# Wilson's Theorem

## Theorem

A (strictly) positive integer $p$ is a prime if and only if:

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

### Corollary $1$

Let $p$ be a prime number.

Then $p$ is the smallest prime number which divides $\paren {p - 1}! + 1$.

### Corollary $2$

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $p$ be a prime factor of $n!$ with multiplicity $\mu$.

Let $n$ be expressed in a base $p$ representation as:

\(\ds n\) | \(=\) | \(\ds \sum_{j \mathop = 0}^m a_j p^j\) | where $0 \le a_j < p$ | |||||||||||

\(\ds \) | \(=\) | \(\ds a_0 + a_1 p + a_2 p^2 + \cdots + a_m p^m\) | for some $m > 0$ |

Then:

- $\dfrac {n!} {p^\mu} \equiv \paren {-1}^\mu a_0! a_1! \dotsb a_m! \pmod p$

## Proof

If $p = 2$ the result is obvious.

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

### Necessary Condition

Let $p$ be a prime number.

Then:

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

### Sufficient Condition

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

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

Then $p$ is a prime number.

## Examples

### $10$ does not divide $\paren {n - 1}! + 1$

For all $n \in \Z_{>0}$, $10$ is not a divisor of $\paren {n - 1}! + 1$.

## 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 $1793$.

## Sources

- 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Example $\text {4-2}$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $24$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $561$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $24$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $563$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Wilson's theorem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Wilson's theorem**