{"paper":{"title":"Arithmetic properties of some permanents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Zhi-Wei Sun","submitted_at":"2021-08-17T16:59:16Z","abstract_excerpt":"In this paper we study arithmetic properties of some permanents, many of which involve trigonometric functions. For any primitive $n$-th root $\\zeta$ of unity, we obtain closed formulas for the permanents $$\\mathrm{per}\\left[1-\\zeta^jx_k\\right]_{1\\le j,k\\le n}\\ \\ \\text{and}\\ \\ \\mathrm{per}\\left[\\frac1{1-\\zeta^{j-k}x}\\right]_{1\\le j,k\\le n}.$$ Another typical result states that for any odd integer $n>1$ we have $$t_n:=\\frac1{\\sqrt n}\\mathrm{per}\\left[\\tan\\pi\\frac{jk}n\\right]_{1\\le j,k\\le (n-1)/2}\\in\\mathbb Z,$$ and that $t_p\\equiv(-1)^{(p+1)/2}\\pmod p$ for any odd prime $p$. We also pose severa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.07723","kind":"arxiv","version":7},"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/2108.07723/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"}