Fibonacci numbers at most one away from a perfect power
type: Item
instance of: scholarly article
title: Fibonacci numbers at most one away from a perfect power
zbMATH Open document ID: 1156.11008
DOI: 10.4171/EM/89
author: Yann Bugeaud
author: Florian Luca
author: Maurice Mignotte
author: Samir Siksek
published in: Elemente der Mathematik
publication date: 2008-11-05T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: In 2006, the first, third and fourth author of the present paper proved [Ann. Math. (2) 163, No. 3, 969--1018 (2006; Zbl 1113.11021)] that the only Fibonacci numbers that are perfect powers different from 0 and 1, are \(F_6=8\) and \(F_{12}=144\); and the only Lucas number that is a perfect power different from 1, is \(L_3=4\). We remind here that the initial terms of the Fibonacci sequence are \(F_0=0\), \(F_1=1\) and those of the Lucas sequence are \(L_0=2\), \(L_1=1\); and both sequences satisfy the second-order recurrence \(u_{n+2}=u_{n+1}+u_n\). In this nicely written paper the authors sketch briefly the proof of the above theorem, by successfully giving a feeling of how the so called \textit{modular approach} to exponential Diophantine equations (with its origins in the proof of \textit{Fermat's Last Theorem} by A.~Wiles) is combined with the lower bounds for linear forms in two or three logarithms of algebraic numbers. Moreover, combining the above strong theorem with only elementary (but not straightforward) arguments they prove that the only solutions \((n,y,p)\) to the equation \(F_n\pm 1=y^p\) with \(p\geq 2\) are furnished by the relations: \(F_0+1=1\), \(F_4+1=2^2\), \(F_6+1=3^2\), \(F_1-1=F_2-1=0\), \(F_3-1=1\) and \(F_5-1=2^2\). In their proof the authors take advantage of the fact that \(F_n+1\) is factored in the form \(F_aL_b\) for appropriate indices \(a,b\) depending on the class of \(n\bmod 4\).
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11J86
Mathematics Subject Classification ID: 11Y50
zbMATH DE Number: 5362007
zbMATH Keywords: Fibonacci numbers
zbMATH Keywords: Lucas numbers
zbMATH Keywords: Frey elliptic curve
zbMATH Keywords: modular approach to Diophantine equations
zbMATH Keywords: linear forms in two or three logarithms of algebraic numbers
reviewed by: Nikos Tzanakis
describes a project that uses: PARI/GP
describes a project that uses: Magma
MaRDI profile type: Publication
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OpenAlex ID: W1997058156
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description: scientific article; zbMATH DE number 5362007