New bounds and conditions for the equation of Nagell-Ljunggren
type: Item
instance of: scholarly article
title: New bounds and conditions for the equation of Nagell-Ljunggren
zbMATH Open document ID: 1127.11021
published in: Journal of Number Theory
publication date: 2007-05-10T00:00:00Z
review text: The equation of Nagell-Ljunggren is \[ {x^p-1\over x-1}=p^e \cdot y^q, \quad x, y \in \mathbb Z^*, \quad e\in \{0,1\}, \] where \(p \not= q\) are odd primes. We do not know whether the set of solutions \((p,q,x,y)\) is finite or not. The author proves very strong bounds and conditions on \(q\), for \(p\) fixed, in particular \(q<(p-1)^2\), and moreover \[ \min \{| x| ,| y| \} < q^{10\,p^2}. \] The proof uses clever methods of cyclotomy, and local and global considerations.
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 5151591
zbMATH Keywords: exponential Diophantine equations
zbMATH Keywords: cyclotomy
reviewed by: Maurice Mignotte
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.10.010
OpenAlex ID: W2104822686
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DOI: 10.1016/J.JNT.2006.10.010
author: Preda Mihaĭlescu
description: scientific article; zbMATH DE number 5151591