Geodesic
Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers
Yann Bugeaud1 ; Maurice Mignotte1 ; Samir Siksek2
1 L'UFR de Mathématique et d'Informatique, Université Louis Pasteur, 67084 Strasbourg, France
2 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Annals of mathematics, Tome 163 (2006) no. 3, pp. 969-1018.

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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 $4$.
DOI : 10.4007/annals.2006.163.969

Yann Bugeaud 1 ; Maurice Mignotte 1 ; Samir Siksek 2

1 L'UFR de Mathématique et d'Informatique, Université Louis Pasteur, 67084 Strasbourg, France
2 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom 
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Yann Bugeaud; Maurice Mignotte; Samir Siksek. Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Annals of mathematics, Tome 163 (2006) no. 3, pp. 969-1018. doi : 10.4007/annals.2006.163.969. http://geodesic.mathdoc.fr/articles/10.4007/annals.2006.163.969/

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