On sums of k-generalized Fibonacci and k-generalized Lucas numbers as first and second kinds of Thabit numbers

Hunar Sherzad Taher and Saroj Kumar Dash
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 3, Pages 448–459
DOI: 10.7546/nntdm.2025.31.3.448-459
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Authors and affiliations

Hunar Sherzad Taher
Department of Mathematics, School of Advanced Sciences
Vellore Institute of Technology, Chennai, 600127, India

Saroj Kumar Dash
Department of Mathematics, School of Advanced Sciences
Vellore Institute of Technology, Chennai, 600127, India

Abstract

Let (F_{r}^{(k)})_{r\geq2-k} and (L_{r}^{(k)})_{r\geq2-k} be generalizations of the Fibonacci and Lucas sequences, where k\geq2. For these sequences the initial k terms are 0, 0, \ldots , 0, 1 and 0, 0, \ldots , 2, 1, and each subsequent term is the sum of the preceding k terms. In this paper, we determined all first and second kinds of Thabit numbers that can be expressed as the sums of k-Fibonacci and k-Lucas numbers. We employed the theory of linear forms in logarithms of algebraic numbers and a reduction method based on the continued fraction.

Keywords

  • Diophantine equations
  • Linear forms in logarithms
  • Generalized Fibonacci numbers
  • Generalized Lucas numbers
  • Reduction method

2020 Mathematics Subject Classification

  • 11D61
  • 11D45
  • 11J70
  • 11J86
  • 11B39

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Manuscript history

  • Received: 13 October 2024
  • Revised: 31 July 2025
  • Accepted: 2 August 2025
  • Online First: 4 August 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

Taher, H. S., & Dash, S. K. (2025). On sums of k-generalized Fibonacci and k-generalized Lucas numbers as first and second kinds of Thabit numbers. Notes on Number Theory and Discrete Mathematics, 31(3), 448-459, DOI: 10.7546/nntdm.2025.31.3.448-459.

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