Padovan numbers which are concatenations of three Padovan or Perrin numbers

Fatih Erduvan
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 1, Pages 137–149
DOI: 10.7546/nntdm.2026.32.1.137-149
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Authors and affiliations

Fatih Erduvan
MEB, Izmit Namık Kemal Anatolia High School
41100, Kocaeli, Türkiye

Abstract

This paper presents all Padovan numbers that can be written as the concatenation of three Padovan or Perrin numbers under a certain constraint. Namely, we consider the Diophantine equations

    \[ P_{k}=10^{d+l}P_{m}+10^{l}P_{n}+P_{r} \]

and

    \[ P_{k}=10^{d+l}R_{m}+10^{l}R_{n}+R_{r} \]

where k,m,n,r,d and l are positive integers satisfying n\leq m. The parameters d and l denote the numbers of digits in the integers P_{n} (or R_{n}) and P_{r} (or R_{r}), respectively. The solutions to these equations can be written in the form P_{18}=\overline{P_{2}P_{2}P_{6}}=114 for all m,n,r\geq2 and, similarly, P_{30}=\overline{R_{3}R_{3}R_{12}}=3329, P_{33}=\overline{R_{7}R_{7}R_{13}}=7739 for all m\geq3 and n,r\geq1.

Keywords

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

2020 Mathematics Subject Classification

  • 11B83
  • 11D61
  • 11J86

References

  1. Alan, M. (2022). On concatenations of Fibonacci and Lucas numbers. Bulletin of the Iranian Mathematical Society, 48(5), 2725–2741.
  2. Alan, M., & Altassan, A. (2025). On b-concatenations of two k-generalized Fibonacci numbers. Acta Mathematica Hungarica, 175(2), 452–471.
  3. Altassan, A., & Alan, M. (2024). Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers. Mathematica Slovaca, 74(3), 563–576.
  4. Banks, W. D., & Luca, F. (2005). Concatenations with binary recurrent sequences. Journal of Integer Sequences, 8(1), Article ID 05.1.3.
  5. Bellaouar, D., Özer, Ö., & Azzouza, N. (2025). Padovan and Perrin numbers of the form 7t − 5z − 3y − 2x. Notes on Number Theory and Discrete Mathematics, 31(1), 191–200.
  6. Bravo, E. (2023). On concatenations of Padovan and Perrin numbers. Mathematical Communications, 28(1), 105–119.
  7. Bravo, J. J., Gomez, C. A., & Luca, F. (2016). Powers of two as sums of two k-Fibonacci numbers. Miskolc Mathematical Notes, 17(1), 85–100.
  8. Bugeaud, Y., Mignotte, M., & Siksek, S. (2006). Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Annals of Mathematic, 163(3), 969–1018.
  9. De Weger, B. M. M. (1989). Algorithms for Diophantine Equations, CWI Tracts 65, Stichting Mathematisch Centrum, Amsterdam.
  10. Deveci, Ö., & Shannon, A. G. (2017). Pell–Padovan-circulant sequences and their  applications. Notes on Number Theory and Discrete Mathematics, 23(3), 100–114.
  11. Duman, M. G. (2025). Padovan numbers that are concatenations of a Padovan number and a Perrin number. Periodica Mathematica Hungarica, 89(1), 139–154.
  12. Duman, M. G. (2025). Perrin numbers that are concatenations of a Perrin number and a Padovan number in base b. Symmetry, 17(3), Article ID 364.
  13. Erduvan, F. (2024). Fibonacci numbers which are concatenations of three Fibonacci or Lucas numbers. Punjab University Journal of Mathematics, 56(10), 603–614.
  14. Irmak, N., & Szalay, L. (2025). On the equation Fn − Fm = Fta. Boletın de la Sociedad Matematica Mexicana, 31(3), Article ID 105.
  15. Legendre, A. M. (1798). Essai sur la Theorie des Nombres. Duprat, Paris, An VI.
  16. Shannon, A. G., Anderson, P. G., & Horadam, A. F. (2006). Properties of Cordonnier, Perrin and Van der Laan Numbers. International Journal of Mathematical Education in Science and Technology, 37(7), 825–831.
  17. 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.
  18. Smart, N. P. (1998). The Algorithmic Resolution of Diophantine Equations: A Computational Cookbook. Cambridge University Press, Vol. 41.
  19. 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.

Manuscript history

  • Received: 12 October 2025
  • Revised: 17 February 2026
  • Accepted: 1 March 2026
  • Online First: 4 March 2026

Copyright information

Ⓒ 2026 by the Author.
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

Erduvan, F. (2026). Padovan numbers which are concatenations of three Padovan or Perrin numbers. Notes on Number Theory and Discrete Mathematics, 32(1), 137-149, DOI: 10.7546/nntdm.2026.32.1.137-149.

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