Proof with congruence and primes

Question

So here it goes.

For any integer $p > 1$, if $(p - 1)!$ is congruent to $-1 \pmod p$ then $p$ is prime.

Any help would be appreciated!

Answer 1

This is Wilson's theorem, and it has been proven by many mathematicians (though perhaps not by Wilson himself, I'm not sure). Proofs can be found in Niven & Zuckerman's An Introduction to the Theory of Numbers and Ethan Bolker's Elementary Number Theory to name just two. On the Web I suggest you go to ProofWiki: https://proofwiki.org/wiki/Wilson%27s_Theorem But avoid Wikipedia at all costs or else you'll be led astray.

Answer 2

For composite number $n$ it is obvious that $\equiv 0$ since all divisors of $n$ are strictly smaller than $n$.

If it is prime. Then $\mathbb{F}_p$ is a field and $x=x^{-1}$ if and only if $x=\pm1$. Hence the other elements has different inverses. Therefore $1\cdots(p-1)\equiv=1.(-1)(a_1a_1^{-1})\cdots \equiv -1$

Note that: $-1 \equiv 1$ in modulo $2$.