Wilson Prime

A Prime satisfying

$$W(p)\equiv 0\ \left({{\rm mod\ } {p}}\right),$$

where $W(p)$ is the Wilson Quotient, or equivalently,

$$(p-1)!\equiv -1\ \left({{\rm mod\ } {p^2}}\right).$$

5, 13, and 563 (Sloane's A007540) are the only Wilson primes less than $5\times 10^8$ (Crandall et al. 1997).

References

Crandall, R.; Dilcher, K; and Pomerance, C. ``A search for Wieferich and Wilson Primes.'' Math. Comput. 66, 433-449, 1997.

Ribenboim, P. ``Wilson Primes.'' §5.4 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 346-350, 1996.

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 73, 1991.