On a problem of Bleicher and Erdős
type: Item
instance of: scholarly article
title: On a problem of Bleicher and Erdős
zbMATH Open document ID: 0652.10015
DOI: 10.1016/0022-314X(88)90017-0
author: H. Yokota
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: Let \(\mathbb Z_ 0\) be the set of positive integers and \(a,N\in\mathbb Z_ 0\) be such that \(a<N\). By an Egyptian fraction expansion of \(a/N\), we mean \(a/N=\sum_{i=1}^{k}1/n_ i\), \(n_ 1<n_ 2<...<n_ k\), where \(n_ i\in\mathbb Z_ 0\). Let \[ D(a,N)=\min \{n_ k: a/N=\sum_{i=1}^{k}1/n_ i,\quad n_ 1<n_ 2<\cdot \cdot \cdot <n_ k,\quad n_ i\in\mathbb Z_ 0\}, \] where the minimum ranges over all Egyptian fraction expansions of \(a/N\) and let \(D(N)=\max \{D(a,N): 1\leq a\leq N\}\). The author proves that \(D(N)/N\leq (\log N)^{1+\delta (N)},\) \(\delta(N) \to 0\) as \(N\to \infty\), establishing a conjecture of M. N. Bleicher and P. Erdős.
Mathematics Subject Classification ID: 11D68
zbMATH DE Number: 4063187
zbMATH Keywords: sum of unit fractions
zbMATH Keywords: Egyptian fraction expansion
Wikidata QID: Q56227640
MaRDI profile type: Publication
cites work: Denominators of Egyptian fractions
cites work: Denominators of Egyptian fractions. II
cites work: Q3872528
cites work: On a Problem of Additive Number Theory
cites work: Approximate formulas for some functions of prime numbers
cites work: On a conjecture of M. N. Bleicher and P. Erdős
cites work: Length and denominators of Egyptian fractions
reviewed by: Ke Zhao
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description: scientific article; zbMATH DE number 4063187