Common values of two k-generalized Pell sequences

Zafer Şiar, Florian Luca and Faith Shadow Zottor
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 2, Pages 256–268
DOI: 10.7546/nntdm.2025.31.2.256-268
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Authors and affiliations

Zafer Şiar
Department of Mathematics, Bingöl University
Bingöl, Türkiye

Florian Luca
Mathematics Division, Stellenbosch University
Stellenbosch, South Africa

Faith Shadow Zottor
Department of Mathematics, University of Johannesburg
Johannesburg, South Africa

Abstract

Let k\geq 2 and let (P_{n}^{(k)})_{n\geq 2-k} be the k-generalized Pell sequence defined by

    \begin{equation*} P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)} \end{equation*}

for n\geq 2 with initial conditions

    \begin{equation*} P_{-(k-2)}^{(k)}=P_{-(k-3)}^{(k)}=\cdot \cdot \cdot =P_{-1}^{(k)}=P_{0}^{(k)}=0,\text{ and }P_{1}^{(k)}=1. \end{equation*}

In this study, we look at the equation P_{n}^{(k)}=P_{m}^{(l)} in positive integers n,m,k,l such that 2\leq l<k and show that it has only trivial solution, namely n=m.

Keywords

  • Baker’s method
  • Exponential Diophantine equation
  • Fibonacci numbers
  • Lucas numbers
  • Linear forms in logarithms

2020 Mathematics Subject Classification

  • 11B39
  • 11D61
  • 11J86

References

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Manuscript history

  • Received: 21 March 2025
  • Revised: 5 May 2025
  • Accepted: 8 May 2025
  • Online First: 9 May 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Şiar, Z., Luca, F., Zottor, F. S. (2025). Common values of two k-generalized Pell sequences. Notes on Number Theory and Discrete Mathematics, 31(2), 256-268, DOI: 10.7546/nntdm.2025.31.2.256-268.

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