{"paper":{"title":"On determinants involving $(\\frac{j+k}p)\\pm(\\frac{j-k}p)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Deyi Chen, Zhi-Wei Sun","submitted_at":"2024-09-12T16:53:53Z","abstract_excerpt":"Let $p=2n+1$ be an odd prime. In this paper, we mainly evaluate determinants involving $(\\frac {j+k}p)\\pm(\\frac{j-k}p)$, where $(\\frac{\\cdot}p)$ denotes the Legendre symbol. When $p\\equiv1\\pmod4$, we determine the characteristic polynomials of the matrices $$\\left[\\left(\\frac{j+k}p\\right)+\\left(\\frac{j-k}p\\right)\\right]_{1\\le j,k\\le n}\\ \\ \\text{and}\\ \\ \\left[\\left(\\frac{j+k}p\\right)-\\left(\\frac{j-k}p\\right)\\right]_{1\\le j,k\\le n},$$ and also establish the general identity \\begin{align*} &\\ \\left|x+\\left(\\frac{j+k}p\\right)+\\left(\\frac{j-k}p\\right)+\\left(\\frac jp\\right)y+\\left(\\frac kp\\right)z+\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.08213","kind":"arxiv","version":4},"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/2409.08213/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"}