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On a Problem of Erd\H{o}s, Herzog and Sch\'onheim
- Source :
- Discrete Applied Mathematics 160(2012) 1501-1506
- Publication Year :
- 2011
-
Abstract
- Let $p_1, p_2,..., p_n$ be distinct primes. 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.<br />Comment: 12 pages
- Subjects :
- Mathematics - Combinatorics
Mathematics - Number Theory
05D05, 11A05
Subjects
Details
- Database :
- arXiv
- Journal :
- Discrete Applied Mathematics 160(2012) 1501-1506
- Publication Type :
- Report
- Accession number :
- edsarx.1102.5574
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.dam.2012.02.013