On the equation \(x^{p^{m-1}}+y^{p^{m-1}}+z^{p^{m-1}} \equiv 0\bmod p^m\). (Q974002)

Two elements of \({\mathbb Z}^3\) which are congruent modulo \((p{\mathbb Z})^3\) are called equivalent. Let \({\mathcal S}_m\) be the set of non-trivial solutions of the Fermat congruence \[ x^{p^{m-1}}+y^{p^{m-1}}+z^{p^{m-1}}\equiv 0\bmod p^m, \] and let \(\Sigma_m:={\mathcal S}_m/\sim \) be the set of equivalence classes of non-trivial solutions, considered as a subset of \(({\mathbb F}_p^*)^3\). Let \(S_m(p)\) be the number of elements in \(\Sigma_m\). For \(n\geq 1\) the \(n\)-th Wendt number is defined by \[ W_n:=\det\left(\binom{n}{|i-j|}\right)_{1\leq i,j\leq n}=\text{Res}_{n,n}(X^n-1,(X+1)^n-1). \] Denote by \(\text{val}_p\) the \(p\)-adic valuation. The author proves: Theorem 1. It holds that \[ \sum_{m\geq 1}S_m(p)=(p-1)\,\text{val}_p W_{p-1}. \] For \(n=6q\) the reduced Wendt number \(W'_q\) is defined by \[ W'_q:=\text{Res}_{n-2,n}\left(\frac{X^n-1}{X^2+X+1},(X+1)^n-1\right). \] A cyclic solution of the Fermat congruence is of the form \(b(1,a,a^2)\) with \(ab\neq 0\) and \((a,b)\in{\mathbb Z}^2\). Denote by \({\mathcal S}'_m\) the set of classes of non-trivial and non-cyclic solutions and by \(S'_m(p)\) its number of elements. Theorem 2. If \(p=1+6q\) then \[ \sum_{m\geq 1}S'_m(p)=6q\,\text{val}_p W'_q. \]