Wilson prime

A Wilson prime is a special kind of prime number. A prime number p is a Wilson prime if (and only if [ ${\displaystyle \iff }$ ])

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

where n is a positive integer (sometimes called natural number). Wilson primes were first described by Emma Lehmer.^[1]

The only known Wilson primes are 5, 13, and 563 (sequence A007540 in the 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 ${\displaystyle [x,y]}$ is about

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

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

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

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

• 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 (2): 252–256. doi:10.1112/jlms/s1-28.2.252.
• Ribenboim, Paulo (1996). The new book of prime number records. Springer-Verlag. p. 346. ISBN 978-0-387-94457-9.
• Crandall, Richard; Dilcher, Karl; Pomerance, Carl (1997). "A search for Wieferich and Wilson primes". Math. Comput. 66 (217): 433–449. Bibcode:1997MaCom..66..433C. doi:10.1090/S0025-5718-97-00791-6.
• Crandall, Richard E.; Pomerance, Carl (2001). Prime Numbers: A Computational Perspective. Springer-Verlag. p. 29. ISBN 978-0-387-94777-8.
• Agoh, Takashi; Dilcher, Karl; Skula, Ladislav (1998). "Wilson quotients for composite moduli". Math. Comput. 67 (222): 843–861. Bibcode:1998MaCom..67..843A. doi:10.1090/S0025-5718-98-00951-X. S2CID 10988592.
• Erna H. Pearson (1963). "On the Congruences (p-1)! ≡ -1 and 2^p-1 ≡ 1 (mod p²)" (PDF). Math. Comput. 17: 194–195. doi:10.2307/2003642. JSTOR 2003642.

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

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