Modular arithmetic and some applications

Question

Show that if $p> 2$ is a prime number, then $p$ divides $(p-2)! - 1$. I have tried using Fermat's Theorem, but I could not solve it.

Answer 1

This follows from Wilson's theorem, which states that

$$(p-1)!\equiv_p -1$$ if and only if $p$ is a prime

If you accept this, then the rest follows easily since $(p-1)!\equiv_p (p-2)!(-1)\equiv_p-1$ implies: $$(p-2)!\equiv_p 1$$ This is the same as saying that $p$ divides $(p-2)!-1$.

Now for the proof of this theorem. In $\mathbb{Z}_p$ every non-zero element has an inverse, in particular $-1$ and $1$ are their own inverses, which means that if you multiply all these numbers you'll get: $$1\cdot 2\cdot\ldots\cdot (p-1)=(p-1)!\equiv_p 1\cdot 1\cdot\ldots\cdot 1\cdot (-1)\equiv_p-1$$

Answer 2

If given $p>2$: $p$ divides $(p-2)! -1$ if and only if there is an $n \in \mathbb{Z}$ for which $$(p-2)! -1 = np \iff (p-2)! = 1 + np$$

Modular arithmetic says an equivalent thing then:

$$(p-2)! \equiv 1 \mod p \iff (p-1)!=(p-1)(p-2)! \equiv p-1 \mod{p} $$ $$\equiv -1 \mod p$$ And this, if and only if, $p$ is a positive prime number. (Wilson's Theorem)

More clearly now:

$p>2$ is a prime number is equivalent to $p>2$ being a divisor of $(p-2)!-1$.