{"paper":{"title":"Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ji-Cai Liu, Victor J. W. Guo","submitted_at":"2015-01-03T15:54:44Z","abstract_excerpt":"The numbers $R_n$ and $W_n$ are defined as \\begin{align*} R_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k}\\frac{1}{2k-1},\\ \\text{and}\\ W_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k}\\frac{3}{2k-3}. \\end{align*} We prove that, for any positive integer $n$ and odd prime $p$, there hold \\begin{align*} \\sum_{k=0}^{n-1}(2k+1)R_k^2 &\\equiv 0 \\pmod{n}, \\\\ \\sum_{k=0}^{p-1}(2k+1)R_k^2 &\\equiv 4p(-1)^{\\frac{p-1}{2}} -p^2 \\pmod{p^3}, \\\\ 9\\sum_{k=0}^{n-1}(2k+1)W_k^2 &\\equiv 0 \\pmod{n}, \\\\ \\sum_{k=0}^{p-1}(2k+1)W_k^2 &\\equiv 12p(-1)^{\\frac{p-1}{2}}-17p^2 \\pmod{p^3}, \\quad\\text{if $p>3$.} \\end{align*} The firs"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00573","kind":"arxiv","version":1},"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"}