Products of factorials in binary recurrence sequences
type: Item
instance of: scholarly article
title: Products of factorials in binary recurrence sequences
zbMATH Open document ID: 0978.11010
author: Florian Luca
published in: Rocky Mountain Journal of Mathematics
publication date: 2002-02-07T00:00:00Z
full work available at URL: CONT29-4.html
review text: The author shows that every non-degenerate binary recurrence sequence contains only finitely many terms which can be written as product of factorials. Further these terms can be effectively computed. In particular for the Lucas sequence \((L_n)_{n\geq 0}\), \(L_0=2\), \(L_1=1\) and \(L_{n+2}= L_{n+1}+ L_n\), the only terms which are products of factorials are \(L_0= 2!\) and \(L_3= (2!)^2\). The corresponding result for the Fibonacci sequence \((F_n)_{n\geq 0}\), \(F_0=0\), \(F_1=1\) and \(F_{n+2}= F_{n+1}+ F_n\), is \(F_3= 2!\), \(F_6= (2!)^3\) and \(F_{12}= (2!)^2 (3!)^2= 3!4!\). The proofs use estimations of linear forms in logarithms of algebraic numbers.
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 1455475
zbMATH Keywords: exponential Diophantine equations
zbMATH Keywords: non-degenerate binary recurrence sequence
zbMATH Keywords: product of factorials
zbMATH Keywords: Lucas sequence
zbMATH Keywords: Fibonacci sequence
zbMATH Keywords: linear forms in logarithms of algebraic numbers
reviewed by: N. Saradha
MaRDI profile type: Publication
OpenAlex ID: W2058844249
cites work: Logarithmic forms and group varieties.
cites work: Q4146788
cites work: Q4296321
cites work: On binary recurrence sequences
DOI: 10.1216/RMJM/1181070412
description: scientific article; zbMATH DE number 1455475