On the exponential Diophantine equation \(F_{n+1}^x - F_{n-1}^x = F_m\)
type: Item
instance of: scholarly article
title: On the exponential Diophantine equation \(F_{n+1}^x - F_{n-1}^x = F_m\)
zbMATH Open document ID: 1490.11042
author: Bijan Kumar Patel
author: Ana Paula Chaves
published in: Mediterranean Journal of Mathematics
publication date: 2021-08-30T00:00:00Z
review text: Let \((F_n)_{n\ge 0}\) be the Fibonacci sequence given by \(F_0=0,~F_1=1\) and \(F_{n+2}=F_{n+1}+F_n\) for all \(n\ge 0\). The authors look at the Diophantine equation in the title in nonnegative integers \((m,n,x)\). The above equation has two parametric families of solutions with \(x=1\) and \(x=2\) because of the identities \(F_{n+1}-F_{n-1}=F_n\) and \(F_{n+1}^2-F_{n-1}^2=F_{2n}\). The authors prove that there are no other solutions \((m,n,x)\) with \(x\ge 3\) and \(n\ge 3\). The proof uses linear forms in logarithms of algebraic numbers and reduction methods based on continued fractions.
reviewed by: Florian Luca
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 7388040
zbMATH Keywords: Fibonacci numbers
zbMATH Keywords: linear forms in logarithms
zbMATH Keywords: continued fraction
zbMATH Keywords: reduction method
MaRDI profile type: Publication
full work available at URL: s00009-021-01831-4
OpenAlex ID: W3188093699
Wikidata QID: Q115609557
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DOI: 10.1007/S00009-021-01831-4
description: scientific article; zbMATH DE number 7388040