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Wilson Prime

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A Wilson prime is a prime p satisfying

W(p)=0 (mod p),

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

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

The known Wilson primes are 5, 13, and 563 (OEIS A007540). Crandall et al. (1997) reported no other Wilson primes below 5×10^8 (McIntosh 2004). Costa et al. (2014) increased the search limit to 2×10^(13).

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." Math. Comput. 83, 3071-3091, 2014. https://doi.org/10.1090/S0025-5718-2014-02800-7.Crandall, R.; Dilcher, K.; and Pomerance, C. "A Search for Wieferich and Wilson Primes." Math. Comput. 66, 433-449, 1997. https://doi.org/10.1090/S0025-5718-97-00791-6.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.Le Lionnais, F. Les nombres remarquables. Paris, France: Hermann, p. 56, 1983.McIntosh, R. email to Paul Zimmermann. 9 Mar 2004. https://members.loria.fr/PZimmermann/records/Wieferich.status.Mersenne Forum. "Wilson-Prime Search Practicalities." https://www.mersenneforum.org/node/11314.Numberphile. "What Do 5, 13 and 563 Have in Common?" Aug. 3, 2014. https://www.youtube.com/watch?v=eZUa5k_VIZg.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 Resource. https://mathworld.wolfram.com/WilsonPrime.html

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