{"paper":{"title":"Supercongruences involving Lucas sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2016-10-11T18:17:12Z","abstract_excerpt":"For $A,B\\in\\mathbb Z$, the Lucas sequence $u_n(A,B)\\ (n=0,1,2,\\ldots)$ are defined by $u_0(A,B)=0$, $u_1(A,B)=1$, and $u_{n+1}(A,B) = Au_n(A,B)-Bu_{n-1}(A,B)$ $(n=1,2,3,\\ldots).$ For any odd prime $p$ and positive integer $n$, we establish the new result $$\\frac{u_{pn}(A,B) - (\\frac{A^2-4B}p) u_n(A,B)}{pn} \\in \\mathbb Z_p,$$ where $(\\frac{\\cdot}p)$ is the Legendre symbol and $\\mathbb Z_p$ is the ring of $p$-adic integers.\n  Let $p$ be an odd prime and let $n$ be a positive integer. For any integer $m\\not\\equiv0\\pmod p$, we show that $$\\frac1{pn}\\bigg(\\sum_{k=0}^{pn-1} \\frac{\\binom{2k}k}{m^k} -"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.03384","kind":"arxiv","version":8},"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/1610.03384/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"}