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. Quelques remarques sur les congruences rp-1 ≡ 1 (mod p2) et (p-1!) ≡ -1 (mod p2). The Messenger of Mathematics 43 (1913–1914). p. 72–84.
- Karl Goldberg. A table of Wilson quotients and the third Wilson prime. J. Lond. Math. Soc. 28 (2) (1953). p. 252–256. doi:10.1112/jlms/s1-28.2.252.
- Ribenboim, Paulo. The new book of prime number records (1996)Springer-Verlag. p. 346. ISBN 978-0-387-94457-9.
- Crandall, Richard. A search for Wieferich and Wilson primes. Math. Comput. 66 (217) (1997). p. 433–449. doi:10.1090/S0025-5718-97-00791-6.
- Crandall, Richard E.. Prime Numbers: A Computational Perspective (2001)Springer-Verlag. p. 29. ISBN 978-0-387-94777-8.
- Agoh, Takashi. Wilson quotients for composite moduli. Math. Comput. 67 (222) (1998). p. 843–861. doi:10.1090/S0025-5718-98-00951-X.
- Erna H. Pearson. On the Congruences (p-1)! ≡ -1 and 2p-1 ≡ 1 (mod p2). Math. Comput. 17 (1963). p. 194–195. doi:10.2307/2003642.[dead link]
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