A $p$-adic ($p\equiv 3\pmod 4$) depth-$5$ supercongruence for Gaussian $p$-th power sums over a square
For an odd prime $p$, define $G_n(p)= \sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+bi)^n \in \mathbb Z[i]$. We study the $p$-adic valuation of these Gaussian power sums for $n\leq p, n=r(p-1)$ and show that it is governed by the interaction of the fourfold symmetry of the square, ordinary power-sum congruences, and Bernoulli numbers. If $p\equiv3\pmod4$ and $p\ge7$, then we prove an unexpectedly deep supercongruence $G_p(p)\equiv -\frac{p^5}{12}(p-1)^2(p-2)(1-i)B_{p-3} \pmod{p^6}$.
math.GM↗