{"paper":{"title":"A Pfaffian Proof and Generalization of a Conjecture of Sun Zhiwei","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Fei Liu, Hong-Ge Chen","submitted_at":"2026-07-02T04:39:20Z","abstract_excerpt":"Let $p$ be an odd prime, let $n=(p-1)/2$, and let $\\chi=(\\frac{\\cdot}{p})$, with $\\chi(0)=0$. For $a\\in\\mathbb F_p^\\times$ define \\[\n  D_a(x)=\\det_{1\\le i,j\\le n}(x+\\chi(i^2-aj)),\n  \\qquad\n  D_a^{(0)}(x)=\\det_{0\\le i,j\\le n}(x+\\chi(i^2-aj)). \\] We prove \\[\n  D_a(0)=0\n  \\quad\\Longleftrightarrow\\quad\n  p\\equiv 3 \\pmod 4\n  \\quad\\text{and}\\quad\n  \\chi(a n!)=1. \\] For $p\\equiv3\\pmod4$ we also give explicit Pfaffian-square factorizations of $D_a(x)$ and $D_a^{(0)}(x)$. Let $s_p=(-1)^{\\lfloor(p+1)/8\\rfloor}$. If $\\chi(a n!)=1$, then $s_pD_a(x)/x=s_pD_a^{(0)}(x)$ is a positive integer square. If $\\chi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01695","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/2607.01695/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"}