{"paper":{"title":"On a Problem of Erd\\H{o}s, Herzog and Sch\\\"onheim","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Cui-Ying Hu, Yong-Gao Chen","submitted_at":"2011-02-28T02:33:35Z","abstract_excerpt":"Let $p_1, p_2,..., p_n$ be distinct primes.\n  In 1970, Erd\\H os, Herzog and Sch\\\"{o}nheim proved that if $\\cal D$ is a set of divisors of $N=p_1^{\\alpha_1}...p_n^{\\alpha_n}$, $\\alpha_1\\ge \\alpha_2\\ge...\\ge \\alpha_n$, no two members of the set being coprime and if no additional member may be included in $\\cal D$ without contradicting this requirement then $ |{\\cal D}|\\ge \\alpha_n \\prod_{i=1}^{n-1} (\\alpha_i +1)$. They asked to determine all sets $\\cal D$ such that the equality holds. In this paper we solve this problem. We also pose several open problems for further research."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.5574","kind":"arxiv","version":2},"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"}