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

arXiv Mathematics eprints
 Pub Date:
 March 2004
 arXiv:
 arXiv:math/0403046
 Bibcode:
 2004math......3046B
 Keywords:

 Mathematics  Number Theory;
 11D61;
 11B39;
 11J86;
 11D59