# Wilson's Theorem/Sufficient Condition

## Theorem

Let $p$ be a (strictly) positive integer such that:

- $\paren {p - 1}! \equiv -1 \pmod p$

Then $p$ is a prime number.

## Proof

Assume $p$ is composite, and $q$ is a prime such that $q \divides p$.

Then both $p$ and $\paren {p - 1}!$ are divisible by $q$.

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

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

Hence for $p$ composite, the congruence $\paren {p - 1}! \equiv -1 \pmod p$ cannot hold.

$\blacksquare$

## Also known as

Some sources refer to **Wilson's Theorem** as the **Wilson-Lagrange theorem**, after Joseph Louis Lagrange, who proved it.

## Source of Name

This entry was named for John Wilson.

## Historical Note

The proof of Wilson's Theorem 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 in $1682$ or $1683$ (accounts differ).

It was in fact finally proved by Lagrange in $1793$.