Wilson prime

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

where n is a positive integer (sometimes called natural number). Wilson primes were first described by Emma Lehmer.On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of. Math. 39(1938), 350-360.

The only known Wilson primes are 5, 13, and 563 ; if any others exist, they must be greater than 5.Status of the search for Wilson primes, also see Crandall et al. 1997 It has been conjecturedThe Prime Glossary: Wilson prime that  there are an infinite number of Wilson primes, and that the number of Wilson primes in an interval  is about

.

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

Notes

References








Other websites
 The Prime Glossary: Wilson prime

 Status of the search for Wilson primes
 Wilson Quotients for composite moduli
 On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson

Category:Classes of prime numbers