  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, 144 and the only perfect powers in the
Lucas sequence are 1, 4.
