{"paper":{"title":"Congruences for Ap\\'ery numbers $\\beta_{n}=\\sum_{k=0}^{n}\\binom{n}{k}^2\\binom{n+k}{k}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hui-Qin Cao, Yuri Matiyasevich, Zhi-Wei Sun","submitted_at":"2018-12-26T15:58:42Z","abstract_excerpt":"In this paper we establish some congruences involving the Ap\\'ery numbers $\\beta_{n}=\\sum_{k=0}^{n}\\binom{n}{k}^2\\binom{n+k}{k}$ $(n=0,1,2,\\ldots)$. For example, we show that $$\\sum_{k=0}^{n-1}(11k^2+13k+4)\\beta_k\\equiv0\\pmod{2n^2}$$ for any positive integer $n$, and $$\\sum_{k=0}^{p-1}(11k^2+13k+4)\\beta_k\\equiv 4p^2+4p^7B_{p-5}\\pmod{p^8}$$ for any prime $p>3$, where $B_{p-5}$ is the $(p-5)$th Bernoulli number. We also present certain relations between congruence properties of the two kinds of A\\'pery numbers, $\\beta_n$ and $A_n=\\sum_{k=0}^n\\binom nk^2\\binom{n+k}k^2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.10351","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1812.10351/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}