{"paper":{"title":"On determinants involving $(\\frac{j^2-k^2}p)$ and $(\\frac{jk}p)$","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":"2024-08-26T16:34:29Z","abstract_excerpt":"Let $p$ be an odd prime and let $(\\frac{\\cdot}p)$ be the Legendre symbol. In this paper, we study the determinant $$\\det\\left[\\left(\\frac{j^2-k^2}p\\right)+\\left(\\frac{jk}p\\right)w\\right]_{\\delta\\le j,k\\le (p-1)/2}$$ with $\\delta\\in\\{0,1\\}$. For example, we prove that the determinant does not depend on $w$ if $p\\equiv3\\pmod4$ and $\\delta=0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.14401","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/2408.14401/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"}