On the number of solutions of \(\sum^{11}_{i=1}\frac {1}{x_i}=1\) in distinct odd natural numbers
type: Item
instance of: scholarly article
title: On the number of solutions of \(\sum^{11}_{i=1}\frac {1}{x_i}=1\) in distinct odd natural numbers
zbMATH Open document ID: 1270.11033
author: R. Arce-Nazario
author: Raúl Figueroa
author: Francis N. Castro
published in: Journal of Number Theory
publication date: 2013-04-15T00:00:00Z
review text: The authors study solutions of the Diophantine equation \(\sum_{i=1}^{11} \frac{1}{x_i}=1\), where the \(x_i\) are odd positive integers. It has been mentioned several times in the literature that \(\sum_{i=1}^{k} \frac{1}{x_i}=1\) only has solutions in odd positive integers \(x_i\), if \(k\) is odd, and \(k\geq 9\). To list all 5 solutions with \(k=9\), is an easy exercise, (at least using a computer). When \(k = 11\) there are very many cases to study, and it is not at all easy to enumerate all solutions, as the number of solutions (and the potential search space) is increasing faster than exponentially: \textit{Y. Chen, C. Elsholtz} and \textit{L. Jiang} [Acta Arith. 154, No. 2, 109--123 (2012; Zbl 1271.11041)] proved the number of solutions is \(>c^{k^2}\) for some constant \(c>1\). For \(k=11\), this paper describes a computation to enumerate all 379 118 solutions. The CPU time was about 133 days. The authors also prove some partial results on the case \(k\geq 13\), and also that for \(k \geq 9\) there are solutions of the special type \(\sum_{i=1}^{k-1} \frac{1}{x_i} +\frac{1}{{\text lcm}(x_1, \ldots , x_{k-1})}=1\) in odd positive integers.
reviewed by: Christian Elsholtz
Mathematics Subject Classification ID: 11D68
Mathematics Subject Classification ID: 11D72
zbMATH DE Number: 6154105
zbMATH Keywords: Egyptian fractions
zbMATH Keywords: sum of unit fractions
Wikidata QID: Q114157583
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.11.011
OpenAlex ID: W1999964822
DOI: 10.1016/J.JNT.2012.11.011
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description: scientific article; zbMATH DE number 6154105