On Pillai's problem with the Fibonacci and Pell sequences
type: Item
instance of: scholarly article
title: On Pillai's problem with the Fibonacci and Pell sequences
zbMATH Open document ID: 1440.11018
author: Santos Hernández Hernández
author: Florian Luca
author: Luis Manuel Rivera
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2019-11-28T00:00:00Z
full work available at URL: http://hdl.handle.net/21.11116/0000-0004-ED02-F
review text: Let \( (F_n)_{n\ge 0} \) and \( (P_n)_{n\ge 0} \) be the sequences of Fibonacci and Pell numbers given by \( F_0=P_0=0, ~ F_1 = P_1=1 \), \( F_{n+2}=F_{n+1}+F_n \), and \( P_{n+2}=2P_{n+1}+P_n \) for all \( n\ge 0 \), respectively. In the paper under review, the authors study the Pillai type problem: \begin{align*} F_n-P_m = F_{n_1}-P_{m_1} = c\tag{1} \end{align*} in non-negative integer pairs \( (n,m)\neq (n_1, m_1) \). They completely solve the Diophantine equation (1). In their main result, they prove that the only integers \( c \) admitting representations as a difference between a Fibonacci number and a Pell number in at least two different ways belong to the set: \begin{align*} c\in\{ -4,-2, -1, 0, 1, 2, 3, 5, 8, 19\}.\tag{2} \end{align*} Furthermore, they explicitly list down the representations for each \(c\) in (2) in the form (1). To prove the main result, the authors use the elementary properties of the Fibonacci and Pell sequences, the Baker's theory for linear forms in logarithms of algebraic numbers to effectively bound their variables, and the Baker-Davenport reduction procedure to reduce the bounds to computable sizes. Computations are done with help of a computer program in Mathematica.
reviewed by: Mahadi Ddamulira
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 7137966
zbMATH Keywords: Fibonacci
zbMATH Keywords: Pell sequences
zbMATH Keywords: Pillai's type problem
zbMATH Keywords: linear form in logarithms
describes a project that uses: Mathematica
MaRDI profile type: Publication
OpenAlex ID: W2900684276
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Wikidata QID: Q128882252
DOI: 10.1007/S40590-018-0223-9
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description: scientific article; zbMATH DE number 7137966