On a few Diophantine equations, in particular, Fermat′s last theorem

C Levesque

doi:10.1155/s0161171203210668

Abstract

This is a survey on Diophantine equations, with the purpose being to give the flavour of some known results on the subject and to describe a few open problems. We will come across Fermat′s last theorem and its proof by Andrew Wiles using the modularity of elliptic curves, and we will exhibit other Diophantine equations which were solved à la Wiles. We will exhibit many families of Thue equations, for which Baker′s linear forms in logarithms and the knowledge of the unit groups of certain families of number fields prove useful for finding all the integral solutions. One of the most difficult conjecture in number theory, namely, the ABC conjecture, will also be described. We will conclude by explaining in elementary terms the notion of modularity of an elliptic curve.

Highlights

On June 23, 1993, at the Isaac Newton Institute of Cambridge (England), Professor Andrew Wiles (Princeton University) made a striking announcement
We offer you an excursion over centuries into this fantastic world of Diophantine equations
Elliptic curves à la Wiles, Darmon and Merel [11] solved some variants of the Fermat equation

Summary

LEVESQUE

This is a survey on Diophantine equations, with the purpose being to give the flavour of some known results on the subject and to describe a few open problems. We will come across Fermat’s last theorem and its proof by Andrew Wiles using the modularity of elliptic curves, and we will exhibit other Diophantine equations which were solved à la Wiles. We will exhibit many families of Thue equations, for which Baker’s linear forms in logarithms and the knowledge of the unit groups of certain families of number fields prove useful for finding all the integral solutions. One of the most difficult conjecture in number theory, namely, the ABC conjecture, will be described. We will conclude by explaining in elementary terms the notion of modularity of an elliptic curve.

Introduction

Zagier also showed that all integral solutions of