{"paper":{"title":"Congruences for Franel numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2011-12-05T19:38:56Z","abstract_excerpt":"The Franel numbers given by $f_n=\\sum_{k=0}^n\\binom{n}{k}^3$ ($n=0,1,2,\\ldots$) play important roles in both combinatorics and number theory. In this paper we initiate the systematic investigation of fundamental congruences for the Franel numbers. We mainly establish for any prime $p>3$ the following congruences:\n  \\begin{align*}\\sum_{k=0}^{p-1}(-1)^kf_k&\\equiv\\left(\\frac p3\\right)\\ \\ (\\mbox{mod}\\ p^2), \\\\ \\sum_{k=0}^{p-1}(-1)^k\\,kf_k&\\equiv-\\frac 23\\left(\\frac p3\\right)\\ \\ (\\mbox{mod}\\ p^2), \\\\ \\sum_{k=1}^{p-1}\\frac{(-1)^k}kf_k &\\equiv0\\ \\ (\\mbox{mod}\\ p^2), \\\\ \\sum_{k=1}^{p-1}\\frac{(-1)^k}{k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.1034","kind":"arxiv","version":11},"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"}