Perfect powers in products of arithmetical progressions with fixed initial term
type: Item
instance of: scholarly article
title: Perfect powers in products of arithmetical progressions with fixed initial term
zbMATH Open document ID: 0874.11034
DOI: 10.1016/S0019-3577(97)89137-9
published in: Indagationes Mathematicae. New Series
publication date: 1997-05-11T00:00:00Z
review text: The author considers the exponential diophantine equation \[ m(m+d)\cdots(m+ (k-1)d)= by^\ell\tag{\(*\)} \] in positive integers \(b\), \(d\), \(k\), \(\ell\), \(m\) satisfying \(P(b)\leq k\), \(\text{gcd}(m,d)= 1\), \(k>2\) and \(\ell\geq 2\). As usual, \(P(x)\) denotes the greatest prime factor of \(x\). The history of this equation goes back to Fermat, and many results have been obtained since. In the list of contributors both the author and his many times over joint author R. Tijdeman occupy prominent positions. P. Erdös conjectured that \((*)\) implies that \(k\) is bounded by an absolute constant. In the present paper, this conjecture is confirmed when \(\ell\geq 7\) and \(m\) fixed. This is an immediate consequence of the following elegant result: If \(\ell\geq 7\) in \((*)\) then \[ m\geq k^{C\log\log k}, \] where \(C>0\) is an effectively computable absolute constant.
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11B25
zbMATH DE Number: 989754
zbMATH Keywords: perfect powers
zbMATH Keywords: products of arithmetical progressions
zbMATH Keywords: exponential diophantine equation
author: T. N. Shorey
reviewed by: Roelof J. Stroeker
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 989754