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Periodica Mathematica Hungarica

, Volume 73, Issue 1, pp 73–82 | Cite as

On the Lucas sequence equations \(V_{n}(P,1)=wkx^{2},\) \(w\in \left\{ 5,7\right\} \)

  • Olcay KaraatlıEmail author
Article

Abstract

Let P be an odd integer and \((V_{n})\) denote the generalized Lucas sequence defined by \(V_{0}=2,\) \(V_{1}=P,\) and \(V_{n+1}=PV_{n}+V_{n-1}\) for \(n\ge 1.\) In this study, we solve the equations \(V_{n}=5kx^{2},\) \(V_{n}=7kx^{2},\) \(V_{n}=5kx^{2}\pm 1,\) and \(V_{n}=7kx^{2}\pm 1\) when k|P with \(k>1.\) Moreover, applying some of the results, we obtain complete solutions to the equations \(V_{n}=\sigma x^{2},\) \(\sigma \in \left\{ 15,21,35\right\} \).

Keywords

Generalized Fibonacci numbers Generalized Lucas numbers Congruences Jacobi symbol 

Mathematics Subject Classification

11B37 11B39 11B50 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.Faculty of Arts and ScienceSakarya UniversityAdapazarıTurkey

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