Wilson’s theorem

Wilson’s theorem, in number theory, theorem that any prime p divides (p − 1)! + 1, where n! is the factorial notation for 1 × 2 × 3 × 4 × ⋯ × n. For example, 5 divides (5 − 1)! + 1 = 4! + 1 = 25. The conjecture was first published by the English mathematician Edward Waring in Meditationes Algebraicae (1770; “Thoughts on Algebra”), where he ascribed it to the English mathematician John Wilson.

The theorem was proved by the French mathematician Joseph-Louis Lagrange in 1771. The converse of the theorem is also true; that is, (n − 1)! + 1 is not divisible by a composite number n. In theory, these theorems provide a test for primes; in practice, the calculations are impractical for large numbers.