On the product of consecutive elements of an arithmetic progression
type: Item
instance of: scholarly article
title: On the product of consecutive elements of an arithmetic progression
zbMATH Open document ID: 0582.10011
DOI: 10.1007/BF01299269
published in: Monatshefte für Mathematik
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178250
review text: \textit{P. Erdős} and \textit{J. L. Selfridge} [Ill. J. Math. 19, 292-301 (1975; Zbl 0295.10017)] proved that the product of \(k\geq 2\) consecutive integers cannot be a proper power and asked for an analogous result concerning the product of consecutive integers of an arithmetic progression. Using the method of the above-mentioned paper, the author proves that the product \((n+d)(n+2d)\cdot \cdot \cdot (n+kd)\) cannot be a proper power provided k is sufficiently large; in fact, for all cases, the largest k is \[ \leq \quad \max \{3\cdot 10^ 4, (3/2) \exp [d(d+2)(d+1)^{1/3}]\}. \]
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11B25
zbMATH DE Number: 3931105
zbMATH Keywords: higher degree diophantine equations
zbMATH Keywords: perfect power
zbMATH Keywords: product of consecutive integers
zbMATH Keywords: arithmetic progression
author: Roman Marszałek
reviewed by: Edward L. Cohen
MaRDI profile type: Publication
cites work: Approximate formulas for some functions of prime numbers
cites work: The product of consecutive integers is never a power
description: scientific article; zbMATH DE number 3931105