{"paper":{"title":"On certain determinants and the square root of some determinants involving Legendre Symbols","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chen-Kai Ren, Xin-Qi Luo","submitted_at":"2024-07-05T14:39:12Z","abstract_excerpt":"Let $p>3$ be a prime and $(\\frac{.}{p})$ be the Legendre symbol. For any integer $d$ with $p\\nmid d$ and any positive integer $m$, Sun introduced the determinants $$T_m(d,p)=\\det\\left[(i^2+dj^2)^m\\left(\\frac{i^2+dj^2}{p}\\right)\\right]_{1\\leqslant i,j \\leqslant (p-1)/2},$$ and $$D_p^{(m)}= \\det\\left[(i^2-j^2)^m\\left(\\frac{i^2-j^2}{p}\\right)\\right]_{1\\leq i,j\\leq (p-1)/2} .$$ In this paper, we obtain some properties of $T_m (d,p)$ and $ \\sqrt{D_p^{(m)}}$ for some $m$. We also confirm some related conjectures posed by Zhi-Wei Sun."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.04556","kind":"arxiv","version":2},"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/2407.04556/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"}