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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 $4$.
Yann Bugeaud 1 ; Maurice Mignotte 1 ; Samir Siksek 2
@article{10_4007_annals_2006_163_969,
author = {Yann Bugeaud and Maurice Mignotte and Samir Siksek},
title = {Classical and modular approaches to exponential {Diophantine} equations {I.} {Fibonacci} and {Lucas} perfect powers},
journal = {Annals of mathematics},
pages = {969--1018},
publisher = {mathdoc},
volume = {163},
number = {3},
year = {2006},
doi = {10.4007/annals.2006.163.969},
mrnumber = {2215137},
zbl = {1113.11021},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2006.163.969/}
}
TY - JOUR AU - Yann Bugeaud AU - Maurice Mignotte AU - Samir Siksek TI - Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers JO - Annals of mathematics PY - 2006 SP - 969 EP - 1018 VL - 163 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2006.163.969/ DO - 10.4007/annals.2006.163.969 LA - en ID - 10_4007_annals_2006_163_969 ER -
%0 Journal Article %A Yann Bugeaud %A Maurice Mignotte %A Samir Siksek %T Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers %J Annals of mathematics %D 2006 %P 969-1018 %V 163 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2006.163.969/ %R 10.4007/annals.2006.163.969 %G en %F 10_4007_annals_2006_163_969
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
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