Wilson prime

A Wilson prime, named after English mathematician John Wilson, is a prime number p such that p² divides (p − 1)! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1.

The only known Wilson primes are 5, 13, and 563 (sequence A007540 in OEIS); if any others exist, they must be greater than 4×10¹¹.^[2] It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval [x, y] is about log(log(y)/log(x)).^[3]

Several computer searches have been done in the hope of finding new Wilson primes. ^[4][5][6] The Ibercivis distributed computing project includes a search for Wilson primes.^[7] Another search is coordinated at the mersenneforum. ^[8]

Near-Wilson primes

• A prime p satisfying the congruence (p − 1)! ≡ − 1 + Bp (mod p²) with small |B| can be called a near-Wilson prime. Near-Wilson primes with B = 0 represent Wilson primes. The following table lists all such primes with |B| ≤ 100 from 10⁶ up to 4×10¹¹:

See also

• Wieferich prime
• Wall–Sun–Sun prime
• Wolstenholme prime

Notes

1. ^ Lehmer, E. (April 1938). "On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson". Annals of Mathematics 39 (2): 350–360. doi:10.2307/1968791. http://gradelle.educanet2.ch/christian.aebi/.ws_gen/14/Emma_Lehmer_1938.pdf. Retrieved 8 March 2011. 
2. ^ Status of a search for Wilson primes (Data collected from mersenneforum.org.) Retrieved on November 3, 2011.
3. ^ The Prime Glossary: Wilson prime
4. ^ McIntosh, R. (9 March 2004). "WILSON STATUS (Feb. 1999)". E-Mail to Paul Zimmermann. http://www.loria.fr/~zimmerma/records/Wieferich.status. Retrieved 6 June 2011. 
5. ^ A search for Wieferich and Wilson primes, p 443
6. ^ Ribenboim, P.; Keller, W. (2006) (in German). Die Welt der Primzahlen: Geheimnisse und Rekorde. Berlin Heidelberg New York: Springer. p. 241. ISBN 3-540-34283-4. http://books.google.de/books?id=-nEM9ZVr4CsC&pg=PA248&dq=die+welt+der+primzahlen+rodenkirch&hl=de&ei=LbLsTfG8G8XdsgamnLXnCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDQQ6AEwAA#v=onepage&q&f=false. 
7. ^ Ibercivis site
8. ^ Distributed search for Wilson primes (at mersenneforum.org)

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. 
• 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. doi:10.1090/S0025-5718-98-00951-X. 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. 

External links

• 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