Wilson prime

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

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

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 $[x,y]$ 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

Notes

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

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[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

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