{"paper":{"title":"On some determinants arising from quadratic residues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chen-Kai Ren, Zhi-Wei Sun","submitted_at":"2024-04-17T16:43:17Z","abstract_excerpt":"Let $p>3$ be a prime, and let $d\\in\\mathbb Z$ with $p\\nmid d$. For the determinants $$S_m(d,p)=\\det\\left[(i^2+dj^2)^{m}\\right]_{1\\leqslant i,j \\leqslant (p-1)/2}\\ \\ \\left(\\frac{p-1}2\\leqslant m\\leqslant p-1\\right),$$ Sun recently determined $S_m(d,p)$ modulo $p$ when $m\\in\\{p-2,p-3\\}$ and $(\\frac {-d}p)=-1$. In this paper, we obtain $S_{p-2}(d,p)$ modulo $p$ in the remaining case $(\\frac{-d}p)=1$, and determine the Legendre symbols $(\\frac{S_{p-3}\\,(d,p)}p)$ and $(\\frac{S_{p-4}\\,(d,p)}p)$ in some special cases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.11547","kind":"arxiv","version":1},"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/2404.11547/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"}