Wilson’s Theorem

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The chapter on Lagrange in E. T. Bell’s Men of Mathematics refers to an elegant theorem named the Wilson’s theorem after the English mathematician, John Wilson. It was proved by Lagrange in 1773.

The theorem states that for any integer p greater than 1, p is prime if and only if (p-1)! + 1 is divisible by p.

In other words, for any prime number p, if all numbers 1,2,…,p-1 are multiplied and 1 be added to the result, the sum is divisible by p.

As an example, consider the prime number 5. According to this theorem: (5-1)! + 1 = 1x2x3x4 +1 = 25 which is divisible by 5.

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