{"paper":{"title":"Jacobsthal sums, Legendre polynomials and binary quadratic forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2012-02-06T18:41:27Z","abstract_excerpt":"Let $p>3$ be a prime and $m,n\\in\\Bbb Z$ with $p\\nmid mn$. Built on the work of Morton, in the paper we prove the uniform congruence: $$&\\sum_{x=0}^{p-1}\\Big(\\frac{x^3+mx+n}p\\Big) \\equiv {-(-3m)^{\\frac{p-1}4} \\sum_{k=0}^{p-1}\\binom{-\\frac 1{12}}k\\binom{-\\frac 5{12}}k (\\frac{4m^3+27n^2}{4m^3})^k\\pmod p&\\t{if $4\\mid p-1$,} \\frac{2m}{9n}(\\frac{-3m}p)(-3m)^{\\frac{p+1}4} \\sum_{k=0}^{p-1}\\binom{-\\frac 1{12}}k\\binom{-\\frac 5{12}}k (\\frac{4m^3+27n^2}{4m^3})^k\\pmod p&\\text{if $4\\mid p-3$,}$$ where $(\\frac ap)$ is the Legendre symbol. We also establish many congruences for $x\\pmod p$, where $x$ is given "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1237","kind":"arxiv","version":3},"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"}