{"paper":{"title":"Some supercongruences modulo $p^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2011-01-05T19:08:46Z","abstract_excerpt":"Let $p>3$ be a prime, and let $m$ be an integer with $p\\nmid m$. In the paper we prove some supercongruences concerning $$\\align &\\sum_{k=0}^{p-1}\\frac{\\binom{2k}k\\binom{3k}k}{54^k},\\ \\sum_{k=0}^{p-1}\\frac{\\binom{2k}k\\binom{4k}{2k}}{128^k},\\ \\sum_{k=0}^{p-1}\\frac{\\binom{3k}k\\binom{6k}{3k}}{432^k}, \n  &\\sum_{k=0}^{p-1}\\frac{\\binom{2k}k^2\\binom{3k}{k}}{m^k}, \\sum_{k=0}^{p-1}\\frac{\\binom{2k}k^2\\binom{4k}{2k}}{m^k},\\ \\sum_{k=0}^{p-1}\\f{\\binom{2k}k\\binom{3k}{k}\\binom{6k}{3k}}{m^k}\\mod {p^2}.\\endalign$$ Thus we solve some conjectures of Zhi-Wei Sun and the author."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.1050","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"}