Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers

By Yann Bugeaud, Maurice Mignotte, Samir Siksek. 2004

489 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/96af95d1c5815b0daef939a96ae21154b6ef2a0c

How researchers see this paper

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In Algebra and Number Theory

1. Samir Siksek
2. Florian Luca
3. Jhon J. Bravo
4. Alain Togbé
5. Volker Ziegler
6. Bo He
7. Maurice Mignotte
8. Yann Bugeaud

In Geometry and Topology

1. Samir Siksek
2. Maurice Mignotte
3. Yann Bugeaud
4. Florian Luca
5. Jhon J. Bravo
6. Alain Togbé
7. Volker Ziegler
8. Bo He

In Statistical and Nonlinear Physics

1. Samir Siksek
2. Florian Luca
3. Jhon J. Bravo
4. Yann Bugeaud
5. Maurice Mignotte
6. Alain Togbé
7. Volker Ziegler
8. Bo He

Citations in the corpus

• There is no Diophantine quintuple (2016, methodology): “The three new key arguments that lead to the proof of our main result are (1) the definition of an operator on Diophantine triples and their classification; (2) the use of sharp lower bounds for linear forms in three logarithms obtained by applying a result due to Mignotte [37] (see also [5, 6]); (3”
• Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation (2004, methodology): “D MODULAR APPROACHES 29 14.1. An upper bound for |∆|. Put M1 = L−1 2 XN j=1 rj, M2 = L−1 2 XN j=1 sj, M3 = L−1 2 XN j=1 tj, and g = 1 4 − N 12RST , G1 = NLR 2 g, G2 = NLS 2 g, G3 = NLT 2 g, then, see [9]: Proposition 14.1. With the previous notation, if K ≥ 3, L ≥ 5 and Λ′ ≤ ρ−KL, for some real numb”
• The Modular Approach to Diophantine Equations (2007, methodology): “This method is central to the recent determination of all perfect powers in the Fibonacci and Lucas sequences [ 8 ].”
• On The Diophantine equation Fn+Fm=2a (2016, background): “4 in [4]) proved the following deep theorem.”