{"paper":{"title":"Some new results on determinants and permanents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bo Jiang, Zhi-Wei Sun","submitted_at":"2026-06-02T17:53:57Z","abstract_excerpt":"In this paper we confirm several conjectures on determinants and permanents. For example, we prove that for any prime $p\\equiv3\\pmod 4$ the number $2\\det[a_{jk}]_{0\\le j,k\\le (p-1)/2}$ is congruent to a square modulo $p$, where $a_{jk}=(\\frac{j+k}{p})+(\\frac{j^2+k^2}{p})$ with $(\\frac{\\cdot}{p})$ the Legendre symbol. We also prove that ${\\rm per}[j^{k-1}]_{1\\leq j,k\\leq n-1}\\equiv0\\pmod n$ for any integer $n>1$ with $n\\not\\equiv2\\pmod 4$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.03970","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/2606.03970/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"}