{"paper":{"title":"Proof of a supercongruence conjectured by Z.-H. Sun","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":"2014-04-28T13:49:05Z","abstract_excerpt":"The Franel numbers are defined by $\nf_n=\\sum_{k=0}^n {n\\choose k}^3. $\nMotivated by the recent work of Z.-W. Sun on Franel numbers, we prove that \\begin{align*} \\sum_{k=0}^{n-1}(3k+1)(-16)^{n-k-1} {2k\\choose k} f_k &\\equiv 0\\pmod{n{2n\\choose n}}, \\\\ \\sum_{k=0}^{p-1}\\frac{3k+1}{(-16)^k} {2k\\choose k} f_k &\\equiv p (-1)^{\\frac{p-1}{2}} \\pmod{p^3}. \\end{align*} where $n>1$ and $p$ is an odd prime. The second congruence modulo $p^2$ confirms a recent conjecture of Z.-H. Sun. We also show that, if $p$ is a prime of the form $4k+3$, then $$ \\sum_{k=0}^{p-1}\\frac{{2k\\choose k} f_k}{(-16)^k} \\equiv 0 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.6978","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"}