The square terms in Lucas sequences
type: Item
instance of: scholarly article
title: The square terms in Lucas sequences
zbMATH Open document ID: 0851.11011
author: Wayne L. McDaniel
author: Paulo Ribenboim
published in: Journal of Number Theory
publication date: 1996-07-09T00:00:00Z
review text: Let \(P\) and \(Q\) be relatively prime odd integers and define the sequences \(\{U_n\}\) and \(\{V_n\}\) by \(U_n = PU_{n - 1} - QU_{n - 2}\) with \(U_0 = 0\), \(U_1 = 1\) and \(V_n = PV_{n - 1} - QV_{n - 2}\) with \(V_0 = 2\), \(V_1 = P\). The main results of the paper are the following. (i) If \(V_n\) is a square, then \(n = 1,3\) or 5. (ii) If \(2V_n\) is a square, then \(n = 0,3\) or 6. (iii) If \(U_n\) is a square, then \(n = 0,1,2,3,6\) or 12. (iv) If \(2U_n\) is a square, then \(n = 0,3\) or 6. These results are nice extensions of some earlier ones concerning special (e.g. Fibonacci and Lucas) sequences.
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11D09
zbMATH DE Number: 883839
zbMATH Keywords: square terms
zbMATH Keywords: Lucas sequence
zbMATH Keywords: recursive sequences
zbMATH Keywords: Fibonacci sequences
Wikidata QID: Q56429718
reviewed by: Péter Kiss
MaRDI profile type: Publication
full work available at URL: jnth.1996.0068
OpenAlex ID: W1968837781
DOI: 10.1006/JNTH.1996.0068
description: scientific article; zbMATH DE number 883839