Definition:Wilson Prime
Jump to navigation
Jump to search
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
- Wilson's Theorem: every prime number $p$ satisfies $p \divides \paren {p - 1}! + 1$
- 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
- Weisstein, Eric W. "Wilson Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WilsonPrime.html