{"paper":{"title":"Upper Bounds for Cyclotomic Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bernhard Schmidt, Ka Hin Leung, Tai Do Duc","submitted_at":"2019-03-18T09:13:47Z","abstract_excerpt":"Let $q$ be a power of a prime $p$, let $k$ be a nontrivial divisor of $q-1$ and write $e=(q-1)/k$. We study upper bounds for cyclotomic numbers $(a,b)$ of order $e$ over the finite field $\\mathbb{F}_q$. A general result of our study is that $(a,b)\\leq 3$ for all $a,b \\in \\mathbb{Z}$ if $p> (\\sqrt{14})^{k/ord_k(p)}$. More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: $(0,0), (0,a), (a,0), (a,a)$ and $(a,b)$, where $a\\neq b$ and $a,b \\in \\{1,\\dots,e-1\\}$. The main idea we use is to transform equations over $\\mathbb{F}_q$ into equation"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.07314","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":""},"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"}