Wilson prime - Definition and Overview

In mathematics, a Wilson prime is a certain kind of prime number. A prime p is called a Wilson prime if p² divides (p − 1)! + 1, where ! denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1.

The only known Wilson primes are 5, 13, and 563 (sequence A007540 in OEIS); if any others exist, they must be greater than 5 · 10⁸. It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval [x, y] is about log(log(y) / log(x)).

Also see

• Wieferich prime
• Wall-Sun-Sun prime
• Wolstenholme prime

External links

• The Prime Glossary: Wilson prime (http://primes.utm.edu/glossary/page.php?sort=WilsonPrime)
• MathWorld: Wilson prime (http://mathworld.wolfram.com/WilsonPrime.html)
• Status of the search for Wilson primes (http://www.loria.fr/~zimmerma/records/Wieferich.status)