On theorems of Fermat, Wilson, and Gegenbauer (Q6634702)

The well-known Fermat's Little Theorem states that for any prime \(p\) and any integer \(a\),\N\[\Na^p \equiv a \pmod{p}.\N\]\NWilson's theorem, on the other hand, states that for any prime p,\N\[\N(p-1)! \equiv -1 \pmod{p}.\N\]\NThe Moser-Moser congruence is an unified extension of the above two theorems,\N\[ \N\sum_{d\mid n}\varphi^{2}(d) \left(\frac{n}{d}\right)!\,d^{n/d}a^{n/d} \equiv 0 \pmod{n^2},\N\]\Nwhere \(\varphi(d)\) is the Euler's totient function. There are two proofs of the this congruence. Evans used the action of the group \(\mathbb Z/n\mathbb Z\) on the set of Hamilton cycles with \(n\) vertices, whereas Moser used the action of the group \(\mathbb Z/n\mathbb Z \times \mathbb Z/n\mathbb Z\) on the set of line permutations of \(\{1, 2, \ldots, n\}\). The paper under review mainly provides a new proof of the Moser-Moser congruence by using the Frobenius-Burnside theorem.\N\NThe paper under review also proves a generalization of the Moser-Moser congruence. Let \(n, a\), and \(b\) be positive integers. Then\N\[\N\sum_{d|n}\varphi^{2}(d) \left(\frac{n}{d}\right)!d^{n/d}a^{n/d}b^{d} \equiv 0 \pmod{n^2}.\N\]