Innumber theory, aWilson prime is aprime number${\displaystyle p}$ such that${\displaystyle p^2}$divides${\displaystyle (p-1)!+1}$, where "${\displaystyle !}$" denotes thefactorial function; compare this withWilson's theorem, which states that every prime${\displaystyle p}$ divides${\displaystyle (p-1)!+1}$. Both are named for 18th-centuryEnglish mathematicianJohn Wilson; in 1770,Edward Waring credited the theorem to Wilson,^[1] although it had been stated centuries earlier byIbn al-Haytham.^[2]

The only known Wilson primes are5,13, and563 (sequenceA007540 in theOEIS). Costa et al. write that "the case${\displaystyle p=5}$ is trivial", and credit the observation that 13 is a Wilson prime toMathews (1892).^[3][4] Early work on these numbers included searches byN. G. W. H. Beeger andEmma Lehmer,^[5][3][6] but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem.^[3][7][8] If any others exist, they must be greater than 2 × 10¹³.^[3] It has beenconjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval${\displaystyle [x,y]}$ is about${\displaystyle \log {\log }_{x}y}$.^[9]

Several computer searches have been done in the hope of finding new Wilson primes.^[10][11][12]TheIbercivisdistributed computing project includes a search for Wilson primes.^[13] Another search was coordinated at theGreat Internet Mersenne Prime Search forum.^[14]

Wilson's theorem can be expressed in general as${\displaystyle (n-1)!(p-n)!\equiv (-1{)}^{n}modp}$ for everyinteger${\displaystyle n\ge 1}$ and prime${\displaystyle p\ge n}$. Generalized Wilson primes of ordern are the primesp such that${\displaystyle p^2}$ divides${\displaystyle (n-1)!(p-n)!-(-1{)}^n}$.

It was conjectured that for everynatural numbern, there are infinitely many Wilson primes of ordern.

The smallest generalized Wilson primes of order${\displaystyle n}$ are:

A prime${\displaystyle p}$ satisfying thecongruence${\displaystyle (p-1)!\equiv -1+Bp(\mathrm{mod}{p}^2)}$ with small${\displaystyle |B|}$ can be called anear-Wilson prime. Near-Wilson primes with${\displaystyle B=0}$ are bona fide Wilson primes. The table on the right lists all such primes with${\displaystyle |B|\le 100}$ from 10⁶ up to 4×10¹¹.^[3]

AWilson number is a natural number${\displaystyle n}$ such that${\displaystyle W(n)\equiv 0(\mathrm{mod}{n}^2)}$, where$${\displaystyle W(n)=\pm 1+\prod _{\stackrel{1\le k\le n}{gcd(k,n)=1}}k,}$$and where the${\displaystyle \pm 1}$ term is positiveif and only if${\displaystyle n}$ has aprimitive root and negative otherwise.^[15] For every natural number${\displaystyle n}$,${\displaystyle W(n)}$ is divisible by${\displaystyle n}$, and the quotients (called generalizedWilson quotients) are listed in (sequenceA157249 in theOEIS). The Wilson numbers are

If a Wilson number${\displaystyle n}$ is prime, then${\displaystyle n}$ is a Wilson prime. There are 13 Wilson numbers up to 5×10⁸.^[16]

• PrimeGrid
• Table of congruences
• Wall–Sun–Sun prime
• Wieferich prime
• Wolstenholme prime

• Crandall, Richard E.; Pomerance, Carl (2001).Prime Numbers: A Computational Perspective. Springer-Verlag. p. 29.ISBN 978-0-387-94777-8.
• Pearson, Erna H. (1963)."On the Congruences (p − 1)! ≡ −1 and 2^p−1 ≡ 1 (mod p²)"(PDF).Math. Comput.17:194–195.

• The Prime Glossary: Wilson prime
• Weisstein, Eric W."Wilson prime".MathWorld.
• Status of the search for Wilson primes

• Classes of prime numbers
• Factorial and binomial topics
• Unsolved problems in number theory

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