{"paper":{"title":"On the divisibility of some truncated hypergeometric series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Guo-Shuai Mao, Hao Pan","submitted_at":"2018-01-07T16:57:43Z","abstract_excerpt":"Let $p$ be an odd prime and $r\\geq 1$. Suppose that $\\alpha$ is a $p$-adic integer with $\\alpha\\equiv2a\\pmod p$ for some $1\\leq a<(p+r)/(2r+1)$. We confirm a conjecture of Sun and prove that $${}_{2r+1}F_{2r}\\bigg[\\begin{matrix}\\alpha&\\alpha&\\ldots&\\alpha\\\\ &1&\\ldots&1\\end{matrix}\\bigg|\\,1\\bigg]_{p-1}\\equiv0\\pmod{p^2},$$ where the truncated hypergeometric series $$ {}_{q+1}F_{q}\\bigg[\\begin{matrix}x_0&x_1&\\ldots&x_{q}\\\\ &y_1&\\ldots&y_q\\end{matrix}\\bigg|\\,z\\bigg]_{n}:=\\sum_{k=0}^n\\frac{(x_0)_k(x_1)_k\\cdots(x_q)_k}{(y_1)_k\\cdot (y_q)_k}\\cdot\\frac{z^k}{k!}. $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.02213","kind":"arxiv","version":2},"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"}