{"paper":{"title":"Minimal zero-sum sequences of length four over cyclic group with order $n=p^\\alpha q^\\beta$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Caixia Shen, Li-meng Xia","submitted_at":"2014-01-30T23:09:22Z","abstract_excerpt":"Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\\cdot...\\cdot(n_kg)$ where $g\\in G$ and $n_1,\\cdots,n_k\\in[1,{\\hbox{\\rm ord}}(g)]$, and the index $\\ind S$ of $S$ is defined to be the minimum of $(n_1+\\cdots+n_k)/\\hbox{\\rm ord}(g)$ over all possible $g\\in G$ such that $\\langle g\\rangle=G$. A conjecture says that if $G$ is finite such that $\\gcd(|G|,6)=1$, then $\\ind(S)=1$ for every minimal zero-sum sequence $S$. In this paper, we prove that the conjecture holds if $|G|$ has two prime factors."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.8021","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"}