Wilson's Theorem

Iff  is a prime, then  is a multiple of , that is

(1)

This theorem was proposed by John Wilson and published by Waring (1770), although it was previously known to Leibniz. It was proved by Lagrange in 1773. Unlike Fermat's little theorem, Wilson's theorem is both necessary and sufficient for primality. For a composite number,  except when .

A corollary to the theorem states that iff a prime  is of the form , then

(2)

The first few primes of the form  are , 13, 17, 29, 37, 41, ... (OEIS A002144), corresponding to , 3, 4, 7, 9, 10, 13, 15, 18, 22, 24, 25, 27, 28, 34, 37, ... (OEIS A005098).

Gauss's generalization of Wilson's theorem considers  the product of integers that are less than or equal to and relatively prime to an integer . For , 2, ..., the first few values are 1, 1, 2, 3, 24, 5, 720, 105, 2240, 189, ... (OEIS A001783). Then defining

(3)

gives the congruence

(4)

for  an odd prime. When , this reduces to  which is equivalent to . The first few values of  are 0, , , , , , , 1, , , , ... (OEIS A103131).

Szántó (2005) notes that defining

			(5)
			(6)

then, taking the minimal residue,

(7)

For , 1, ..., the first terms are then 0, , 1, 1, 0, , 1, 0, , , 0, ... (OEIS A112448).

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.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 167, 2003.

Hilton, P.; Holton, D.; and Pedersen, J. Mathematical Reflections in a Room with Many Mirrors. New York: Springer-Verlag, pp. 41-42, 1997.

Nagell, T. "Wilson's Theorem and Its Generalizations." Introduction to Number Theory. New York: Wiley, pp. 99-101, 1951.

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

Séroul, R. "Wilson's Theorem." §2.9 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 16-17, 2000.

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

Sloane, N. J. A. Sequences A001783/M0921, A002144/M3823, A005098, A103131, and A112448 in "The On-Line Encyclopedia of Integer Sequences."

Szántó, S. "The Proof of Szántó's Note." https://www.dkne.hu/Proof.html.

Waring, E. Meditationes Algebraicae. Cambridge, England: University Press, 1770.