Topology of Numbers · Problem 6.4.1

The Class Group for Quadratic Forms — Allen Hatcher

Mathematics — Number Theory · wilson-theorem-converse

Problem

As a sort of converse to Wilson's Theorem, show that if nn is not a prime then (n−1)!(n-1)! is not congruent to −1-1 mod nn. More precisely, when n>4n > 4 and nn is not prime, show that nn divides (n−1)!(n-1)!, so (n−1)!≡0 mod n(n-1)! \equiv 0 \bmod n. What happens when n=4n = 4?

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