tailieunhanh - Đề tài " Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers "

Annals of Mathematics 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. | Annals of Mathematics Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers By Yann Bugeaud Maurice Mignotte and Samir Siksek Annals of Mathematics 163 2006 969 1018 Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers By Yann Bugeaud Maurice Mignotte and Samir Siksek 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. 1. Introduction Wiles proof of Fermat s Last Theorem 53 49 is certainly the most spectacular recent achievement in the field of Diophantine equations. The proof uses what may be called the modular approach initiated by Frey 19 20 which has since been applied to many other Diophantine equations mostly though not exclusively of the form 1 axp byp czp axp byp cz2 axp byp cz3 . p prime . The strategy of the modular approach is simple enough associate to a putative solution of such a Diophantine equation an elliptic curve called a Frey curve in a way that the discriminant is a p-th power up to a factor which depends only on the equation being studied and not on the solution. Next apply Ribet s level-lowering theorem 43 to show that the Galois representation on the p-torsion of the Frey curve arises from a newform of weight 2 and a fairly small level N say. If there are no such newforms then there are no nontrivial solutions to the original Diophantine equation. A solution is said to be trivial S. Siksek s work is funded by a grant from Sultan