{"paper":{"title":"New Congruences on Multiple Harmonic Sums and Bernoulli Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2015-04-13T15:49:16Z","abstract_excerpt":"Let ${\\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p \\ge 11$ and integer $r\\ge 2$, we prove that $$ \\sum\\limits_{\\begin{smallmatrix}\n  {{l}_{1}}+{{l}_{2}}+\\cdots +{{l}_{6}}={{p}^{r}}\n  {{l}_{1}},\\cdots ,{{l}_{6}}\\in {\\mathcal{P}_{p}} \\end{smallmatrix}}{\\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}{l}_{6}}}\\equiv - \\frac{{5!}}{18}p^{r-1}B_{p-3}^{2} \\pmod{{{p}^{r}}}. $$ This extends a family of curious congruences. We also obtain other interesting congruences involving multiple harmonic sums and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03227","kind":"arxiv","version":4},"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"}