{"paper":{"title":"Super Congruences Involving Multiple Harmonic Sums and Bernoulli Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jianqiang Zhao, Kevin Chen","submitted_at":"2017-02-27T17:48:25Z","abstract_excerpt":"Let $m$, $r$ and $n$ be positive integers. We denote by ${\\bf k}\\vdash n$ any tuple of odd positive integers ${\\bf k}=(k_1,\\dots,k_t)$ such that $k_1+\\dots+k_t=n$ and $k_j\\ge 3$ for all $j$. In this paper we prove that for every sufficiently large prime $p$ $$ \\sum_{\\substack{l_1+l_2+\\cdots+l_n=mp^r p\\nmid l_1 l_2 \\cdots l_n }} \\frac1{l_1l_2\\cdots l_n} \\equiv p^{r-1} \\sum_{{\\bf k}\\vdash n} C_{m,{\\bf k}} B_{p-{\\bf k}} \\pmod{p^r} $$ where $B_{p-{\\bf k}}=B_{p-k_1}B_{p-k_2}\\cdots B_{p-k_t}$ are products of Bernoulli numbers and the coefficients $C_{m,{\\bf k}}$ are polynomials of $m$ independent of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.08401","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}