{"paper":{"title":"Evaluation of a determinant involving Legendre symbols","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":"2025-07-24T17:14:21Z","abstract_excerpt":"Let $p>3$ be a prime, and let $(\\frac{\\cdot}p)$ be the Legendre symbol. Let $A_p(x)$ denote the matrix $[x+a_{ij}]_{1\\leqslant i,j\\leqslant (p-1)/2}$, where $$ a_{ij}=\\begin{cases} (\\frac{j}{p}) &\\text{if} \\ i=1, \\$\\frac{i+j}{p}) &\\text{if} \\ i>1. \\end{cases}$$ In 2018 Z.-W. Sun conjectured that $\\det A_p(0)=-2^{(p-3)/2}$ if $p\\equiv 3 \\pmod{4}$, which was later confirmed by G. Zaimi. In this paper we evaluate $\\det A_p(x)$ completely."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.18589","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/2507.18589/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"}