Residue Classes with Order 1 or 2 and a Generalisation of Wilson’s Theorem

Proof. Suppose first that (m− 1)! ≡ −1 (mod m) for some positive integer m. If m is not prime then there exists a divisor d of m with 1 < d < m, so d|(m− 1)!. But d|m, so d| − 1, a contradiction. Thus, m must be prime. Suppose now that m is prime. If some residue class x modulo m has got a multiplicative inverse x with x 6≡ x (mod m) then they both drop out of (m−1)!. Hence, (m−1)! is congruent to the product of all integers x with 1 ≤ x ≤ m− 1 and x ≡ 1 (mod m). However, since m is prime, x ≡ 1 (mod m) ⇔ (x− 1)(x+ 1) ≡ 0 (mod m) ⇔ x ≡ 1 (mod m) or x ≡ m− 1 (mod m).