{"paper":{"title":"Products of $k$ atoms in Krull monoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.AC","authors_text":"Qinghai Zhong, Yushuang Fan","submitted_at":"2015-08-14T13:41:17Z","abstract_excerpt":"Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. For $k\\in \\mathbb N$, let $\\mathcal U_k(H)$ denote the set of all $m\\in \\mathbb N$ with the following property: There exist atoms $u_1, \\ldots, u_k, v_1, \\ldots , v_m\\in H$ such that $u_1\\cdot\\ldots\\cdot u_k=v_1\\cdot\\ldots\\cdot v_m$. It is well-known that the sets $\\mathcal U_k (H)$ are finite intervals whose maxima $\\rho_k(H)=\\max \\mathcal U_k(H) $ depend only on $G$. If $|G|\\le 2$, then $\\rho_k (H) = k$ for every $k \\in \\mathbb N$. Suppose that $|G| \\ge 3$. An elementary counting argument sh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.03500","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"}