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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 $p$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2107.08984","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-07-19T15:58:31Z","cross_cats_sorted":[],"title_canon_sha256":"4c9310cc8f6a68f2e7c2356d8b69bcca2da3ed6a42156d1247567cd2764590a1","abstract_canon_sha256":"5cf347966e27ae6f7b2f7e0c1c9d21b04cfb709024cf78a4431f6b59e2762fec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:04:02.193734Z","signature_b64":"pj0Ok92G7umEHSTiGSPSGd8udY5mpK5bTSPl46lJcyqC2ZZ2G6tQeqzJSoMMlfDRetdhD/42FCDS7ZoSm24rAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b1b272b07b254fa0ac36cbe37f75f18ab9693b01b878bbe521211c9dd6499f0","last_reissued_at":"2026-07-05T05:04:02.193401Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:04:02.193401Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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