{"paper":{"title":"Two congruences involving harmonic numbers with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Guo-Shuai Mao, Zhi-Wei Sun","submitted_at":"2014-11-28T14:07:57Z","abstract_excerpt":"The harmonic numbers $H_n=\\sum_{0<k\\le n}1/k\\ (n=0,1,2,\\ldots)$ play important roles in mathematics. Let $p>3$ be a prime. With helps of some combinatorial identities, we establish the following two new congruences: $$\\sum_{k=1}^{p-1}\\frac{\\binom{2k}k}kH_k\\equiv\\frac13\\left(\\frac p3\\right)B_{p-2}\\left(\\frac13\\right)\\pmod{p}$$ and $$\\sum_{k=1}^{p-1}\\frac{\\binom{2k}k}kH_{2k}\\equiv\\frac7{12}\\left(\\frac p3\\right)B_{p-2}\\left(\\frac13\\right)\\pmod{p},$$ where $B_n(x)$ denotes the Bernoulli polynomial of degree $n$. As an application, we determine $\\sum_{n=1}^{p-1}g_n$ and $\\sum_{n=1}^{p-1}h_n$ modulo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.0523","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"}