Wilson's Theorem
Iff 
is a Prime, then 
is a multiple of , that is
This theorem was proposed by John Wilson in 1770 and proved by Lagrange
in 1773. Unlike Fermat's LittleTheorem, Wilson's theorem is both Necessary and Sufficient for primality. For a Composite Number,
except when 
.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.
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