Powerful arithmetic progressions
type: Item
instance of: scholarly article
title: Powerful arithmetic progressions
zbMATH Open document ID: 1186.11017
DOI: 10.1016/S0019-3577(09)00012-3
published in: Indagationes Mathematicae. New Series
publication date: 2009-10-22T00:00:00Z
review text: A \textit{powerful arithmetic progression} is a progression whose \(k\)th term is a perfect \(k\)th power for all \(k\). An arithmetic progression \(a_1\), \(a_2\), \dots\ is called \textit{primitive} if \(\gcd(a_1,a_2)=1\). In 1998, K. D. Bolkan asked the following question: what is the length of the longest nonconstant powerful arithmetic progression? In 2000, J. P. Robertson proved that the answer is five. In the present paper, the author proves in particular the following results: -- The only primitive powerful arithmetic progression of length five is the trivial one 1,~1,~1,~1,~1. -- The are infinitely many powerful arithmetic progressions of length four. -- The are infinitely many pairwise nonproportional powerful arithmetic progressions of length five. The proof of the first of these results is obtained by a deep study of the Diophantine equation \(3X^4-Y^2=2Z^5\), involving in particular the elliptic Chabauty method.
reviewed by: Maurice Mignotte
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11Y50
zbMATH DE Number: 5619293
zbMATH Keywords: perfect powers
zbMATH Keywords: arithmetic progression
author: Lajos Hajdu
describes a project that uses: Magma
MaRDI profile type: Publication
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full work available at URL: s0019-3577(09)00012-3
OpenAlex ID: W2008283246
description: scientific article; zbMATH DE number 5619293