Wilson's Theorem

Iff $p$ is a Prime, then $(p-1)!+1$ is a multiple of $p$, that is

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

This theorem was proposed by John Wilson in 1770 and proved by Lagrange in 1773. Unlike Fermat's Little Theorem, Wilson's theorem is both Necessary and Sufficient for primality. For a Composite Number, $(n-1)!\equiv 0\ \left({{\rm mod\ } {n}}\right)$ except when $n=4$.

See also Fermat's Little Theorem, Wilson's Theorem Corollary, Wilson's Theorem (Gauss's Generalization)

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 61, 1987.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 142-143 and 168-169, 1996.

Ore, Ø. Number Theory and Its History. New York: Dover, pp. 259-261, 1988.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 37-38, 1993.