{"paper":{"title":"Some q-analogues of (super)congruences of Beukers, Van Hamme and Rodriguez-Villegas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Jiang Zeng, Victor J. W. Guo","submitted_at":"2014-08-03T17:03:06Z","abstract_excerpt":"For any odd prime p we obtain q-analogues of Van Hamme's supercongruence: $$ \\sum_{k=0}^{\\frac{p-1}{2}}{2k\\choose k}^3\\frac{1}{64^k} \\equiv 0 \\pmod{p^2} \\quad\\text{for}\\quad p\\equiv 3\\pmod 4, $$ and Rodriguez-Villegas' Beukers-like supercongruences involving products of three binomial coefficients. For example, we prove that \\begin{align*} \\sum_{k=0}^{\\frac{p-1}{2}} {2k\\brack k}_{q^2}^3 \\frac{q^{2k}}{(-q^2;q^2)_k^2 (-q;q)_{2k}^2} &\\equiv 0\\pmod{[p]^2} \\quad\\text{for}\\quad p\\equiv 3\\pmod 4, \\\\ \\sum_{k=0}^{\\frac{p-1}{2}}{2k\\brack k}_{q^3}\\frac{(q;q^3)_k (q^{2};q^3)_{k} q^{3k} }{ (q^{6};q^{6})_k^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.0512","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"}