Equal sums of like powers, both positive and negative
type: Item
instance of: scholarly article
title: Equal sums of like powers, both positive and negative
zbMATH Open document ID: 1243.11049
DOI: 10.1216/RMJ-2011-41-3-737
author: Ajai Choudhry
published in: Rocky Mountain Journal of Mathematics
publication date: 2011-08-02T00:00:00Z
review text: In the literature there are several results concerning the problem of finding two sets of integers \(x_1,\dots,x_s\) and \(y_1,\dots,y_s\) such that \(\sum_{i=1}^s x_i^r=\sum_{i=1}^s y_i^r\) for certain exponents \(r=k_1,\dots,k_n\). In the introduction the author mentions several related results; now we only recall the famous Tarry-Escott problem, where \(r=1,\dots,n\). The author takes up the scarcely investigated problem where the exponent \(r\) can take negative values, as well. More generally, the author considers so-called chains, i.e. the problem of finding not only two, but \(t\geq 2\) different sets of integers having the same sums of \(r\)-th powers, where \(r\) runs through a fixed finite set of (possibly both positive and negative) integers. Among other results, it is proved that for any fixed set of exponents, one can find an \(s\), and \(t\) sets of integers \(x_{i1},\dots,x_{it}\) \((i=1,\dots,s)\), forming a chain. In certain cases parametric and numerical solutions are also given.
reviewed by: Lajos Hajdu
Mathematics Subject Classification ID: 11D72
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 5934867
zbMATH Keywords: equal sums of powers
zbMATH Keywords: multigrade equations
zbMATH Keywords: multigrade chains
zbMATH Keywords: Tarry-Escott problem
MaRDI profile type: Publication
cites work: The Prouhet-Tarry-Escott problem revisited
cites work: A Geometric Approach to Equal Sums of Sixth Powers
cites work: A geometric approach to equal sums of fifth powers
cites work: Ideal Solutions of the Tarry-Escott Problem
cites work: On equal sums of sixth powers.
cites work: Ideal solutions of the Tarry-Escott problem of degree four and a related Diophantine system
cites work: Ideal solutions of the Tarry-Escott problem of degrees four and five and related Diophantine systems.
cites work: The Tarry-Escott Problem
cites work: Equal sums of like powers
description: scientific article; zbMATH DE number 5934867