Wilson's theorem

In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is (using the notations of modular arithmetic), the factorial ${\displaystyle (n-1)!=1\times 2\times 3\times \cdots \times (n-1)}$ satisfies

${\displaystyle (n-1)!\ \equiv \;-1{\pmod {n}}}$

exactly when n is a prime number. In other words, any integer n > 1 is a prime number if, and only if, (n − 1)! + 1 is divisible by n.^[1]