Almost perfect powers in consecutive integers. II
type: Item
instance of: scholarly article
title: Almost perfect powers in consecutive integers. II
zbMATH Open document ID: 1223.11039
DOI: 10.1016/S0019-3577(08)80027-4
author: N. Saradha
author: T. N. Shorey
published in: Indagationes Mathematicae. New Series
publication date: 2009-10-22T00:00:00Z
review text: Let \(k\) be an integer. The authors find all integers of the form \(by^\ell\), where \(\ell\geq2\), and the greatest prime factor of \(b\) is at most \(k\) such that they are all products of \(k\) consecutive integers with two terms omitted. Namely, if these integers are \(n+d_1\), \(n+d_2\), \dots, \(n+d_{k-2}\), \(0\leq d_1<d_2<d_{k-2}<k\), then the only solutions are obtained for \(\ell=3\), \(k=4\), \(n=125\) or \(k=7\) and the greatest prime factor of \(b\) is \(7\). This result can be compared to a previous result of the same authors (with \textit{G. Hanrot}) [Acta Arith. 99, No. 1, 13--25 (2001; Zbl 0971.11017)] where exactly one term was omitted. The proof involves the solution of difficult exponential and Thue equations.
reviewed by: Maurice Mignotte
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11D59
zbMATH DE Number: 5619300
zbMATH Keywords: arithmetic progressions
zbMATH Keywords: exponential Diophantine equations
zbMATH Keywords: generalized Fermat equations
zbMATH Keywords: modular methods
zbMATH Keywords: Thue equations
MaRDI profile type: Publication
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full work available at URL: s0019-3577(08)80027-4
OpenAlex ID: W2123694992
description: scientific article; zbMATH DE number 5619300