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

2006/05/01 by Yann Bugeaud, Maurice Mignotte, Samir Siksek · 1 citation
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Advanced Mathematical Theories and Applications #Analytic Number Theory Research

paper · pdf · doi:10.4007/annals.2006.163.969

openalex publication_date 2006/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

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.

Citations

• Bounds for the solutions of Thue-Mahler equations and norm form equations
• On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises
• Perfect powers in second order linear recurrences
• Ring-Theoretic Properties of Certain Hecke Algebras
• Modular Elliptic Curves and Fermat's Last Theorem

Cited by

• Fibonacci sequence [wikipedia]