Definition:Wilson Prime

A Wilson prime is a prime number $p$ such that:

$p^2 \backslash \left({p-1}\right)! + 1$

where $\backslash$ signifies "divides" and $!$ is the factorial operator.

The only Wilson primes known to exist are:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 5\)	\(\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 13\)	\(\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 563\)	\(\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

If there are any more, they must be greater than $5 \times 10^8$.

This sequence is A007540 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Also see

Wilson's Theorem states that every prime number $p$ satisfies:

$p \backslash \left({p-1}\right)! + 1$

However, the condition that its square also divides that same number is satisfied much more rarely.

Source of Name

This entry was named for John Wilson.