On the Diophantine equation \({x^ n-1\over x-1}=y^ q\).
type: Item
instance of: scholarly article
title: On the Diophantine equation \({x^ n-1\over x-1}=y^ q\).
zbMATH Open document ID: 1064.11030
author: Yann Bugeaud
author: Maurice Mignotte
author: Yves Roy
published in: Pacific Journal of Mathematics
publication date: 2000-12-03T00:00:00Z
review text: The paper deals with the Diophantine equation \[ \frac{x^n-1}{x-1}=y^q\;\text{ in \;integers}\;x>1,\;y>1,\;n>2,\;q\geq 2.\tag{1} \] It has been conjectured that this equation has only the solutions \[ \frac{3^5-1}{3-1}=11^2, \quad \frac{7^4-1}{7-1}=20^2\quad \text{and}\quad \frac{18^3-1}{18-1}=7^3. \] This conjecture is still open. Ljunggren solved it for \(q=2\) and Nagell when \(3| n\) and \(4| n.\) The equation is completely solved when \(x\) is a square. This latter result depends on several methods in Diophantine approximation together with heavy computer calculations. These methods and long computations were used in this paper to show that the equation (1) implies that there exists a prime divisor of \(x\) which does not divide \(y-1.\) It is known that every solution of (1) with \(y\) an odd prime satisfies \(n=\text{ord}_y x.\) Using the above result the authors were able to show that \(q=\text{ord}_x y\) answering a question of Edgar. Irrationality of certain numbers is also obtained.
reviewed by: N. Saradha
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 1537814
zbMATH Keywords: exponential Diophantine equations
zbMATH Keywords: Diophantine approximations
zbMATH Keywords: \(p\)-adic logarithms
MaRDI profile type: Publication
full work available at URL: pjm.2000.193.257
OpenAlex ID: W825387207
DOI: 10.2140/PJM.2000.193.257
description: scientific article; zbMATH DE number 1537814