of Real Number|absolute value]] of $x$.
+\end{theorem}
+
+\begin{proof}
+Let $x = x + 0 i \in \R$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \cmod {x + 0 i}
+ | r = \sqrt {x^2 + 0^2}
+ | c = {{Defof|Complex Modulus}}
+}}
+{{eqn | r = \sqrt {x^2}
+ | c =
+}}
+{{eqn | r = \size x
+ | c = {{Defof|Absolute Value|index = 2}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Ceiling Function}
+Tags: Ceiling Function
+
+\begin{theorem}
+Let $x$ be a [[Definition:Real Number|real number]].
+{{TFAE|def = Ceiling Function}}
+\end{theorem}
+
+\begin{proof}
+=== Definition 1 equals Definition 2 ===
+Follows from [[Infimum of Set of Integers equals Smallest Element]].
+{{qed|lemma}}
+=== Definition 1 equals Definition 3 ===
+Let $S$ be the [[Definition:Set|set]]:
+:$S = \left\{ {m \in \Z: m \ge x}\right\}$
+Let $n = \inf S$.
+By [[Infimum of Set of Integers is Integer]], $n \in \Z$.
+By [[Infimum of Set of Integers equals Smallest Element]], $n\in S$.
+Because $n\in S$, we have $n \geq x$.
+Because $n-1 < n$, we have by definition of [[Definition:Infimum of Subset of Real Numbers|supremum]]:
+:$n-1 \notin S$
+Thus $n-1 < x$.
+Thus $n$ is an [[Definition:Integer|integer]] such that:
+:$n-1 < x \leq n$
+So $n$ is the [[Definition:Ceiling Function/Definition 3|ceiling function by definition 3]].
+{{qed|lemma}}
+=== Definition 3 equals Definition 2 ===
+Let $n$ be an [[Definition:Integer|integer]] such that:
+:$n-1 < x \leq n$
+We show that $n$ is the [[Definition:Smallest Element|smallest element]] of the [[Definition:Set|set]]:
+:$S = \left\{ {m \in \Z: m \ge x}\right\}$
+Let $m \in \Z$ such that $m \ge x$.
+We show that $n\leq m$.
+{{AimForCont}} $m < n$.
+By [[Weak Inequality of Integers iff Strict Inequality with Integer minus One]]:
+:$m \le n - 1$
+and so from the definition of $g$ it follows that $m < x$.
+By [[Proof by Contradiction]] it follows that $m \ge n$.
+Because $m \in S$ was arbitrary, $n$ is the [[Definition:Smallest Element|smallest element]] of $S$.
+Thus $n$ is the [[Definition:Ceiling Function/Definition 2|ceiling function by definition 2]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Characterizing Property of Infimum of Subset of Real Numbers}
+Tags: Infima, Real Numbers
+
+\begin{theorem}
+Let $S \subset \R$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Numbers|real numbers]].
+Let $S$ be [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
+Let $\alpha \in \R$.
+{{TFAE}}
+:$(1): \quad \alpha$ is the [[Definition:Infimum of Subset of Real Numbers|infimum]] of $S$
+:$(2): \quad \alpha$ is a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] for $S$
+::and:
+:::$\forall \epsilon \in \R_{> 0}$ there exists $x \in S$ with $x < \alpha + \epsilon$
+\end{theorem}
+
+\begin{proof}
+=== $(1)$ implies $(2)$ ===
+Let $\alpha$ be the [[Definition:Infimum of Subset of Real Numbers|infimum]] of $S$.
+Then by definition, $\alpha$ is a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] for $S$.
+Let $\epsilon>0$.
+Because $\alpha+\epsilon>\alpha$, it is not a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] for $S$.
+Thus there exists $x\in S$ with $x < \alpha + \epsilon$.
+{{qed|lemma}}
+=== $(2)$ implies $(1)$ ===
+Let $\alpha$ be a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] of $S$ such that $\forall \epsilon > 0$ there exists $x \in S$ with $x < \alpha + \epsilon$.
+Let $d \in \R$ be a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] of $S$.
+We have to prove that $d \le \alpha$.
+{{AimForCont}} $d > \alpha$.
+Let $\epsilon = d - \alpha > 0$.
+Then there exists $x \in S$ such that $x < \alpha + \left({d - \alpha}\right) = d$.
+But this [[Definition:Contradiction|contradicts]] our assumption that $d$ is a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] of $S$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Characterizing Property of Supremum of Subset of Real Numbers}
+Tags: Suprema, Real Numbers
+
+\begin{theorem}
+Let $S \subset \R$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Numbers|real numbers]].
+Let $S$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
+Let $\omega \in \R$.
+{{TFAE}}
+:$(1): \quad \omega$ is the [[Definition:Supremum of Subset of Real Numbers|supremum]] of $S$
+:$(2): \quad \omega$ is an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]] for $S$
+::and:
+:::$\forall \epsilon \in \R_{> 0}$ there exists $x \in S$ with $x > \omega - \epsilon$
+\end{theorem}
+
+\begin{proof}
+=== $(1)$ implies $(2)$ ===
+Let $\omega$ be the [[Definition:Supremum of Subset of Real Numbers|supremum]] of $S$.
+Then by definition, $\omega$ is an [[Definition:upper Bound of Subset of Real Numbers|upper bound]] for $S$.
+Let $\epsilon > 0$.
+Because $\omega - \epsilon < \omega$, it is not an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]] for $S$.
+Thus there exists $x\in S$ with $x > \omega - \epsilon$.
+{{qed|lemma}}
+=== $(2)$ implies $(1)$ ===
+Let $\omega$ be an [[Definition:upper Bound of Subset of Real Numbers|upper bound]] of $S$ such that $\forall \epsilon > 0$ there exists $x \in S$ with $x > \omega - \epsilon$.
+Let $d \in \R$ be an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]] of $S$.
+We have to prove that $d \ge \omega$.
+{{AimForCont}} $d < \omega$.
+Let $\epsilon = \omega - d > 0$.
+Then there exists $x \in S$ such that $x > \omega - \left({\omega - d}\right) = d$.
+But this [[Definition:Contradiction|contradicts]] our assumption that $d$ is an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]] of $S$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Limit of Positive Real Sequence is Positive}
+Tags: Convergence
+
+\begin{theorem}
+Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Positive Real Number|positive real numbers]].
+Let $x_n$ [[Definition:Convergent Real Sequence|converge]] to $L$.
+Then $L \ge 0$.
+\end{theorem}
+
+\begin{proof}
+{{AimForCont}} $L < 0$.
+Then for any $n \in \N$:
+{{begin-eqn}}
+{{eqn | l = \size {x_n - L}
+ | r = x_n - L
+ | c = $x_n \ge 0 > L$
+}}
+{{eqn | o = \ge
+ | r = -L
+ | c = $> 0$
+}}
+{{end-eqn}}
+This contradicts {{Defof|Convergent Real Sequence}}.
+Hence we must have $L \ge 0$.
+{{qed}}
+[[Category:Convergence]]
+pxp5gu8p4oqesdba1ntolkfn6lqw4fz
+\end{proof}<|endoftext|>
+\section{Real Sequence with Nonzero Limit is Eventually Nonzero}
+Tags: Convergence
+
+\begin{theorem}
+Let $\sequence {x_n}$ be a [[Definition:Real Sequence|real sequence]].
+Let $\sequence {x_n}$ [[Definition:Convergent Real Sequence|converge]] to $a \ne 0$.
+Then:
+:$\exists N \in \N: \forall n \ge N: x_n \ne 0$
+That is, eventually every term of $\sequence {x_n}$ becomes non-zero.
+\end{theorem}
+
+\begin{proof}
+Suppose $a > 0$.
+By [[Sequence Converges to Within Half Limit]]:
+:$\exists N \in \N: \forall n > N: x_n > \dfrac a 2 > 0$
+Now suppose $a < 0$.
+By [[Sequence Converges to Within Half Limit]]:
+:$\exists N \in \N: \forall n > N: x_n < \dfrac a 2 < 0$
+This shows that if $a \ne 0$:
+:$\exists N \in \N: \forall n > N: x_n \ne 0$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Mittag-Leffler Expansion for Cotangent Function}
+Tags: Cotangent Function, Mittag-Leffler Expansions, Mittag-Leffler Expansion for Cotangent Function
+
+\begin{theorem}
+:$\displaystyle \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
+where:
+:$z \in \C$ is not an [[Definition:Integer|integer]]
+:$\cot$ is the [[Definition:Complex Cotangent Function|cotangent function]].
+\end{theorem}
+
+\begin{proof}
+Let $\map \zeta s$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
+Let $\displaystyle \map g z = \sum_{n \mathop = 1}^\infty z^n \map \zeta {2 n}$ be the [[Definition:Generating Function| generating function]] of $\map \zeta {2 n}$
+By [[Power Series Expansion for Cotangent Function]], for $\size z < 1$:
+{{begin-eqn}}
+{{eqn | l = \pi \map \cot {\pi z}
+ | r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \pi^{2 n} 2^{2 n} B_{2 n} \, z^{2 n - 1} } {\paren {2 n}!}
+}}
+{{eqn | ll= \leadsto
+ | l = \pi z \map \cot {\pi z}
+ | r = 1 + \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \pi^{2 n} 2^{2 n} B_{2 n} \, z^{2 n} } {\paren {2 n}!}
+}}
+{{eqn | ll= \leadsto
+ | l = \frac {\pi z \map \cot {\pi z} - 1} {-2}
+ | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} \pi^{2 n} 2^{2 n - 1} B_{2 n} \, z^{2 n} } {\paren {2 n}!}
+}}
+{{end-eqn}}
+By [[Riemann Zeta Function at Even Integers]]:
+:$\map \zeta {2 n} = \dfrac {\paren {-1}^{n + 1} \pi^{2 n} 2^{2 n - 1} B_{2 n} } {\paren {2 n}!}$
+Thus:
+{{begin-eqn}}
+{{eqn | l = \dfrac {\pi z \map \cot {\pi z} - 1} {-2}
+ | r = \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n}
+}}
+{{eqn | r = \map g {z^2}
+}}
+{{end-eqn}}
+By [[Analytic Continuation of Generating Function of Dirichlet Series]] and [[Uniqueness of Analytic Continuation]]:
+:$\displaystyle \dfrac {\pi z \map \cot {\pi z} - 1} {-2} = \sum_{n \mathop = 1}^\infty \dfrac {z^2} {n^2 - z^2}$
+for all of $\C$, as this is the overlap of their domains.
+Thus:
+{{begin-eqn}}
+{{eqn | l = \pi z \map \cot {\pi z} - 1
+ | r = 2 \sum_{n \mathop = 1}^\infty \frac {z^2} {z^2 - n^2}
+}}
+{{eqn | ll= \leadsto
+ | l = \pi z \map \cot {\pi z}
+ | r = 1 + 2 \sum_{n \mathop = 1}^\infty \frac {z^2} {z^2 - n^2}
+}}
+{{eqn | ll= \leadsto
+ | l = \pi \map \cot {\pi z}
+ | r = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}
+
+\begin{proof}
+Let $\mathcal L$ denote the [[Definition:Logarithmic Derivative of Meromorphic Function|logarithmic derivative]].
+On the [[Definition:Open Subset of Complex Plane|open set]] $\C \setminus \Z$ we have:
+{{begin-eqn}}
+{{eqn | l = \pi \cot \pi z
+ | r = \mathcal L \left({\sin \left({\pi z}\right)}\right)
+ | c = [[Primitive of Cotangent Function]], or a complex version thereof
+}}
+{{eqn | r = \mathcal L \left({\pi z \prod_{n \mathop = 1}^\infty \left({1 - \frac {z^2} {n^2} }\right)}\right)
+ | c = [[Euler Formula for Sine Function]]
+}}
+{{eqn | r = \mathcal L \left({\pi z}\right) + \sum_{n \mathop = 1}^\infty \mathcal L \left({1 - \frac {z^2} {n^2} }\right)
+ | c = [[Logarithmic Derivative of Infinite Product of Analytic Functions]]
+}}
+{{eqn | r = \frac \pi {\pi z} + \sum_{n \mathop = 1}^\infty \frac 1 {1 - \frac {z^2} {n^2} } \cdot \frac {\mathrm d} {\mathrm d z} \left({1 - \frac {z^2} {n^2} }\right)
+ | c = {{Defof|Logarithmic Derivative of Meromorphic Function}}
+}}
+{{eqn | r = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \frac z {n^2 - z^2}
+ | c = [[Derivative of Power]]
+}}
+{{eqn | r = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Laurent Series Expansion for Cotangent Function}
+Tags: Laurent Series Expansion for Cotangent Function, Laurent Series Expansions, Cotangent Function
+
+\begin{theorem}
+:$\displaystyle \pi \cot \pi z = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}$
+where:
+:$z \in \C$ such that $\cmod z < 1$
+:$\zeta$ is the [[Definition:Riemann Zeta Function|Riemann Zeta function]].
+\end{theorem}
+
+\begin{proof}
+From [[Mittag-Leffler Expansion for Cotangent Function]]:
+:$\displaystyle \pi \cot \pi z = \frac 1 z + 2 \sum_{k \mathop = 1}^\infty \frac z {z^2 - k^2}$
+Factoring $-\dfrac 1 {k^2}$:
+:$\displaystyle \pi \cot \pi z = \frac 1 z - 2 \sum_{k \mathop = 1}^\infty \frac z {k^2} \cdot \frac 1 {1 - \frac {z^2} {k^2} }$
+Taking $\cmod z < 1$, and noting that $k \ge 1$, we have, by [[Sum of Infinite Geometric Sequence]]:
+:$\displaystyle \pi \cot \pi z = \frac 1 z - 2 \sum_{k \mathop = 1}^\infty \frac z {k^2} \cdot \sum_{n \mathop = 1}^\infty \paren {\frac {z^2} {k^2} }^{n - 1}$
+from which:
+{{begin-eqn}}
+{{eqn | l = \pi \cot \pi z
+ | r = \frac 1 z - 2 \sum_{k \mathop = 1}^\infty \sum_{n \mathop = 1}^\infty \frac {z^{2 n - 2} \cdot z} {k^{2 n - 2} \cdot k^2}
+}}
+{{eqn | r = \frac 1 z - 2 \sum_{k \mathop = 1}^\infty \sum_{n \mathop = 1}^\infty \frac 1 {k^{2 n} } \cdot z^{2 n - 1}
+}}
+{{eqn | r = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}
+ | c = {{Defof|Riemann Zeta Function}}
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Laurent Series Expansion for Cotangent Function]]
+[[Category:Laurent Series Expansions]]
+[[Category:Cotangent Function]]
+tmigxwb63elg6ab0xcb4011tt72zbp1
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Removable Discontinuity of Real Function}
+Tags: Continuity
+
+\begin{theorem}
+Let $A \subseteq \R$ be a [[Definition:Subset|subset]] of the [[Definition:Real Numbers|real numbers]].
+Let $f : A \to \R$ be a [[definition:Real Function|real function]].
+Let $f$ be [[Definition:Discontinuous Point of Real Function|discontinuous]] at $a\in A$.
+{{TFAE|def = Removable Discontinuity of Real Function|view = removable discontinuity}}
+\end{theorem}<|endoftext|>
+\section{Equivalence of Definitions of Polynomial Function on Subset of Ring}
+Tags: Polynomial Theory
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
+Let $S \subset R$ be a [[Definition:Subset|subset]].
+{{TFAE|def = Polynomial Function/Ring |view = polynomial function}}
+\end{theorem}
+
+\begin{proof}
+=== 1 implies 2 ===
+Let $\map P X \in R \sqbrk X$ be the [[Definition:Polynomial|polynomial]]:
+:$P = \ds \sum_{k \mathop = 0}^n a_k \cdot X^k$
+where $\sum$ denotes [[Definition:Indexed Summation|indexed summation]].
+We show that $\map P \iota = f$.
+{{finish}}
+=== 2 implies 1 ===
+Let $P \in R \sqbrk X$ be a [[Definition:Polynomial|polynomial]] and $f = \map P \iota \in R^S$.
+By [[Polynomial is Linear Combination of Monomials]], there exist:
+:$n \in \N$
+:$a_0, \ldots, a_n \in R$
+such that:
+:$P = \ds \sum_{k \mathop = 0}^n a_k \cdot X^k$
+where $\sum$ denotes [[Definition:Indexed Summation|indexed summation]].
+Let $\operatorname{ev}_{\iota}$ denote the [[Definition:Polynomial Evaluation Homomorphism|evaluation homomorphism]] at $\iota$.
+Then $\map {\operatorname{ev}_{\iota} } P = f$.
+We have:
+{{begin-eqn}}
+{{eqn | l = f
+ | r = \map {\operatorname{ev}_{\iota} } {\sum_{k \mathop = 0}^n a_k \cdot X^k}
+}}
+{{eqn | r = \sum_{k \mathop = 0}^n \map {\operatorname{ev}_{\iota} } {a_k \cdot X^k}
+ | c = [[Ring Homomorphism Preserves Indexed Summations]]
+}}
+{{eqn | r = \sum_{k \mathop = 0}^n a_k \cdot \map {\operatorname{ev}_{\iota} } {X^k}
+ | c = {{Defof|Polynomial Evaluation Homomorphism}}
+}}
+{{eqn | r = \sum_{k \mathop = 0}^n a_k \cdot \map {\operatorname{ev}_{\iota} } {X^k }
+ | c = {{Defof|Polynomial Evaluation Homomorphism}}
+}}
+{{end-eqn}}
+{{finish|but move the calculation to [[Explicit Formula for Polynomial Evaluation Homomorphism]]}}
+[[Category:Polynomial Theory]]
+a7c8wt4tngw3mr8y1r3e43mx6dcmw9e
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Polynomial Ring in Multiple Variables}
+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/Monoid Ring on Free Monoid on Set|Definition 1: As the monoid ring on a free monoid on a set]] ===
+{{:Definition:Polynomial Ring/Monoid Ring on Free Monoid on Set}}
+{{ExpandList}}
+\end{theorem}<|endoftext|>
+\section{Monomials form Basis of Polynomial Ring/One Variable}
+Tags: Polynomial Rings, Monomials
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+Let $R \sqbrk X$ be a [[Definition:Polynomial Ring|polynomial ring]] over $R$ in the [[Definition:Variable of Polynomial Ring|variable]] $X$.
+Then the [[Definition:Monomial of Polynomial Ring|monomials]] of $R \sqbrk X$ are a [[Definition:Basis of Module|basis]] of $R \sqbrk X$ [[Definition:Module Structure of Polynomial Ring|as a module]] over $R$.
+\end{theorem}
+
+\begin{proof}
+Follows from:
+* [[Polynomial is Linear Combination of Monomials]]
+* [[Monomials of Polynomial Ring are Linearly Independent]]
+{{qed}}
+[[Category:Polynomial Rings]]
+[[Category:Monomials]]
+ibj1ycvcvta6451o5b2hkeptfdnrhkb
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Polynomial in Ring Element}
+Tags: Polynomial Theory
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
+Let $S$ be a [[Definition:Subring|subring]] [[Definition:Ring with Unity|with unity]] of $R$.
+Let $x\in R$.
+{{TFAE|def = Polynomial in Ring Element}}
+\end{theorem}
+
+\begin{proof}
+{{proof wanted}}
+[[Category:Polynomial Theory]]
+imnf88w7v83rtn8p2kas888887ipv94
+\end{proof}<|endoftext|>
+\section{Union of Blocks is Set of Points}
+Tags: Design Theory
+
+\begin{theorem}
+Let $\struct {X, \mathcal B}$ be a [[Definition:Pairwise Balanced Design|pairwise balanced design]].
+That is, let $\struct {X, \mathcal B}$ be a [[Definition:Design|design]], with $\size X \ge 2$, and the number of occurrences of each pair of distinct points in $\mathcal B$ be $\lambda$ for some $\lambda > 0$ [[Definition:Constant|constant]].
+Then the [[Definition:Set Union|set union]] of all the [[Definition:Subset|subset]] [[Definition:Element|elements]] in $\mathcal B$ is precisely $X$.
+\end{theorem}
+
+\begin{proof}
+Let $X = \set {x_1, x_2, \ldots, x_v}$.
+Let $\mathcal B = \multiset {y_1, y_2,\ldots, y_b}$, where the notation denotes a [[Definition:Multiset|multiset]].
+Let $Y = \displaystyle \bigcup_{i \mathop = 1}^b y_i$.
+We shall show that $Y \subseteq X$ and $X \subseteq Y$.
+=== $Y \subseteq X$: ===
+By definition, $\mathcal B$ is a [[Definition:Multiset|multiset]] of [[Definition:Subset|subsets]] of $X$.
+This means that each $y_i \in Y$ is a [[Definition:Subset|subset]] of $X$ for $i \in \closedint 1 b$.
+By [[Union of Subsets is Subset/Family of Sets|Union of Subsets is Subset]], $Y$ itself is a [[Definition:Subset|subset]] of $\mathcal B$.
+{{qed|lemma}}
+=== $X \subseteq Y$: ===
+Let $x_i \in X$ be arbitrary.
+Choose any $x_j \in X$ such that $i \ne j$.
+Such an $x_j$ necessarily exists because [[Definition:By Hypothesis|by hypothesis]] $\card X \ge 2$.
+Then $\set {x_i, x_j} \subseteq Y$ by the definition of [[Definition:Block|blocks]], as $\lambda > 0$ [[Definition:By Hypothesis|by hypothesis]].
+But by the definition of [[Definition:Subset|subset]], this implies that $x_i$ is an element in $Y$.
+Hence the result, as $x_i$ was arbitrary.
+{{qed}}
+=== Proof of Lemma ===
+By the definition of [[Definition: Balanced Incomplete Block Design|balanced incomplete block design]], $\struct {X, \mathcal B}$ is itself a type of [[Definition:Pairwise Balanced Design|pairwise balanced design]].
+Thus the lemma follows from the main result.
+{{qed}}
+[[Category:Design Theory]]
+tpi7f069keudxo9nh0u1i5giiqhl2bg
+\end{proof}<|endoftext|>
+\section{Equality of Monomials of Polynomial Ring in One Variable}
+Tags: Monomials
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+Let $R \sqbrk X$ be a [[Definition:Polynomial Ring in One Variable|polynomial ring in one variable]] $X$ over $R$.
+Let $k, l \in \N$ be [[Definition:Distinct|distinct]] [[Definition:Natural Number|natural numbers]].
+Then the [[Definition:Monomial of Polynomial Ring|mononomials]] $X^k$ and $X^l$ are [[Definition:Distinct|distinct]], where $X^k$ denotes the $k$th [[Definition:Power of Ring Element|power]] of $X$.
+\end{theorem}
+
+\begin{proof}
+By:
+* [[Uniqueness of Polynomial Ring in One Variable]]
+* [[Homomorphism Preserves Indexed Products]]
+we may assume $R \sqbrk X$ is the [[Definition:Ring of Sequences of Finite Support|ring of sequences of finite support]] over $R$, and $X$ is the [[Definition:Sequence|sequence]] $\sequence {0, 1, 0, 0 \ldots}$.
+One verifies that, for $k \ge 0$, $X^k$ is the [[Definition:Sequence|sequence]] with $\map {X^k} l = \delta_{k l}$, where $\delta$ is the [[Definition:Kronecker Delta|Kronecker delta]].
+{{theoremWanted|verify this somewhere else}}
+Thus $X^k \ne X^l$ if $k \ne l$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Ring Isomorphic to Polynomial Ring is Polynomial Ring/One Variable}
+Tags: Polynomial Rings
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+Let $R \sqbrk X$ be a [[Definition:Polynomial Ring in One Variable|polynomial ring in one variable]] $X$ over $R$.
+Let $\iota : R \to R \sqbrk X$ denote the [[Definition:Embedding into Polynomial Ring|canonical embedding]].
+Let $S$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]] and $f: R \sqbrk X \to S$ be a [[Definition:Ring Isomorphism|ring isomorphism]].
+Then $\struct {S, f \circ \iota, \map f X}$ is a [[Definition:Polynomial Ring in One Variable|polynomial ring]] in one [[Definition:Variable of Polynomial Ring|variable]] $\map f X$ over $R$.
+\end{theorem}
+
+\begin{proof}
+{{proof wanted}}
+[[Category:Polynomial Rings]]
+4awnnfe99b8800jwaxzarzp3xfiq024
+\end{proof}<|endoftext|>
+\section{Equality of Monomials of Polynomial Ring in Multiple Variables}
+Tags: Monomials
+
+\begin{theorem}
+Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
+Let $I$ be a [[Definition:Set|set]].
+Let $R \sqbrk {\sequence {x_i}_{i \mathop \in I} }$ be a [[Definition:Polynomial Ring in Multiple Variables|polynomial ring in $I$ variables]] $\sequence {x_i}_{i \mathop \in I}$ over $R$.
+Let $a, b : I \to \N$ be [[Definition:Distinct|distinct]] [[Definition:Mapping|mappings]] with [[Definition:Support of Mapping to Algebraic Structure|finite support]].
+Then the [[Definition:Monomial of Polynomial Ring|monomials]] $\ds \prod_{i \mathop \in I} X_i^{a_i}$ and $\ds \prod_{i \mathop \in I} X_i^{b_i}$ are [[Definition:Distinct|distinct]], where:
+:$X_i^k$ denotes the $k$th [[Definition:Power of Ring Element|power]] of $X_i$
+:$\prod$ denotes [[Definition:Product over Set with Finite Support|product with finite support]]
+\end{theorem}<|endoftext|>
+\section{Mittag-Leffler Expansion for Cosecant Function}
+Tags: Cosecant Function, Mittag-Leffler Expansions, Mittag-Leffler Expansion for Cosecant Function
+
+\begin{theorem}
+:$\displaystyle \pi \cosec \pi z = \frac 1 z + 2\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac z {z^2 - n^2}$
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}}
+\end{proof}<|endoftext|>
+\section{Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions}
+Tags: Real Analysis
+
+\begin{theorem}
+Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
+Let $f$ be a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Real Function|real function]] defined on $\closedint a b$.
+Let $P$ and $Q$ be [[Definition:Finite Subdivision|finite subdivisions]] of $\closedint a b$.
+Let $\map L P$ be the [[Definition:Lower Sum|lower sum]] of $f$ on $\closedint a b$ with respect to $P$.
+Let $\map U Q$ be the [[Definition:Upper Sum|upper sum]] of $f$ on $\closedint a b$ with respect to $Q$.
+Then $\map L P \le \map U Q$.
+\end{theorem}
+
+\begin{proof}
+Let $P' = P \cup Q$.
+We observe:
+:$P'$ is either equal to $P$ or [[Definition:Finer Subdivision|finer]] than $P$
+:$P'$ is either equal to $Q$ or [[Definition:Finer Subdivision|finer]] than $Q$
+We find:
+:$\map L P \le \map L {P'}$ by the definition of [[Definition:Lower Sum|lower sum]] and $P'$ [[Definition:Finer Subdivision|refining]] $P$
+:$\map L {P'} \le \map U {P'}$ by [[Upper Sum Never Smaller than Lower Sum]]
+:$\map U {P'} \le \map U Q$ by the definition of [[Definition:Upper Sum|upper sum]] and $P'$ [[Definition:Finer Subdivision|refining]] $Q$
+By combining these inequalities, we conclude:
+:$\map L P \le \map U Q$
+{{qed}}
+[[Category:Real Analysis]]
+pl2sdzoid9vqslje2yplrl97iqur6q1
+\end{proof}<|endoftext|>
+\section{Set Finite iff Surjection from Initial Segment of Natural Numbers}
+Tags: Set Theory, Surjections
+
+\begin{theorem}
+Let $S$ be a [[Definition:Set|set]].
+Then $S$ is [[Definition:Finite Set|finite]] {{iff}} for some $n \in \N$ there exists a [[Definition:Surjection|surjection]] $f: \N_{< n} \to S$.
+Here, $\N_{< n}$ denotes an [[Definition:Initial Segment of Natural Numbers|initial segment of $\N$]].
+\end{theorem}
+
+\begin{proof}
+=== Necessary Condition ===
+Suppose that $S$ is [[Definition:Finite Set|finite]].
+By definition, this means there exists a [[Definition:Bijection|bijection]] $f: \N_{< n} \to S$.
+Then $f$ is [[Definition:A Fortiori|a fortiori]] also the sought [[Definition:Surjection|surjection]].
+{{qed|lemma}}
+=== Sufficient Condition ===
+Let $f: \N_{< n} \to S$ be a [[Definition:Surjection|surjection]].
+Define $g: S \to \N_{< n}$ by:
+:$g (s) := \min f^{-1} (s)$
+where $f^{-1} (s)$ is the [[Definition:Preimage of Element under Mapping|preimage of $s$ under $f$]].
+Note that $f^{-1} (s)$ is not [[Definition:Empty Set|empty]] because $f$ is a [[Definition:Surjection|surjection]].
+By the [[Well-Ordering Principle]], $f^{-1} (s) \subseteq \N$ has a [[Definition:Smallest Element|smallest element]].
+Hence $g$ is [[Definition:Well-Defined Mapping|well-defined]].
+Next we show that $g$ is [[Definition:Injection|injective]].
+So suppose that $g(s) = g(s')$ for some $s, s' \in S$:
+{{begin-eqn}}
+{{eqn|l = g(s)
+ |r = g(s')
+}}
+{{eqn|ll= \implies
+ |l = f \left({ g(s) }\right)
+ |r = f \left({ g(s') }\right)
+}}
+{{eqn|ll= \implies
+ |l = s
+ |r = s'
+ |c = Definition of $g$
+}}
+{{end-eqn}}
+Hence $g$ is [[Definition:Injection|injective]].
+Then by [[Injection to Image is Bijection]], $S$ is [[Definition:Set Equivalence|equivalent]] to a [[Definition:Subset|subset]] of $\N_{
+\section{Set Finite iff Injection to Initial Segment of Natural Numbers}
+Tags: Set Theory, Injections
+
+\begin{theorem}
+Let $S$ be a [[Definition:Set|set]].
+Then $S$ is [[Definition:Finite Set|finite]] {{iff}} for some $n \in \N$ there exists an [[Definition:Injection|injection]] $f: S \to \N_{< n}$.
+Here, $\N_{< n}$ denotes an [[Definition:Initial Segment of Natural Numbers|initial segment of $\N$]].
+\end{theorem}
+
+\begin{proof}
+=== Necessary Condition ===
+Suppose that $S$ is [[Definition:Finite Set|finite]].
+By definition, this means there exists a [[Definition:Bijection|bijection]] $f: S \to \N_{< n}$.
+Then $f$ is [[Definition:A Fortiori|a fortiori]] also the sought [[Definition:Injection|injection]].
+{{qed|lemma}}
+=== Sufficient Condition ===
+Let $f: S \to \N_{< n}$ be an [[Definition:Injection|injection]].
+Then by [[Injection to Image is Bijection]], $S$ is [[Definition:Set Equivalence|equivalent]] to a [[Definition:Subset|subset]] of $\N_{
+\section{Largest Integer whose Digits taken in Pairs all form Distinct Primes}
+Tags: Prime Numbers, Recreational Mathematics
+
+\begin{theorem}
+The largest [[Definition:Integer|integer]] which has the property that every pair of its [[Definition:Digit|digits]] taken together is a [[Definition:Distinct|distinct]] [[Definition:Prime Number|prime number]] is $619 \, 737 \, 131 \, 179$.
+\end{theorem}
+
+\begin{proof}
+Let $p$ be such an [[Definition:Integer|integer]].
+Apart from the first [[Definition:Digit|digit]], each of its [[Definition:Digit|digits]] forms the second [[Definition:Digit|digit]] of a $2$-[[Definition:Digit|digit]] [[Definition:Prime Number|prime number]].
+Thus, apart from the first [[Definition:Digit|digit]], $p$ cannot contain $2$, $4$, $5$, $6$, $8$ or $0$.
+$0$ cannot of course be the first [[Definition:Digit|digit]] either.
+Thus, apart from the first pair of its [[Definition:Digit|digits]], the $2$-[[Definition:Digit|digit]] [[Definition:Prime Number|prime numbers]] contained in $p$ can only ever be in the [[Definition:Set|set]]:
+:$S = \left\{ {11, 13, 17, 19, 31, 37, 71, 73, 79, 97}\right\}$
+Let it be assumed that the largest such $p$ contains all of the above.
+There are $10$ [[Definition:Element|elements]] of $S$.
+Each of the [[Definition:Element|elements]] of $S$ apart from one shares both [[Definition:Digit|digits]] with one other [[Definition:Element|element]] of $S$, apart from the final one in $p$.
+Thus the [[Definition:Element|elements]] of $S$ contribute $11$ [[Definition:Digit|digits]] of $p$ in total.
+The first [[Definition:Digit|digit]] of the first pair contributes an extra $1$ [[Definition:Digit|digit]] to $p$.
+Thus $p$ contains $12$ [[Definition:Digit|digits]].
+Only $1$ [[Definition:Element|element]] of $S$ begins with $9$, but $2$ of them end with $9$.
+Therefore $9$ must be the last digit of $p$.
+Only $3$ [[Definition:Element|elements]] of $S$ end in $1$, but $4$ of them begin with $1$.
+Thus the first pair of [[Definition:Digit|digits]] of $p$ must be a [[Definition:Prime Number|prime number]] which does not appear in $S$ whose $2$nd [[Definition:Digit|digit]] is $1$.
+Thus the first $2$ [[Definition:Digit|digits]] of $p$ can only be $41$ or $61$.
+Let us construct $p$ by starting from the left and use the [[Definition:Greedy Algorithm|greedy algorithm]] consisting of select the largest [[Definition:Digit|digit]] available.
+:$61$ is larger than $41$, so $p$ can be assumed to start with $61$.
+Using this algorithm, we select the [[Definition:Element|elements]] of $S$ in the following order:
+:$61$, $19$, $97$ (which is forced), $73$, $37$, $71$
+The next [[Definition:Element|element]] of $S$ cannot be $17$, because that would force $79$ to follow.
+But as $79$ is the only remaining [[Definition:Element|element]] of $S$ ending in $9$, $79$ must be at the end of $p$.
+So $71$ must be followed by $11$ or $13$.
+Continuing to use the [[Definition:Greedy Algorithm|greedy algorithm]]:
+:$61$, $19$, $97$ $73$, $37$, $71$, $13$, $31$ (forced)
+For the same reason as above, we cannot then select $17$, as this will be followed by $79$ which cannot be followed.
+So we continue with the [[Definition:Greedy Algorithm|greedy algorithm]]:
+:$61$, $19$, $97$ $73$, $37$, $71$, $13$, $31$, $11$, $17$, $79$
+and $p$ is complete:
+:$619 \, 737 \, 131 \, 179$
+As a bonus, $p$ is itself a [[Definition:Prime Number|prime number]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Finite Product of Finite Sets is Finite}
+Tags: Cartesian Product
+
+\begin{theorem}
+Let $\sequence {S_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Finite Set|finite sets]].
+Let $\displaystyle \prod_{k \mathop = 1}^n S_k$ be their [[Definition:Finite Cartesian Product|Cartesian product]].
+Then $\displaystyle \prod_{k \mathop = 1}^n S_k$ is also a [[Definition:Finite Set|finite set]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted|Boring induction from [[Product of Finite Sets is Finite]]}}
+\end{proof}<|endoftext|>
+\section{Smallest 22 Primes in Arithmetic Sequence}
+Tags: Prime Numbers, Arithmetic Sequences
+
+\begin{theorem}
+The smallest $22$ [[Definition:Prime Number|primes]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] are:
+:$11 \, 410 \, 337 \, 850 \, 553 + 4 \, 609 \, 098 \, 694 \, 200 n$
+for $n = 0, 1, \ldots, 21$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 0 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 11 \, 410 \, 337 \, 850 \, 553
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 1 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 16 \, 019 \, 436 \, 544 \, 753
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 2 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 20 \, 628 \, 535 \, 238 \, 953
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 3 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 25 \, 237 \, 633 \, 933 \, 153
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 4 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 29 \, 846 \, 732 \, 627 \, 353
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 5 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 34 \, 455 \, 831 \, 321 \, 553
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 6 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 39 \, 064 \, 930 \, 015 \, 753
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 7 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 43 \, 674 \, 028 \, 709 \, 953
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 8 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 48 \, 283 \, 127 \, 404 \, 153
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 9 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 52 \, 892 \, 226 \, 098 \, 353
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 10 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 57 \, 501 \, 324 \, 792 \, 553
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 11 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 62 \, 110 \, 423 \, 486 \, 753
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 12 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 66 \, 719 \, 522 \, 180 \, 953
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 13 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 71 \, 328 \, 620 \, 875 \, 153
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 14 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 75 \, 937 \, 719 \, 569 \, 353
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 15 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 80 \, 546 \, 818 \, 263 \, 553
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 16 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 85 \, 155 \, 916 \, 957 \, 753
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 17 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 89 \, 765 \, 015 \, 651 \, 953
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 18 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 94 \, 374 \, 114 \, 346 \, 153
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 19 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 98 \, 983 \, 213 \, 040 \, 353
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 20 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 103 \, 592 \, 311 \, 734 \, 553
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 21 \times 4 \, 609 \, 098 \, 694 \, 200
+ | r = 108 \, 201 \, 410 \, 428 \, 753
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+But note that $11 \, 410 \, 337 \, 850 \, 553 + 22 \times 4 \, 609 \, 098 \, 694 \, 200 = 112 \, 810 \, 509 \, 122 \, 953 = 61 \times 107 \times 1907 \times 9063277$ 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{Existence of Integers with Multiplicative Persistence Greater than 11}
+Tags: Multiplicative Persistence, Open Questions
+
+\begin{theorem}
+It is not known whether there exists a number $n$ such that:
+:$P \left({n}\right) = 12$
+where $P \left({n}\right)$ denotes the [[Definition:Multiplicative Persistence|multiplicative persistence]] of $n$.
+\end{theorem}<|endoftext|>
+\section{Smallest Strong Fibonacci Pseudoprime of Type I}
+Tags: Strong Fibonacci Pseudoprimes, 443,372,888,629,441
+
+\begin{theorem}
+The smallest [[Definition:Strong Fibonacci Pseudoprime of Type I|strong Fibonacci pseudoprime of type I]] is $443 \, 372 \, 888 \, 629 \, 441$.
+\end{theorem}
+
+\begin{proof}
+Let $N := 443 \, 372 \, 888 \, 629 \, 441$, to save writing it out in full each time.
+From the definition of a [[Definition:Strong Fibonacci Pseudoprime of Type I|strong Fibonacci pseudoprime of type I]]:
+{{:Definition:Strong Fibonacci Pseudoprime of Type I}}
+We have that:
+:$N = 17 \times 31 \times 41 \times 43 \times 89 \times 97 \times 167 \times 331$
+Of these, we see that:
+{{begin-eqn}}
+{{eqn | l = 17
+ | r = 4 \times 4 + 1
+ | c = which is a [[Definition:Prime Number|prime number]] of the form $4 n + 1$
+}}
+{{eqn | l = 31
+ | r = 4 \times 8 - 1
+ | c = which is a [[Definition:Prime Number|prime number]] of the form $4 n - 1$
+}}
+{{eqn | l = 41
+ | r = 4 \times 10 + 1
+ | c = which is a [[Definition:Prime Number|prime number]] of the form $4 n + 1$
+}}
+{{eqn | l = 43
+ | r = 4 \times 11 - 1
+ | c = which is a [[Definition:Prime Number|prime number]] of the form $4 n - 1$
+}}
+{{eqn | l = 89
+ | r = 4 \times 22 + 1
+ | c = which is a [[Definition:Prime Number|prime number]] of the form $4 n + 1$
+}}
+{{eqn | l = 97
+ | r = 4 \times 24 + 1
+ | c = which is a [[Definition:Prime Number|prime number]] of the form $4 n + 1$
+}}
+{{eqn | l = 167
+ | r = 4 \times 42 - 1
+ | c = which is a [[Definition:Prime Number|prime number]] of the form $4 n - 1$
+}}
+{{eqn | l = 331
+ | r = 4 \times 83 - 1
+ | c = which is a [[Definition:Prime Number|prime number]] of the form $4 n - 1$
+}}
+{{end-eqn}}
+Thus there are $4$ (an [[Definition:Even Integer|even number]]) of [[Definition:Prime Factor|prime factors]] of $N$ of the form $4 n - 1$.
+We note that:
+{{begin-eqn}}
+{{eqn | l = N - 1
+ | r = 288 \times 1 \, 539 \, 489 \, 196 \, 630
+ | c = and so $\paren {17^2 - 1} \divides \paren {N - 1}$
+}}
+{{eqn | r = 960 \times 461 \, 846 \, 758 \, 989
+ | c = and so $\paren {31^2 - 1} \divides \paren {N - 1}$
+}}
+{{eqn | r = 1680 \times 263 \, 912 \, 433 \, 708
+ | c = and so $\paren {41^2 - 1} \divides \paren {N - 1}$
+}}
+{{eqn | r = 1848 \times 239 \, 920 \, 394 \, 280
+ | c = and so $\paren {43^2 - 1} \divides \paren {N - 1}$
+}}
+{{eqn | r = 7920 \times 55 \, 981 \, 425 \, 332
+ | c = and so $\paren {89^2 - 1} \divides \paren {N - 1}$
+}}
+{{eqn | r = 9408 \times 47 \, 127 \, 220 \, 305
+ | c = and so $\paren {97^2 - 1} \divides \paren {N - 1}$
+}}
+{{eqn | r = 27 \, 888 \times 15 \, 898 \, 339 \, 380
+ | c = and so $\paren {167^2 - 1} \divides \paren {N - 1}$
+}}
+{{eqn | r = 109 \, 560 \times 4 \, 046 \, 850 \, 024
+ | c = and so $\paren {331^2 - 1} \divides \paren {N - 1}$
+}}
+{{end-eqn}}
+Thus for all $p \divides N$, we have that $\paren {p^2 - 1} \divides \paren {N - 1}$.
+From [[Difference of Two Squares]] we have that:
+:$p^2 - 1 = \paren {p + 1} \paren {p - 1}$
+and so for all $p \divides N$:
+:$\paren {p - 1} \divides \paren {N - 1}$
+We also have that $N$ is [[Definition:Square-Free Integer|square-free]].
+Thus by [[Korselt's Theorem]], $N$ is a [[Definition:Carmichael Number|Carmichael number]].
+Now we also have that:
+{{begin-eqn}}
+{{eqn | l = \paren {p^2 - 1}
+ | o = \divides
+ | r = \paren {N - 1}
+ | c =
+}}
+{{eqn | ll= \leadsto
+ | l = \paren {p + 1} \paren {p - 1}
+ | o = \divides
+ | r = \paren {N - 1}
+ | c = [[Difference of Two Squares]]
+}}
+{{eqn | ll= \leadsto
+ | l = 2 \paren {p + 1} \paren {\frac {p - 1} 2}
+ | o = \divides
+ | r = \paren {N - 1}
+ | c = as $p - 1$ is [[Definition:Even Integer|even]]
+}}
+{{eqn | ll= \leadsto
+ | l = 2 \paren {p + 1}
+ | o = \divides
+ | r = \paren {N - 1}
+ | c =
+}}
+{{end-eqn}}
+Thus we have that:
+:$\forall p \divides N: 2 \paren {p + 1} \divides \paren {N - 1}$
+This is actually stronger than the conditions for which $N$ is a [[Definition:Strong Fibonacci Pseudoprime of Type I|strong Fibonacci pseudoprime of type I]]:
+{{:Definition:Strong Fibonacci Pseudoprime of Type I}}
+It can be established by an exhaustive search that there are no smaller [[Definition:Carmichael Number|Carmichael numbers]] with this property.
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Infimum of Upper Sums Never Smaller than Lower Sum}
+Tags: Real Analysis
+
+\begin{theorem}
+:$\inf_P \map U P \ge \map L S$
+\end{theorem}
+
+\begin{proof}
+From [[Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions]], $\map L S$ is a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] for the [[Definition:Real Number|real]] [[Definition:Set|set]]:
+:$T = \leftset {\map U P: P}$ is a [[Definition:Finite Subdivision|finite subdivision]] of $\rightset {\closedint a b}$
+Since $\inf_P \map U P$ is the [[Definition:Infimum of Subset of Real Numbers|infumum]] of $T$:
+:$\inf_P \map U P \ge \map L S$
+Hence the result.
+{{qed}}
+[[Category:Real Analysis]]
+jq9jbzo0wiugc4xg797a485pbx5zxaq
+\end{proof}<|endoftext|>
+\section{Supremum of Lower Sums Never Greater than Upper Sum}
+Tags: Real Analysis
+
+\begin{theorem}
+:$\sup_P \map L P \le \map U S$
+\end{theorem}
+
+\begin{proof}
+From [[Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions]], $\map U S$ is an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]] for the [[Definition:Real Number|real]] [[Definition:Set|set]]:
+:$T = \leftset {\map L P: P}$ is a [[Definition:Finite Subdivision|finite subdivision]] of $\rightset {\closedint a b}$
+Since $\sup_P \map L P$ is the [[Definition:Supremum of Subset of Real Numbers|supremum]] of $T$:
+:$\sup_P \map L P \le \map U S$
+Hence the result.
+{{qed}}
+[[Category:Real Analysis]]
+1eh713jwmo7suop90lls86p3n0c3inu
+\end{proof}<|endoftext|>
+\section{Smallest Cunningham Chain of the Second Kind of Length 13}
+Tags: Cunningham Chains
+
+\begin{theorem}
+The smallest [[Definition:Cunningham Chain of the Second Kind|Cunningham chain of the second kind]] of [[Definition:Length of Sequence|length]] $13$ is:
+:$758 \, 083 \, 947 \, 856 \, 951$, $1 \, 516 \, 167 \, 895 \, 713 \, 901$, $3 \, 032 \, 335 \, 791 \, 427 \, 801$, $6 \, 064 \, 671 \, 582 \, 855 \, 601$, $12 \, 129 \, 343 \, 165 \, 711 \, 201$, $24 \, 258 \, 686 \, 331 \, 422 \, 401$, $48 \, 517 \, 372 \, 662 \, 844 \, 801$, $97 \, 034 \, 745 \, 325 \, 689 \, 601$, $194 \, 069 \, 490 \, 651 \, 379 \, 201$, $388 \, 138 \, 981 \, 302 \, 758 \, 401$, $776 \, 277 \, 962 \, 605 \, 516 \, 801$, $1 \, 552 \, 555 \, 925 \, 211 \, 033 \, 601$, $3 \, 105 \, 111 \, 850 \, 422 \, 067 \, 201$
+\end{theorem}
+
+\begin{proof}
+Let $C$ denote the [[Definition:Sequence|sequence]] in question.
+We have that $758 \, 083 \, 947 \, 856 \, 951$ is [[Definition:Prime Number|prime]].
+First note that:
+:$\dfrac {758 \, 083 \, 947 \, 856 \, 951 + 1} 2 = 379 \, 041 \, 973 \, 928 \, 476 = 2^2 \times 94 \, 760 \, 493 \, 482 \, 119$
+and so is not [[Definition:Prime Number|prime]].
+Thus $758 \, 083 \, 947 \, 856 \, 951$ fulfils the requirement for $C$ to be a [[Definition:Cunningham Chain of the Second Kind|Cunningham chain of the second kind]].
+Then:
+{{begin-eqn}}
+{{eqn | n = 1
+ | l = 2 \times 758 \, 083 \, 947 \, 856 \, 951 - 1
+ | r = 1 \, 516 \, 167 \, 895 \, 713 \, 901
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 2
+ | l = 2 \times 1 \, 516 \, 167 \, 895 \, 713 \, 901 - 1
+ | r = 3 \, 032 \, 335 \, 791 \, 427 \, 801
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 3
+ | l = 2 \times 3 \, 032 \, 335 \, 791 \, 427 \, 801 - 1
+ | r = 6 \, 064 \, 671 \, 582 \, 855 \, 601
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 4
+ | l = 2 \times 6 \, 064 \, 671 \, 582 \, 855 \, 601 - 1
+ | r = 12 \, 129 \, 343 \, 165 \, 711 \, 201
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 5
+ | l = 2 \times 12 \, 129 \, 343 \, 165 \, 711 \, 201 - 1
+ | r = 24 \, 258 \, 686 \, 331 \, 422 \, 401
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 6
+ | l = 2 \times 24 \, 258 \, 686 \, 331 \, 422 \, 401 - 1
+ | r = 48 \, 517 \, 372 \, 662 \, 844 \, 801
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 7
+ | l = 2 \times 48 \, 517 \, 372 \, 662 \, 844 \, 801 - 1
+ | r = 97 \, 034 \, 745 \, 325 \, 689 \, 601
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 8
+ | l = 2 \times 97 \, 034 \, 745 \, 325 \, 689 \, 601 - 1
+ | r = 194 \, 069 \, 490 \, 651 \, 379 \, 201
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 9
+ | l = 2 \times 194 \, 069 \, 490 \, 651 \, 379 \, 201 - 1
+ | r = 388 \, 138 \, 981 \, 302 \, 758 \, 401
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 10
+ | l = 2 \times 388 \, 138 \, 981 \, 302 \, 758 \, 401 - 1
+ | r = 776 \, 277 \, 962 \, 605 \, 516 \, 801
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 11
+ | l = 2 \times 776 \, 277 \, 962 \, 605 \, 516 \, 801 - 1
+ | r = 1 \, 552 \, 555 \, 925 \, 211 \, 033 \, 601
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 12
+ | l = 2 \times 1 \, 552 \, 555 \, 925 \, 211 \, 033 \, 601 - 1
+ | r = 3 \, 105 \, 111 \, 850 \, 422 \, 067 \, 201
+ | c = which is [[Definition:Prime Number|prime]]
+}}
+{{eqn | n = 13
+ | l = 2 \times 3 \, 105 \, 111 \, 850 \, 422 \, 067 \, 201 - 1
+ | r = 6 \, 210 \, 223 \, 700 \, 844 \, 134 \, 401
+ | c = which is $41 \times 691 \times 739 \times 4957 \times 59 \, 838 \, 677$ and so not [[Definition:Prime Number|prime]]
+}}
+{{end-eqn}}
+Establishing that this is indeed the smallest such [[Definition:Cunningham Chain of the Second Kind|Cunningham chain of the second kind]] of [[Definition:Length of Sequence|length]] $13$ can be done by a computer search.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Properties of 5,559,060,566,555,523}
+Tags: 5,559,060,566,555,523, Recreational Mathematics
+
+\begin{theorem}
+$3^{33} = 5 \, 559 \, 060 \, 566 \, 555 \, 523$ has the following properties:
+: It has a remarkably large number of $5$s (half of its [[Definition:Digit|digits]]).
+: Multiply it by $2$, $4$ or $6$, and the result has $10$ of a particular [[Definition:Digit|digit]].
+: Multiply it by $8$, and $9$ of the [[Definition:Digit|digits]] of the result are $4$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 5 \, 559 \, 060 \, 566 \, 555 \, 523 \times 2
+ | r = 11 \, 118 \, 121 \, 133 \, 111 \, 046
+ | c = $10$ $1$s
+}}
+{{eqn | l = 5 \, 559 \, 060 \, 566 \, 555 \, 523 \times 4
+ | r = 22 \, 236 \, 242 \, 266 \, 222 \, 092
+ | c = $10$ $2$s
+}}
+{{eqn | l = 5 \, 559 \, 060 \, 566 \, 555 \, 523 \times 6
+ | r = 33 \, 354 \, 363 \, 399 \, 333 \, 138
+ | c = $10$ $3$s
+}}
+{{eqn | l = 5 \, 559 \, 060 \, 566 \, 555 \, 523 \times 8
+ | r = 44 \, 472 \, 484 \, 532 \, 444 \, 184
+ | c = $9$ $4$s
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{First Occurrence of Prime Gap of 864}
+Tags: Prime Gaps
+
+\begin{theorem}
+The first occurrence of a [[Definition:Prime Gap|prime gap]] of $864$ is between $6 \, 505 \, 941 \, 701 \, 960 \, 039$ and $6 \, 505 \, 941 \, 701 \, 960 \, 903$.
+\end{theorem}
+
+\begin{proof}
+$6 \, 505 \, 941 \, 701 \, 960 \, 039$ is a [[Definition:Prime Number|prime number]].
+$6 \, 505 \, 941 \, 701 \, 960 \, 903$ is a [[Definition:Prime Number|prime number]].
+It can be checked that all numbers between these two are [[Definition:Composite Number|composite]].
+It can also be checked by computer that there is no [[Definition:Prime Gap|prime gap]] of $864$ between smaller [[Definition:Prime Number|prime numbers]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Hardy-Ramanujan Number/Examples/48,988,659,276,962,496}
+Tags: Hardy-Ramanujan Numbers, 48,988,659,276,962,496
+
+\begin{theorem}
+The $5$th [[Definition:Hardy-Ramanujan Number|Hardy-Ramanujan number]] $\map {\operatorname {Ta} } 5$ is $48 \, 988 \, 659 \, 276 \, 962 \, 496$:
+{{begin-eqn}}
+{{eqn | l = 48 \, 988 \, 659 \, 276 \, 962 \, 496
+ | r = 38 \, 787^3 + 365 \, 757^3
+ | c =
+}}
+{{eqn | r = 107 \, 839^3 + 362 \, 753^3
+ | c =
+}}
+{{eqn | r = 205 \, 292^3 + 342 \, 952^3
+ | c =
+}}
+{{eqn | r = 221 \, 424^3 + 336 \, 588^3
+ | c =
+}}
+{{eqn | r = 231 \, 518^3 + 331 \, 954^3
+ | c =
+}}
+{{end-eqn}}
+\end{theorem}<|endoftext|>
+\section{Reciprocal of 19 from Sum of Powers of 2 Backwards}
+Tags: 19, Examples of Reciprocals
+
+\begin{theorem}
+The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $19$ can be constructed by [[Definition:Integer Addition|summing]] the [[Definition:Integer Power|powers]] of $2$, offset progressively backwards by $1$ digit:
+
+ 1
+ 2
+ 4
+ 8
+ 16
+ 32
+ 64
+ 128
+ 256
+ 512
+ 1024
+ 2048
+ 4096
+ 8192
+ 16384
+ 32768
+ 65536
+ 131072
++ 262144
+ ......
+--------------------------
+.......1052631578947368421
+
+while:
+:$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$
+\end{theorem}
\ No newline at end of file