{"paper":{"title":"Some congruences related to a congruence of Van Hamme","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Ji-Cai Liu, Victor J. W. Guo","submitted_at":"2019-03-09T08:57:30Z","abstract_excerpt":"We establish some supercongruences related to a supercongruence of Van Hamme, such as \\begin{align*} \\sum_{k=0}^{(p+1)/2} (-1)^k (4k-1)\\frac{(-\\frac{1}{2})_k^3}{k!^3} &\\equiv p(-1)^{(p+1)/2}+p^3(2-E_{p-3})\\pmod{p^{4}},\\\\ \\sum_{k=0}^{(p+1)/2} (4k-1)^5 \\frac{(-\\frac{1}{2})_k^4}{k!^4} &\\equiv 16p\\pmod{p^{4}}, \\end{align*} where $p$ is an odd prime and $E_{p-3}$ is the $(p-3)$-th Euler number. Our proof uses some congruences of Z.-W. Sun, the Wilf--Zeilberger method, Whipple's $_7F_6$ transformation, and the software package {\\tt Sigma} developed by Schneider. We also put forward two related conje"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.03766","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"}