Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers

Abstract

This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are $0$, $1$, $8$ and $144$ and the only perfect powers in the Lucas sequence are $1$ and $4$.