Wilson's Theorem

Theorem
A positive integer $p$ is a prime iff $\left({p-1}\right)! \equiv -1 \pmod {p}$.

Proof
If $p = 2$ the result is obvious.

Therefore we assume that $p$ is an odd prime.

If part
Consider $n \in \Z, 1 \le n < p$.

From Law of Inverses (Modulo Arithmetic), we have:
 * $\exists n' \in \Z, 1 \le n' < p: n n' \equiv 1 \pmod p$

as $p$ prime and therefore $n \perp p$.

By Congruence by Factors of Modulo, for each $n$ there is exactly one such $n'$, and $\left({n'}\right)' = n$.

So, provided $n \ne n'$, we can pair any given $n$ from $1$ to $p$ with another $n'$ from $1$ to $p$.

We are then left with the numbers such that $n = n'$.

Then we have $n^2 \equiv 1 \pmod p$.

Consider $n^2 - 1 = \left({n+1}\right) \left({n-1}\right)$ from Difference of Two Squares.

So either $n+1$ divides $p$ or $n-1$ does.

(They can't both do, as they have a difference of $2$, and $p$ is an odd prime.

From Congruence Modulo Negative Number‎, we have that $p-1 \equiv -1 \pmod p$.

Hence $n=1$ or $n = p-1$.

So, we have that $\left({p-1}\right)!$ consists of numbers multiplied together as follows:
 * in pairs whose product is congruent to $1 \pmod p$
 * the numbers $1$ and $p-1$.

The product of all these numbers is therefore congruent to $1 \times 1 \times \cdots \times 1 \times p-1 \pmod p$ by modulo multiplication.

From Congruence Modulo Negative Number we therefore have that $\left({p-1}\right)! \equiv -1 \pmod {p}$.

Only if part
Now consider $p$ is a composite, and $q$ is a prime such that $q \backslash p$.

Then both $p$ and $\left({p-1}\right)!$ are divisible by $q$.

If the congruence $\left({p-1}\right)! \equiv -1 \pmod{p}$ was satisfied, we would have $\left({p-1}\right)! \equiv -1 \pmod {q}$.

This would means $0 \equiv -1 \pmod {q}$ which is false.

Hence for $p$ composite, the congruence $\left({p-1}\right)! \equiv -1 \pmod {p}$ cannot hold.

Historical Note
This proof was attributed to John Wilson by Edward Waring in his 1770 edition of Meditationes Algebraicae.

It was first stated by Ibn al-Haytham ("Alhazen").

It appears also to have been known to Gottfried Leibniz before 1663.

It was in fact finally proved by Lagrange in 1793.