Padovan and Perrin numbers of the form 7^{t}-5^{z}-3^{y}-2^{x}

Djamel Bellaouar, Özen Özer and Noureddine Azzouza
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 1, Pages 191–200
DOI: 10.7546/nntdm.2025.31.1.191-200
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Authors and affiliations

Djamel Bellaouar
Department of Mathematics, University 8 Mai 1945 Guelma
B.P. 401 Guelma 24000, Algeria

Özen Özer
Department of Mathematics, University of Kırklareli
39000, Kirklareli, Türkiye

Noureddine Azzouza
Department of Mathematics, University 8 Mai 1945 Guelma
B.P. 401 Guelma 24000, Algeria

Abstract

Consider the Padovan sequence \left( p_{n}\right) _{{n\geq 0}} given by p_{n+3}=p_{n+1}+p_{n} with p_{0}=p_{1}=p_{2}=1. Its companion sequence, the Perrin sequence \left( \wp _{n}\right) _{{n\geq 0}}, follows the same recursive formula as the Padovan numbers, but with different initial values: p_{0}=3, p_{1}=0 and p_{2}=2. In this paper, we leverage Baker’s theory concerning nonzero linear forms in logarithms of algebraic numbers along with a reduction procedure that employs the theory of continued fractions. This enables us to explicitly identify all Padovan and Perrin numbers that conform to the representation 7^{t}-5^{z}-3^{y}-2^{x}, where x,y,z and t are positive integers with 0\leq x,y,z\leq t.

Keywords

  • Exponential Diophantine equations
  • Padovan numbers
  • Perrin numbers
  • Linear forms in logarithms

2020 Mathematics Subject Classification

  • 11B37
  • 11D61
  • 11J86

References

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Manuscript history

  • Received: 5 May 2024
  • Revised: 20 October 2024
  • Accepted: 28 April 2025
  • Online First: 30 April 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

Bellaouar, D., Özer, Ö, & Azzouza, N. (2025). Padovan and Perrin numbers of the form 7^{t}-5^{z}-3^{y}-2^{x}. Notes on Number Theory and Discrete Mathematics, 31(1), 191-200, DOI: 10.7546/nntdm.2025.31.1.191-200.

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