{"paper":{"title":"Congruences concerning Legendre polynomials II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2010-12-17T15:00:29Z","abstract_excerpt":"Let $p>3$ be a prime, and let $m$ be an integer with $p\\nmid m$. In the paper we solve some conjectures of Z.W. Sun concerning $\\sum_{k=0}^{p-1}\\binom{2k}k^3/m^k\\pmod{p^2}$, $\\sum_{k=0}^{p-1}\\binom{2k}k\\b{4k}{2k}/m^k\\pmod p$ and $\\sum_{k=0}^{p-1}\\binom{2k}k^2\\b{4k}{2k}/m^k\\pmod {p^2}.$ In particular, we show that $\\sum_{k=0}^{\\frac{p-1}{2}}\\binom{2k}k^3\\equiv 0\\pmod {p^2}$ for $p\\equiv 3,5,6\\pmod 7$. Let $P_n(x)$ be the Legendre polynomials. In the paper we also show that $ P_{[\\frac {p}{4}]}(t)\\equiv -\\big(\\frac{-6}{p}\\big)\\sum_{x=0}^{p-1} \\big(\\frac{x^3-3/2(3t+5)x-9t-7}{p}\\big)\\pmod p$ and d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3898","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"}