Definition:Wilson Prime

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Definition

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

$p^2 \divides \paren {p - 1}! + 1$

where:

$\divides$ signifies divisibility
$!$ is the factorial operator.


Sequence

The sequence of Wilson primes begins:

$5, 13, 563$

The next term in the sequence, if there is one, is greater than $2 \times 10^{13}$.


Also see

  • Results about Wilson primes can be found here.


Source of Name

This entry was named for John Wilson.


Historical Note

The third Wilson prime was discovered by Karl Goldberg in $1953$, as a result of a computer search in which all numbers up to $10 \, 000$ were tested.

This was one of the first examples of a problem in number theory being attacked by a computer.

Subsequent searches have reached $2 \times 10^{13}$ without finding a $4$th such prime.


Sources