{"paper":{"title":"Lehmer's totient problem over $\\mathbb{F}_q[x]$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hourong Qin, Qingzhong Ji","submitted_at":"2013-12-11T10:35:01Z","abstract_excerpt":"In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\\in\\mathbb{F}_q[x]$ and $\\varphi(q,p(x))$ be the Euler's totient function of $p(x)$ over $\\mathbb{F}_q[x],$ where $\\mathbb{F}_q$ is a finite field with $q$ elements. We prove that $\\varphi(q,p(x))|(q^{{\\rm deg}(p(x))}-1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3, \\; p(x)$ is the product of any $2$ non-associate irreducibes of degree $1;$ or (iii) $q=2,\\; p(x)$ is the product of all irreducibles of degree $1,$ all irreducibles of degree $1$ and $2,$ and the product of any $3$ irreducib"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3107","kind":"arxiv","version":3},"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"}