Wilson prime
A Wilson prime is a special kind of prime number. A prime number p is a Wilson prime if (and only if [ [math]\displaystyle{ \iff }[/math] ])
[math]\displaystyle{ \frac{(p-1)!+1}{p^2}=n }[/math]
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 [math]\displaystyle{ [x,y] }[/math] is about
[math]\displaystyle{ \frac{\log \left ( \log y \right )}{\log x} }[/math].
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 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 (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 2p-1 ≡ 1 (mod p2)" (PDF). Math. Comput. 17: 194–195. doi:10.2307/2003642. JSTOR 2003642.
Other websites
- 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