Wilson prime

A Wilson prime is a special kind of prime number. A prime number p is a Wilson prime if (and only if [ $⟺$ ])

$\frac{(p-1)!+1}{p^2}=n$

where n is a positive integer (sometimes called natural number). Wilson primes were first described by Emma Lehmer.

The only known Wilson primes are 5, 13, and 563 Template:OEIS; if any others exist, they must be greater than 5Template:E. It has been conjectured that there are an infinite number of Wilson primes, and that the number of Wilson primes in an interval $[x,y]$ is about

$\frac{\log (\log y)}{\log x}$.

Compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1.

Related pages

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

References

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Other websites

• The Prime Glossary: Wilson prime
• Template:Mathworld
• Status of the search for Wilson primes
• Wilson Quotients for composite moduli
• On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson