Wilson's Theorem
for every prime .
Theorem 4.35 (Wilson's Theorem)
Let be a prime. Then .
Proof
As an edge case the result is true for ; assume . Apart from self-inverse elements, the units modulo come in pairs where each pair multiplies to modulo ; a self-inverse element satisfies . By the lemma above, the only self-inverse elements are and , so the remaining units of come in inverse pairs. Hence
Related
Stated in
- Theorem 4.35 (Wilson's Theorem)ยง4.7 Prime Modular Arithmetic
