A $p$-adic ($p\equiv 3\!\!\pmod 4$) depth-$5$ supercongruence for Gaussian $p$-th power sums over a square
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911417511182336 |
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| author | Kalinin, Nikita Zottor, Faith Shadow |
| author_facet | Kalinin, Nikita Zottor, Faith Shadow |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_00206 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A $p$-adic ($p\equiv 3\!\!\pmod 4$) depth-$5$ supercongruence for Gaussian $p$-th power sums over a square Kalinin, Nikita Zottor, Faith Shadow General Mathematics 11B68, 11A07, 11F33, 11T30, 11R11 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. |
| title | A $p$-adic ($p\equiv 3\!\!\pmod 4$) depth-$5$ supercongruence for Gaussian $p$-th power sums over a square |
| topic | General Mathematics 11B68, 11A07, 11F33, 11T30, 11R11 |
| url | https://arxiv.org/abs/2602.00206 |