Squares from blocks of consecutive integers: a problem of Erdős and Graham
type: Item
instance of: scholarly article
title: Squares from blocks of consecutive integers: a problem of Erdős and Graham
zbMATH Open document ID: 1270.11026
author: Michael A. Bennett
author: Ronald van Luijk
published in: Indagationes Mathematicae. New Series
publication date: 2012-02-25T00:00:00Z
review text: Let \(n, k\in\mathbb{N}\) and put \[ f(n,k)=n(n+1)\cdots(n+k-1). \] An old result of \textit{P. Erdős} and \textit{J. Selfridge} [Ill. J. Math. 19, 292--301 (1975; Zbl 0295.10017)] states that the \(f(n,k)\) is never a perfect power of an integer, provided \(n\geq 1\) and \(k\geq 2.\) \textit{P. Erdős} and \textit{R. Graham} [Old and new problems and results in combinatorial number theory. Gene\`{v}e: L'Enseignement Mathématique, Université de Gene\`{v}e (1980; Zbl 0434.10001)] asked whether a more general Diophantine equation \[ \prod_{j=1}^{r}f(n_{j},k_{j})=x^2 \tag{1} \] could have, for fixed \(r\geq 1\) and \(\{k_{1},k_{2},\ldots,k_{r}\}\) with \(k_{j}\geq 4\) for \(j=1,2,\ldots,r\), at most finitely many solutions in positive integers \((n_{1},n_{2},\ldots,n_{r},x)\) satisfying the condition \[ n_{j}+k_{j}\leq n_{j+1}\tag{2} \] for each \(j\in\{1,\ldots,r-1\}\). The last condition guarantees that the corresponding blocks of integers are disjoint. The authors concentrate on the case \(k_{1}=k_{2}=\ldots=k_{r}=5\) and prove that if \(r\geq 5\) then there are infinitely many integers solutions of equation (1) satisfying (2), and thus answering the question of Erdős and Graham in the negative.
reviewed by: Maciej Ulas
Mathematics Subject Classification ID: 11D09
zbMATH DE Number: 6009292
zbMATH Keywords: Diophantine equations
zbMATH Keywords: arithmetic progressions
zbMATH Keywords: product of consecutive integers
MaRDI profile type: Publication
full work available at URL: j.indag.2011.11.002
OpenAlex ID: W2040953949
cites work: On a question of Erdős and Graham
cites work: Q3872528
cites work: The product of consecutive integers is never a power
cites work: On a diophantine equation related to a conjecture of Erdös and Graham
cites work: On products of disjoint blocks of consecutive integers
DOI: 10.1016/J.INDAG.2011.11.002
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description: scientific article; zbMATH DE number 6009292