{"paper":{"title":"Quadratic residues and quartic residues modulo primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2018-10-22T15:56:13Z","abstract_excerpt":"In this paper we study some products related to quadratic residues and quartic residues modulo primes. Let $p$ be an odd prime and let $A$ be any integer. We mainly determine completely the product $$f_p(A):=\\prod_{1\\le i,j\\le(p-1)/2\\atop p\\nmid i^2-Aij-j^2}(i^2-Aij-j^2)$$ modulo $p$; for example, if $p\\equiv1\\pmod4$ then $$f_p(A)\\equiv\\begin{cases}-(A^2+4)^{(p-1)/4}\\pmod p&\\text{if}\\ (\\frac{A^2+4}p)=1, \\\\(-A^2-4)^{(p-1)/4}\\pmod p&\\text{if}\\ (\\frac{A^2+4}p)=-1,\\end{cases}$$ where $(\\frac{\\cdot}p)$ denotes the Legendre symbol. We also determine $$\\prod^{(p-1)/2}_{i,j=1\\atop p\\nmid 2i^2+5ij+2j^2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.12102","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/1810.12102/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"}