On the equation \(\sum ^{s}_{j=1}(1/x_ j)+(1/(x_ 1\,\dots \,x_ s))=1\) and Znám's problem
type: Item
instance of: scholarly article
title: On the equation \(\sum ^{s}_{j=1}(1/x_ j)+(1/(x_ 1\,\dots \,x_ s))=1\) and Znám's problem
zbMATH Open document ID: 0621.10013
DOI: 10.1016/0022-314X(87)90062-X
author: Rui Liu
author: Liangrui Zhang
author: Zhenfu Cao
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: Consider the equations \[ (1)\quad \sum^{s}_{j=1}(1/x_ j)+(1/(x_ 1...x_ s))=1,\quad 1<x_ 1<x_ 2<...<x_ s \] and \[ (2)\quad \sum^{s}_{j=1}(1/x_ j)-(1/(x_ 1...x_ s))=1,\quad 1<x_ 1<x_ 2<...<x_ s,\quad s>2. \] Znám asked whether there exists an integer \(x_ i\) for every integer \(s>1\) such that \(x_ i\) is a proper factor of \(x_ 1...x_{i-1}x_{i+1}...x_ s+1\) for \(i=1,...,s.\) Let \(\Omega\) (s) be the number of solutions of (1), Z(s) be the number of solutions of Znám's problem and A(s) be the number of solutions of (2). In this paper, the authors give all the 23 integer solutions of (1) when \(s=7\). They also prove that (i) if \(s\geq 11\), then \(\Omega (s+1)\geq \Omega (s)+8\) and if 2 \(\nmid s\), \(s\geq 11\), then \(\Omega (s+1)\geq \Omega (s)+9;\) (ii) if \(s\geq 12\), then Z(s)\(\geq 8\) and if 2 \(| s\), \(s\geq 12\), then Z(s)\(\geq 9;\) (iii) if \(s\geq 10\), then \(A(s+1)\geq \Omega (s)+\Omega (s-1)+16\) and if 2 \(| s\), \(s\geq 12\), then \(A(s+1)\geq \Omega (s)+\Omega (s-1)+18\).
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 4006338
zbMATH Keywords: sums of unit fractions
MaRDI profile type: Publication
Wikidata QID: Q56504609
cites work: Q3310697
cites work: Q3208717
cites work: Q3658048
reviewed by: Ke Zhao
description: scientific article; zbMATH DE number 4006338