A $p$-adic ($p\equiv 3(\mathrm{mod}4)$) depth-$5$ supercongruence for Gaussian $p$-th power sums over a square

Nikita Kalinin, Faith Shadow Zottor

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…