On Fibonacci numbers of the form \(q^ky^k\).
type: Item
instance of: scholarly article
title: On Fibonacci numbers of the form \(q^ky^k\).
zbMATH Open document ID: 1113.11010
author: Yann Bugeaud
author: Samir Siksek
author: Maurice Mignotte
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2004-11-23T00:00:00Z
review text: The authors study the equations (1) \(F_n=q^ky^p\) and (2) \(L_n=q^ky^p\), where \(k>0\) and \(p\), \(q\) are primes, and \(F_n\) and \(L_n\) denote the \(n\)th Fibonacci and Lucas numbers, respectively. They prove that: (i) if \(q=2\), then the only solutions of (1) occur for \(n=3, 6, 12\); (ii) If \(q\) is odd and \(n\) is even, then similarly \(n=0,4,12\); (iii) if \(q\) and \(n\) are odd, then necessarily \(q\equiv 1\pmod 4\). Furthermore, either \(q=n=5\), or the Fibonacci entry point of \(q\) must be odd, and (1) has a solution such that \(q\nmid n\). The authors also show that if \(q=2\), then (2) has a solution only when \(n=3\). The authors use the software package MAGMA to get solutions of the Diophantine equation arising in the solution of (2) for \(q=2\) and \(y=3\).
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 2118257
zbMATH Keywords: Ribenboim's question
describes a project that uses: Magma
describes a project that uses: PARI/GP
MaRDI profile type: Publication
full work available at URL: j.crma.2004.06.007
OpenAlex ID: W2052526094
cites work: The Magma algebra system. I: The user language
cites work: Classical and modular approaches to exponential Diophantine equations. I: Fibonacci and Lucas perfect powers
cites work: Lucas and fibonacci numbers and some diophantine Equations
cites work: Sur les équations xp+2βyp= z2et xp+2βyp= 2z2
cites work: The terms Cxh(h≥3) in Lucas sequences: an algorithm and applications to diophantine equations
DOI: 10.1016/J.CRMA.2004.06.007
description: scientific article; zbMATH DE number 2118257