On the {L}ucas sequence equations {\(V_n(P,1)=wkx^2\)}, {\(w\in\{5,7\}\)}
type: Item
instance of: scholarly article
title: On the {L}ucas sequence equations {\(V_n(P,1)=wkx^2\)}, {\(w\in\{5,7\}\)}
zbMATH Open document ID: 1389.11037
author: Olcay Karaatli
published in: Periodica Mathematica Hungarica
publication date: 2017-11-10T00:00:00Z
review text: In this paper, the author considers the Lucas sequence of the second kind \(\{V_n\}_{n\geq 0}\) given by \(V_0=2,~V_1=P,~V_{n+2}=PV_n+V_{n-1}\), where \(P\) is an odd positive integer and solves the equations \(V_n=5k x^2,~7kx^2,~5kx^2\pm 1\), where \(k\) is any divisor of \(P\). For example, the first main result asserts all solutions to the first of above equations \(V_n=5kx^2\) have \(n=1,3\). He applies these results to prove that if \(V_n=Ax^2\) and \(A\in \{15,21,35\}\), then \(n=1,3\). Except for one equation for which the author used MAGMA to find all the integral points on a certain elliptic curve, all the other equations are dealt with by elementary but clever manipulations with Jacobi symbols.
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11B50
zbMATH DE Number: 6806099
zbMATH Keywords: generalized Fibonacci numbers
zbMATH Keywords: generalized Lucas numbers
zbMATH Keywords: congruences
zbMATH Keywords: Jacobi symbol
reviewed by: Florian Luca
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: s10998-016-0130-7
OpenAlex ID: W2340554875
cites work: The Magma algebra system. I: The user language
cites work: Q5332677
cites work: Squares in some recurrent sequences
cites work: Q3550102
cites work: The Fibonacci Numbers: Exposed
cites work: On the Lucas sequence equations \(V_n=7\,\square \) and \(V_n=7V_m\,\square \)
cites work: GENERALIZED LUCAS NUMBERS OF THE FORM 5kx2AND 7kx2
cites work: On the equations \(U_{n}=5\square\) and \(V_{n}=5\square\)
cites work: Generalized Fibonacci and Lucas numbers of the form 5x2
cites work: Generalized Fibonacci and Lucas numbers of the form \(wx^2\) and \(wx^2\mp 1\)
cites work: Generalized Fibonacci and Lucas Sequences and Rootfinding Methods
cites work: Q4883354
cites work: The square terms in Lucas sequences
cites work: On Lucas sequence terms of the form \(kx^2\)
cites work: Q5401319
cites work: THE SQUARE TERMS IN GENERALIZED LUCAS SEQUENCES
DOI: 10.1007/S10998-016-0130-7
description: scientific article; zbMATH DE number 6806099