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.

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 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.

Related pages

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

References

• N. G. W. H. Beeger (1913-1914). "Quelques remarques sur les congruences r^p-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. doi:10.1112/jlms/s1-28.2.252.
• Paulo Ribenboim (1996). The new book of prime number records. Springer-Verlag. pp. 346. ISBN 0-387-94457-5.
• Richard E. Crandall; Karl Dilcher, Carl Pomerance (1997). "A search for Wieferich and Wilson primes". Math. Comput. 66 (217): 433–449. doi:10.1090/S0025-5718-97-00791-6.
• Richard E. Crandall; Carl Pomerance (2001). Prime Numbers: A Computational Perspective. Springer-Verlag. p. 29. ISBN 0-387-94777-9.
• Takashi Agoh; Karl Dilcher, Ladislav Skula (1998). "Wilson quotients for composite moduli". Math. Comput. 67 (222): 843-861. http://en.wikipedia.org/w/index.php?title=Wilson_prime&action=edit&section=4.
• Erna H. Pearson (1963). "On the Congruences (p-1)! ≡ -1 and 2^p-1 ≡ 1 (mod p²)". Math. Comput. 17: 194-195. http://www.ams.org/journals/mcom/1963-17-082/S0025-5718-1963-0159780-0/S0025-5718-1963-0159780-0.pdf.

Other websites

• The Prime Glossary: Wilson prime
• Weisstein, Eric W., "Wilson prime" from 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