The Diophantine equation \((x^{k} - 1)(y^{k} - 1) = (z^{k} - 1)^{t}\)
type: Item
instance of: scholarly article
title: The Diophantine equation \((x^{k} - 1)(y^{k} - 1) = (z^{k} - 1)^{t}\)
zbMATH Open document ID: 1169.11015
DOI: 10.1016/S0019-3577(07)80059-0
author: Michael A. Bennett
published in: Indagationes Mathematicae. New Series
publication date: 2008-08-12T00:00:00Z
review text: The main results of this paper, which improve on \textit{Y. Bugeaud}'s [Indag. Math., New Ser. 15, No. 1, 21--28 (2004; Zbl 1098.11022)], are Theorem 1. The Diophantine equation \[ (x^k-1)(y^k-1)=(z^k-1) \] has only the solutions \[ (x,y,z,k)=(-1,4,-5,3)\quad \text{and}\quad (4,-1,-5,3)\; \] in integers \(x\), \(y\), \(z\), and \(k\) whith \(|z|\geq 2\) and \(k\geq3\). and Theorem 2. The Diophantine equation \[ (x^k-1)(y^k-1)=(z^k-1)^2 \] has no solutions in integers \(x\), \(y\), \(z\), and \(k\) with \(|x|\not=|y|\), \(|z|\geq 2\) and \(k\geq 4\). Notice that theorem 1 is best possible. The proofs use the hypergeometric method of Thue-Siegel, several gap principles and also the modular method.
reviewed by: Maurice Mignotte
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11J82
Mathematics Subject Classification ID: 11J86
Mathematics Subject Classification ID: 11D45
zbMATH DE Number: 5311046
zbMATH Keywords: exponential Diophantine equation
zbMATH Keywords: hypergeometric method
zbMATH Keywords: gap principle
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5311046