Almost fifth powers in arithmetic progression
type: Item
instance of: scholarly article
title: Almost fifth powers in arithmetic progression
zbMATH Open document ID: 1253.11045
author: Lajos Hajdu
author: Tünde Kovács
published in: Journal of Number Theory
publication date: 2011-07-22T00:00:00Z
review text: The Diophantine equation \[ x(x+d)\cdots (x+(k-1)d)=y^n \] has been studied by many authors. Recently important results were obtained using the very powerful modular method, but this method needs \(n\geq 7\). Many papers considered the cases \(n=2\) and \(3\), they used elliptic curves and quadratic residues. Here the authors consider the special case \(n=5\) and they use genus 2 curves and apply the Chabauty method. In particular, they prove that the product of \(k\) consecutive terms of a primitive arithmetic progression is never a fifth power when \(3\leq k \leq 54\).
reviewed by: Maurice Mignotte
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11B25
zbMATH DE Number: 5931164
zbMATH Keywords: perfect powers
zbMATH Keywords: arithmetic progression
zbMATH Keywords: genus 2 curves
zbMATH Keywords: Chabauty method
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: j.jnt.2011.04.009
OpenAlex ID: W1976425371
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DOI: 10.1016/J.JNT.2011.04.009
description: scientific article; zbMATH DE number 5931164