{"paper":{"title":"A new theorem on quadratic residues modulo primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hao Pan, Qing-Hu Hou, Zhi-Wei Sun","submitted_at":"2021-07-19T15:58:31Z","abstract_excerpt":"Let $p>3$ be a prime, and let $(\\frac{\\cdot}p)$ be the Legendre symbol. Let $b\\in\\mathbb Z$ and $\\varepsilon\\in\\{\\pm 1\\}$. We mainly prove that $$\\left|\\left\\{N_p(a,b):\\ 1<a<p\\ \\text{and}\\ \\left(\\frac ap\\right)=\\varepsilon\\right\\}\\right|=\\frac{3-(\\frac{-1}p)}2,$$ where $N_p(a,b)$ is the number of positive integers $x<p/2$ with $\\{x^2+b\\}_p>\\{ax^2+b\\}_p$, and $\\{m\\}_p$ with $m\\in\\mathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.08984","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2107.08984/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"}