Suppose that $n$ is a factor of $(n-1)!+1$. Prove that $n$ is prime

Question

This is in an Algebra and Combinatorics module and I don't know how to prove this. The full question is,

Let $n$ be a natural number greater than $1$. Suppose that $n$ is a factor of $(n-1)!+1$. Prove that $n$ is prime.

Any help would be greatly appreciated -thanks.

I disagree that this is a duplicate of the question linked. Indeed, the other question asks for a proof of one implication in Wilson's theorem ("if $p$ is prime then..."), while this question is about the other implication ("if ... then $p$ is prime"). — Wojowu, Jan 30 '16 at 18:26
There is a better target, which is a canonical reference for both directions. — , Jan 30 '16 at 21:44

score 3 · Answer 1 · answered Jan 30 '16 at 18:12

3

Suppose $n=ab$ with $1<a<n$. Then $a\mid (n-1)!$ and $a\nmid (n-1)!+1$, hence $n\nmid (n-1)!+1$.

answered Jan 30 '16 at 18:12

Hagen von Eitzen

1 Answers1