A $p$-adic ($p\equiv 3\!\!\pmod 4$) depth-$5$ supercongruence for Gaussian $p$-th power sums over a square
Nikita Kalinin, Faith Shadow Zottor
TL;DR
The paper establishes a p-adic depth for Gaussian power sums at prime square arguments by analyzing \\mathbf{G}_p(p) in \\mathbb{Z}[i]. It proves a split-prime congruence \\mathbf{G}_p(p) ≡ p^2(1+i) \\pmod{p^3} when p ≡ 1 (mod 4), and a refined inert-prime congruence involving the Bernoulli number B_{p−3} modulo p^6 for p ≥ 7 with p ≡ 3 (mod 4): \\mathbf{G}_p(p) ≡ -\frac{p^5}{12}(p−1)^2(p−2) B_{p−3} (1−i) \\pmod{p^6}. The approach relies on a binomial–power–sum factorization, a Bernoulli-number truncation modulo p^6, and a careful partition of endpoint, near-endpoint, and bulk contributions, together with extensive computations that motivate several conjectures about p-adic valuations, exceptional blocks, and irregular indices. The results link p-adic depth to splitting behavior in \\mathbb{Z}[i] and to classical Bernoulli obstructions, offering a Gaussian analogue of Carlitz–Staudt phenomena and highlighting rich arithmetic structure in finite Gaussian sums with potential applications to higher-order supercongruences.
Abstract
Let $p$ be an odd prime. Define the Gaussian power sum \[ \G_n(p)=\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+b\ii)^n\in\ZZ[\ii]. \] We determine $\G_p(p)$ modulo high powers of $p$: if $p\equiv 1\pmod 4$ then $$\G_p(p)\equiv p^2(1+\ii)\pmod{p^3},$$ while for $p\equiv 3\pmod 4, p\ge 7$ we prove the supercongruence \[ \G_p(p)\equiv -\frac{p^5}{12}(p-1)^2(p-2)\,B_{p-3}\,(1-\ii)\pmod{p^6}, \] where $B_m$ denotes the $m$-th Bernoulli number. We also formulate several conjectures suggested by extensive computations.