Diophantine equations after Fermat's last theorem
type: Item
instance of: scholarly article
title: Diophantine equations after Fermat's last theorem
zbMATH Open document ID: 1222.11045
author: Samir Siksek
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-10-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10889
review text: The author considers the following two questions: {\parindent=7mm \begin{itemize}\item[(i)]Given a Diophantine equation, what information can be obtained by following the strategy of Wiles' proof of Fermat's last theorem? \item[(ii)]Is it useful to combine this approach with the traditional approaches to Diophantine equations: Diophantine approximation, arithmetical geometry, \dots? \end{itemize}} Moreover, the problem is to give an explicit resolution of these Diophantine equations. The use of Frey curves is explained in detail. Three examples are presented to illustrate the method: -- the generalized Ramanujan--Nagell equations \[ x^2+D=y^m, \;m \geq 3, \] for \(1\leq D \leq 100\), -- perfect powers in the Fibonacci sequence \[ F_n = y^m, \;m\geq2, \] in which case it was proved that the only solutions are \(F_n=0,1,8,144\), -- some multiply exponential Diophantine equations such as \[ q^u x^n - 2^r y^n = \pm 1, \;n\geq3, \] where \(q\) is an odd prime \({}<100\). For the two first examples, the author gives many concrete explanations and shows the combination of the modular method with more classical methods (elementary arguments and heavy use of Baker's theory). The paper ends with open problems for which -- up to now -- the combination of the previous methods fails, the most exiting ones being \[ x^3+y^3 = z^n, \;xyz\not=0, \;n\geq3, \] and \[ x^2-2=y^m, \;m\geq3. \]
reviewed by: Maurice Mignotte
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 5620659
zbMATH Keywords: exponential Diophantine equations
zbMATH Keywords: Wiles theory
zbMATH Keywords: Baker's theory
describes a project that uses: PARI/GP
describes a project that uses: Magma
describes a project that uses: ecdata
MaRDI profile type: Publication
OpenAlex ID: W2020789638
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DOI: 10.5802/JTNB.679
description: scientific article; zbMATH DE number 5620659