On a generalization of a problem of Erdős and Graham
type: Item
instance of: scholarly article
zbMATH Open document ID: 1324.11040
DOI: 10.5486/PMD.2014.5862
published in: Publicationes Mathematicae Debrecen
publication date: 2014-08-14T00:00:00Z
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11Y50
zbMATH DE Number: 6330564
zbMATH Keywords: Diophantine equations
zbMATH Keywords: Runge's method
author: Szabolcs Tengely
author: Nóra Varga
MaRDI profile type: Publication
title: On a generalization of a problem of Erdős and Graham
review text: Let \(f(x,k,d)\) be the product of \(k\) terms in an arithmetic progression of difference \(d\) and first term \(x\). \textit{P. Erdős} and \textit{J. L. Selfridge} [Ill. J. Math. 19, 292--301 (1975; Zbl 0295.10017)] proved that \(f(x,k,1)\) is never a perfect power of exponent \(\geq 2\) if \(x\geq 1\) and \(k\geq 2\) are integers. Their result spurred a lot of activity especially by Shorey and his collaborators on Diophantine equations involving perfect powers in \(f(x,k,d)\). In the paper under review, the authors study for fixed distinct integers \(a,b\) the integers \(x\) such that \(x(x+1)(x+2)(x+3)/(x+a)(x+b)\) is a square of a rational number. The main observation is instead of dividing out \((x+a)(x+b)\) from \(x(x+1)(x+2)(x+3)\) and getting a square of a rational, one might as well multiply the two expressions getting a square of an integer. The resulting equation is of the form \(f(x)=y^2\), where \(f\) is a monic polynomial with integer coefficients of degree \(6\), so one can use Runge's method to bound the integer solutions. The author's main results are explicit upper bounds on \(| x| \) in terms of \(a\) and \(b\), and a complete list of solutions for all distinct \(a,b\in \{-4,-3,-2.-1,4,5,6,7\}\).
description: scientific article; zbMATH DE number 6330564