On the distance between products of consecutive Fibonacci numbers and powers of Fibonacci numbers
type: Item
instance of: scholarly article
title: On the distance between products of consecutive Fibonacci numbers and powers of Fibonacci numbers
zbMATH Open document ID: 1282.11010
author: Jhon J. Bravo
author: Takao Komatsu
author: Florian Luca
published in: Indagationes Mathematicae. New Series
publication date: 2013-01-03T00:00:00Z
review text: Let \((F_n)_{n \geq 0}\) be the Fibonacci sequence given by \(F_0=0, F_1=1\) and \(F_{n+2}=F_{n+1}+F_n\) for all \(n \geq 0\). In the paper under review the authors study a lower bound on the quantity \[ | F_n \cdots F_{n+k-1}-F_m^l | \] where \(k \geq 1, l \geq 1, n \geq 3, m\geq 3\). The main result of the paper is the following { Theorem 1.} Let \((k,l,m,n)\) be integers with the properties that \(k \geq 1, l \geq 1, n \geq 3\) and \(m\geq 3\). Put \(X:=\max\{k,l,m,n\}\). Then the inequality \[ | F_n \cdots F_{n+k-1}-F_m^l | >10^{-3/2}X^{1/{40}} \] holds always except when either \(l=k=1\) and \(m=n(=X)\) or \((k,l,m,n)=(1,3,3,6)\), for which the left-hand side of the above equation is 0. As a consequence of Theorem 1 the authors derive the following numerical corollary. Corollary 1. The largest solution of the inequality \[ | F_n \cdots F_{n+k-1}-F_m^l | \leq 100 \] with \(k \geq 1, l \geq 1, n \geq 3, m\geq 3\) and \(m \neq n\) when \(k=l=1\) is \[ | F_9 \cdots F_{13}-F_{11}^5|=89. \] Here, by the largest solution we mean the solution with the maximal value of \(\max\{F_n \cdots F_{n+k-1}, F_m^l\}\) among all the possible solutions. This maximal value equals \(5 584 059 449\). For the proof of Theorem 1, the authors combine Baker's method with some known facts about Fibonacci numbers and with some deep, earlier results, such as \textit{R. D. Carmichael}'s Primitive Divisor Theorem [Ann. Math. (2) 15, 30--48 (1913; JFM 44.0216.01), Ann. Math. (2) 15, 49--70 (1914; JFM 45.1259.10)], a result of \textit{Y. Bugeaud} et al. [Ann. Math. (2) 163, No. 3, 969--1018 (2006; Zbl 1113.11021)] on Fibonacci and Lucas perfect powers and a result of \textit{F. Luca} and \textit{T. N. Shorey} [J. Number Theory 114, No. 2, 298--311 (2005; Zbl 1081.11023)] concerning Diophantine equations with products of consecutive terms in Lucas sequences. For the proof of Corollary 1, the initial upper bound for \(X=\max\{k,l,m,n\}\) is \(X<1.2\cdot 10^{100}\), which is a consequence of Theore…
reviewed by: István Pink
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 6121468
zbMATH Keywords: Fibonacci numbers
zbMATH Keywords: application of linear forms in logarithms of algebraic numbers
MaRDI profile type: Publication
full work available at URL: j.indag.2012.08.004
OpenAlex ID: W2094091287
DOI: 10.1016/J.INDAG.2012.08.004
Recommended article: On the equation \(\sum_{j = 1}^k j F_j^p = F_n^q\)
Recommended article: Q4483706
Recommended article: Q3490071
Recommended article: Q4441165
Recommended article: Sums of Fibonacci numbers close to a power of 2
Recommended article: Q5042408
Recommended article: On the spacing between terms of generalized Fibonacci sequences
Recommended article: Q4441185
Recommended article: On the distance between generalized Fibonacci numbers
Recommended article: Diophantine approximations with Fibonacci numbers
description: scientific article; zbMATH DE number 6121468