Almost powers in the Lucas sequence
type: Item
instance of: scholarly article
title: Almost powers in the Lucas sequence
zbMATH Open document ID: 1204.11030
author: Yann Bugeaud
author: Florian Luca
author: Maurice Mignotte
author: Samir Siksek
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-06-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10852
review text: The \textit{Lucas sequence} \((L_n)_{n\geq 0}\) is defined by \(L_0=2, L_1=1\) and \(L_n=L_{n-1}+L_{n-2}\) for \(n\geq 2\). The first, third and fourth authors have proved, among other things, that the only perfect powers in the Lucas sequence are \(L_1=1\) and \(L_3=4\) [\textit{Y. Bugeaud, M. Mignotte} and \textit{S. Siksek}, Ann. Math. (2) 163, No. 3, 969--1018 (2006; Zbl 1113.11021)]. The present paper deals with the equation \(L_n=q^ay^p\), where \(p\geq 2\) and \(q\) are primes, and \(a,y\) are positive integers. In recent years, many interesting papers have been devoted to the study of analogous problems, in which the sequences of Lucas and/or Fibonacci mainly figure. The solution of such problems results from the combination of a variety of tools, like elementary tricks, the so called \textit{Modular Method} and the \textit{Double Frey Method} -- both inspired by and developed after the proof of Fermat's Last Theorem --, sharp bounds of two or three logarithms (``two'' or ``three'', that makes a big difference from the practical point of view!) to mention the most important. The present paper uses all these tools, in combination with various tricks which reduce the amount of computations, in order to prove the following Theorem: The only solutions to \(L_n=q^ay^p\) with \(q<10^6\) (under the assumptions on \(p,q,a,y\) mentioned above) and \(q\) different from 1087, 2207, 4481, 14503, 19207, 21503, 34303, 48767, 119809, 232049, 524287, 573569, 812167, are \(L_0=2\), \(L_2=3\), \(L_3=2^2\), \(L_4=7\), \(L_5=11\), \(L_6=2\cdot 3^2\), \(L_7=29\), \(L_8=47\), \(L_9=19\cdot 2^2\), \(L_{11}=199\), \(L_{13}=521\), \(L_{17}=3571\) and \(L_{19}=9349\). The reason for the exclusion of the thirteen values of \(q\) given above is, according to the authors, because ``the necessary modular forms computations needed for these values are too demanding for the currently available hardware''. It is worth noting that, using elementary arguments and aided by the computer, the au…
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11D59
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11J86
Mathematics Subject Classification ID: 11Y50
zbMATH DE Number: 5572693
zbMATH Keywords: Lucas sequence
zbMATH Keywords: modular method
zbMATH Keywords: double Frey method
zbMATH Keywords: linear form in two or three logarithms
zbMATH Keywords: Thue equation
reviewed by: Nikos Tzanakis
describes a project that uses: PARI/GP
describes a project that uses: Magma
describes a project that uses: ecdata
MaRDI profile type: Publication
OpenAlex ID: W2086081153
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DOI: 10.5802/JTNB.642
description: scientific article; zbMATH DE number 5572693