Almost perfect powers in products of consecutive integers
type: Item
instance of: scholarly article
title: Almost perfect powers in products of consecutive integers
zbMATH Open document ID: 1096.11014
published in: Monatshefte für Mathematik
publication date: 2005-06-14T00:00:00Z
review text: A first main object of the present paper is the study of the equation \[ m(m+1)\cdots (m+k-1)=by^n \tag{1} \] in the integer unknowns \(m,b,y\geq 1\) and \(n\geq 2\), where \(k\geq 2\) is an integer parameter. Since the publication, in 1975, of a famous result by Erdős and Selfridge, which says that the product of consecutive positive integers is never a perfect power, a rather extensive research around this and similar equations has been done and a number of very interesting relevant papers has been published. Let us denote by \(P(x)\) the greatest prime divisor of the integer \(x\) and by \(p_k\) the \(k\)-th prime. The first main result of the paper (Theorem 1) is the following: Under the restrictions \(P(b)\leq p_k\) and (only for the case \(k=2\)) \(n>2\), equation (1) can have solutions only in the following cases: (i) \(k=2\) and \(m\in\{1,2,3,8\}\), (ii) \(k=3\) and \(m\in\{1,2,3,4,8\}\), (iii) \(k=4\) and \(m\in\{1,\ldots, 7\}\), (iv) \(k=5\) and \(m\in\{1,\ldots, 8\}\). A second main object of the paper is the study of the equation (below the reviewer somewhat modifies the authors' notation) \[ m(m+d_1)\cdots (m+d_{k-2})=by^n\tag{2,} \] where \(k\geq 3\), in the posititve integer unknowns \(m,b\), \(d_1<\cdots <d_{k-2}\) and \(n\geq 2\) (with \(n>2\) when \(k=3\)). If \(k\in\{3,4,5\}\), then Theorem 2 gives explicitly in a table all cases for which equation (2) is possible under the restrictions \(d_{k-2}\leq k-1\) and \(P(b)\leq p_{k-1}\). For example, when \(k=4\), equation (2) is possible only with \((d_1,d_2)=(1,3)\) and \(m\in\{1,2,3,5,9,15,24\}\). For the proof of Theorem 2 the authors need to prove Theorem 3 which reads as follows: Let \(n\) be equal to 4 or to an odd prime. Then the equation \(m(m+3)=2^{\alpha}3^{\beta}y^n\) in integers \(m,y\geq 1\) and \(\alpha,\beta\geq 0\) is possible only for \(m=1,3,6,9,24\), except possibly if \(n\in I=\{19,23,29,31,37,41,43,47,73\}\) in which case further values of \(m\), with \(m\equiv 1,5\pmod{6}\) and \(…
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11D59
zbMATH DE Number: 2175898
zbMATH Keywords: exponential Diophantine equations
zbMATH Keywords: binomial Thue equations
zbMATH Keywords: product of consecutive integers
author: Kálmán Győry
author: Ákos Pintér
reviewed by: Nikos Tzanakis
MaRDI profile type: Publication
full work available at URL: s00605-004-0278-8
OpenAlex ID: W1996084695
DOI: 10.1007/S00605-004-0278-8
description: scientific article; zbMATH DE number 2175898