Wilson Theorem

Wilson’s Theorem says that when p is prime, the product of all nonzero residue classes modulo p, namely 1·2·3·…·(p−1), leaves a remainder of p−1, which is the same as −1 modulo p. The reason is that every number from 1 to p−1 has a unique multiplicative inverse modulo p, and these inverses pair up to give factors of 1, except for the self-inverse elements 1 and p−1; their product is −1. This makes the theorem both a deep structural fact about the arithmetic of primes and a useful one-line test for primality.

Use this when you need a theoretical primality criterion or when proving properties of modular inverses and residue classes. It is especially useful in advanced number theory proofs, though it is rarely practical for large-number primality testing because computing (p−1)! directly is expensive.

• Forgetting that the theorem is stated modulo p, so the result p−1 is the same as −1 only after reducing mod p
• Trying to use the theorem as a fast computational primality test for large p, even though factorial growth makes it inefficient
• Assuming the theorem works for composite p; it is an iff statement, so failure of the congruence means p is not prime