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ZixuanZhang
ZixuanZhang
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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

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