Diophantine equations after Fermat’s last theorem

Abstract

The author considers the following two questions: {\parindent=7mm \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? \par} Moreover, the problem is to give an explicit resolution of these Diophantine equations. \par The use of Frey curves is explained in detail. Three examples are presented to illustrate the method: \par -- the generalized Ramanujan--Nagell equations $$ x^2+D=y^m, m \ge 3, $$ for $1\le D \le 100$, \par -- perfect powers in the Fibonacci sequence $$ F_n = y^m, m\ge2, $$ in which case it was proved that the only solutions are $F_n=0,1,8,144$, \par -- some multiply exponential Diophantine equations such as $$ q^u x^n - 2^r y^n = \pm 1, n\ge3, $$ where $q$ is an odd prime ${}<100$. \par 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). \par The paper ends with open problems for which -- up to now -- the combination of the previous methods fails, the most exiting ones being $$ x^3+y^3 = z^n, xyz\not=0, n\ge3, $$ and $$ x^2-2=y^m, m\ge3. $$