Abstract:
After a short survey of results obtained by Thue and Siegel about one century ago, we explain how the theory of linear forms in the logarithms of algebraic numbers, developed by Alan Baker in the 60s, applies to Diophantine equations and provides us with explicit upper bounds for the size of the integer solutions of certain families of equations. Unfortunately, these upper bounds are huge, and we cannot hope to list all the integer solutions by brutal enumeration. However, many important progress have been accomplished during the last twenty years and, by combining various methods, it is now possible to completely solve some famous exponential Diophantine equations. For instance, jointly with Mignotte and Siksek, we have proved in 2006 that 1, 8 and 144 are the only perfect powers in the Fibonacci sequence.