Wilson's Theorem

    This theorem is named after one of Edward Waring's students, John Wilson. But actually, Wilson only observed the result to be true and did not provide its proof. Later Euler proved Wilson's theorem and then Gauss gave a generalization. Basically, the theorem states that the factorial of a prime number minus one is congruent to minus one modulo the prime. Interestingly, its converse is also well-known and gives a sufficient condition for an integer to be a prime.  

Proposition (Wilson's Theorem) If is prime, then

    Proof. The cases and are evident. Choose any integer from the list Recall that the linear congruence has a unique solution since Notice that if and then and so there are exactly pairs left. If we omit the numbers and the effect is to group the remaining integers into pairs and where such that their product When these congruences are multiplied together and the factors are rearranged, we get
    


and thus whence as desired.   

Example (Wilson's Theorem) Illustrate the proof of Wilson's theorem with

    Solution. For we have which we can prove by using the congruences











When these congruences are multiplied together and the factors are rearranged, we get
    


and thus whence as desired.   

Proposition (Converse of Wilson's Theorem) If is a positive integer such that then is prime.

    Proof. Assume that is not prime; and so there is some nontrivial factor of for which   ; whence since for some integer Therefore, can not exist and so is prime.

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