International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 71, Pages 4473-4500
On a few Diophantine equations, in particular, Fermat's last theorem
Département de Mathématiques et de Statistique, Université Laval, Québec G1K 7P4, Canada
Received 20 October 2002
Copyright © 2003 C. Levesque. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.