# Wilson prime

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

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

where n is a positive integer (sometimes called natural number). Wilson primes were first described by Emma Lehmer.[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.[2] It has been conjectured[3] that there are an infinite number of Wilson primes, and that the number of Wilson primes in an interval [x , y] is about

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

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

## Notes

1. On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of. Math. 39(1938), 350-360.
2. Status of the search for Wilson primes, also see Crandall et al. 1997
3. The Prime Glossary: Wilson prime

## References

• N. G. W. H. Beeger (1913-1914). "Quelques remarques sur les congruences rp-1 ≡ 1 (mod p2) et (p-1!) ≡ -1 (mod p2)". The Messenger of Mathematics 43: 72-84.
• Karl Goldberg (1953). "A table of Wilson quotients and the third Wilson prime". J. Lond. Math. Soc. 28: 252–256.
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• Richard E. Crandall; Karl Dilcher, Carl Pomerance (1997). "A search for Wieferich and Wilson primes". Math. Comput. 66 (217): 433–449.
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• Richard E. Crandall; Carl Pomerance (2001). Prime Numbers: A Computational Perspective. Springer-Verlag. p. 29.
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