{"paper":{"title":"Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Victor J. W. Guo","submitted_at":"2012-01-03T12:40:50Z","abstract_excerpt":"For all nonnegative integers n, the Franel numbers are defined as $$ f_n=\\sum_{k=0}^n {n\\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on congruences for Franel numbers: \\sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\\equiv 0 \\pmod{2n^2}, \\sum_{k=0}^{p-1}(3k+2)(-1)^k f_k &\\equiv 2p^2 (2^p-1)^2 \\pmod{p^5}, where n is a positive integer and p>3 is a prime."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.0617","kind":"arxiv","version":3},"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"}