Wilson's Theorem

Theorem

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

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

Let $p$ be a prime number.

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

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:

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

If $p = 2$ the result is immediate.

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

Let $p$ be a prime number.

Then:

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

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

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

Then $p$ is a prime number.

$5$ is a divisor of $\paren {5 - 1}! + 1$.

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

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

This entry was named for John Wilson.

The proof of 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 proved by Lagrange in $1771$.

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): Wilson's theorem
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Wilson's theorem