A $p$-adic ($p\equiv 3\!\!\pmod 4$) depth-$5$ supercongruence for Gaussian $p$-th power sums over a square

Abstract

Let $p$ be an odd prime. Define the Gaussian power sum \[ G_n(p)=\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+bi)^n\in\mathbb Z[i]. \] We determine $G_p(p)$ modulo high powers of $p$: if $p\equiv 1\pmod 4$ then $$G_p(p)\equiv p^2(1+i)\pmod{p^3},$$ while for $p\equiv 3\pmod 4, p\ge 7$ we prove the supercongruence \[ G_p(p)\equiv -\frac{p^5}{12}(p-1)^2(p-2)\,B_{p-3}\,(1-i)\pmod{p^6}, \] where $B_m$ denotes the $m$-th Bernoulli number. We also formulate several conjectures suggested by extensive computations.