Wilson's theorem

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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 ( n − 1 ) ! = 1 × 2 × 3 × ⋯ × ( n − 1 ) {\displaystyle (n-1)!=1\times 2\times 3\times \cdots \times (n-1)} satisfies ( n − 1 ) ! ≡ − 1 ( mod n ) {\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.

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