{"paper":{"title":"Legendre symbols related to certain determinants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xin-Qi Luo, Zhi-Wei Sun","submitted_at":"2022-10-26T14:19:47Z","abstract_excerpt":"Let $p$ be an odd prime. For $b,c\\in\\mathbb Z$, Sun introduced the determinant $$D_p(b,c)=\\left|(i^2+bij+cj^2)^{p-2}\\right|_{1\\leqslant i,j \\leqslant p-1},$$ and investigated the Legendre symbol $(\\frac{D_p(b,c)}p)$. Recently Wu, She and Ni proved that $(\\frac{D_p(1,1)}p)=(\\frac {-2}p)$ if $p\\equiv2\\pmod 3$, which confirms a previous conjecture of Sun. In this paper we determine $(\\frac{D_p(1,1)}p)$ in the case $p\\equiv1\\pmod3$. Sun proved that $D_p(2,2)\\equiv0\\pmod p$ if $p\\equiv3\\pmod4$, in contrast we prove that $(\\frac{D_p(2,2)}p)=1$ if $p\\equiv1\\pmod8$, and $(\\frac{D_p(2,2)}p)=0$ if $p\\eq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.14741","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/2210.14741/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"}