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Wilson Prime

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A Wilson prime is a prime satisfying

W(p)=0 (mod p),

where W(p) is the Wilson quotient, or equivalently,

(p-1)!=-1 (mod p^2).

The first few Wilson primes are 5, 13, and 563 (OEIS A007540). Crandall et al. (1997) showed there are no others less than 5×10^8 (McIntosh 2004), a limit that has subsequently been increased to 2×10^(13) (Costa et al. 2012).

See also

Brown Numbers, Wilson Quotient, Wilson's Theorem

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References

Costa, E.; Gerbicz, R.; and Harvey, D. "A Search for Wilson Primes." 5 Dec 2012. http://arxiv.org/abs/1209.3436.Crandall, R.; Dilcher, K; and Pomerance, C. "A search for Wieferich and Wilson Primes." Math. Comput. 66, 433-449, 1997.Gonter, R. H. and Kundert, E. G. "All Numbers Up to 18876041 Have Been Tested without Finding a New Wilson Prime." Preprint, 1994.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 167, 2003.McIntosh, R. email to Paul Zimmermann. 9 Mar 2004. http://www.loria.fr/~zimmerma/records/Wieferich.status.Mersenne Forum. "Wilson-Prime Search Practicalities." http://www.mersenneforum.org/showthread.php?t=16028.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983.Ribenboim, P. "Wilson Primes." §5.4 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 346-350, 1996.Sloane, N. J. A. Sequence A007540/M3838 in "The On-Line Encyclopedia of Integer Sequences."Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 73, 1991.

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Wilson Prime

Cite this as:

Weisstein, Eric W. "Wilson Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WilsonPrime.html

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