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.

Necessary Condition
Let $p$ be prime.

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

As $p$ is prime, $n \perp p$.

From Law of Inverses (Modulo Arithmetic), we have:


 * $\exists n' \in \Z, 1 \le n' < p: n n' \equiv 1 \pmod p$

By Solution of Linear Congruence, 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 $n + 1 \mathop \backslash p$ or $n - 1 \mathop \backslash p$.

Observe that these cases do not occur simultaneously, as their difference is $2$, 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 $\left({p - 1}\right)! \equiv -1 \pmod p$.

Sufficient Condition
Now assume $p$ is composite, and $q$ is a prime such that $q \mathop \backslash p$.

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

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

However, this amounts to $0 \equiv -1 \pmod q$, a contradiction.

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

The proof was attributed to by  in his 1770 edition of Meditationes Algebraicae.

It was first stated by.

It appears also to have been known to before 1663.

It was in fact finally proved by in 1793.