TITLE: A conjecture: for all $n\in\mathbb{N}$, the least $k>1$ such that $\phi(k)\geqslant n$ is a prime
QUESTION [15 upvotes]: I came across a problem in a book that asked us to find the first number $n$ such that $\phi(n)\geqslant 1,000$ it turns out that the answer is 1009, which is a prime number. There were several questions of this type, and our professor conjectured that it will always be the next prime. However, no one has been able to come up with a proof for this conjecture. So, more formally the conjecture is:
For all $n\in\mathbb{N}$ the least positive integer $k\in\mathbb{N}$, with $k>1$ such that $\phi(k)\geqslant n$ is a prime.
I have worked with the Möbius Inversion function, and other minimal element contradictory  proofs, but nothing has worked so far. Does anyone have any good ideas?

REPLY [5 votes]: If $n$ is not prime, then it has a prime factor $\le \sqrt{n}$, and so
$$
\varphi(n) = n \prod_{p\;|\;n}\left(1-\frac{1}{p}\right)\le n\left(1-\frac{1}{\sqrt{n}}\right)=n-\sqrt{n};
$$
while $\varphi(n)=n-1$ if $n$ is prime.
So, fix $M$.  Let $p$ be the next prime number after $M$, so $\varphi(p)=p-1\ge M$; and let $p'$ be the largest prime number $\le M$.  (That is, $p$ and $p'$ are adjacent primes.)  Finally, suppose that a composite $n < p$ also satisfies $\varphi(n)\ge M$.  Then
$$
p-\sqrt{p}>n-\sqrt{n}\ge\varphi(n)\ge M \ge p',
$$
or
$$
p-p'\ge \sqrt{p}.
$$
If this were known to never happen for sufficiently large $p$ (i.e., if eventually $g_k < \sqrt{p_{k+1}}$, where $p_k$ is the $k$-th prime, and the prime gap $g_k\equiv p_{k+1}-p_{k}$), then your conjecture could be proven.  This is a strong conjecture (although almost certainly true)... I do not believe it is even known to follow from the Riemann hypothesis.  Looking at the table of prime gaps, though, there are very few counterexamples, and the only near misses for your conjecture are:
$$
M=2 \qquad \varphi(3)=\varphi(4)=2; \\
M=6 \qquad \varphi(7)=\varphi(9)=6; \\
M=20 \qquad \varphi(23)=22; \varphi(25)=20; \\
M=110 \qquad \varphi(113)=112; \varphi(121)=110.
$$