Floor and ceiling functions for Pell numbers

İsmail Sulan and Mustafa Aşçı
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 2, Pages 326–334
DOI: 10.7546/nntdm.2025.31.2.326-334
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Authors and affiliations

İsmail Sulan
Department of Mathematics, Faculty of Science, Pamukkale University
20160, Kınıklı, Denizli, Türkiye

Mustafa Aşçı
Department of Mathematics, Faculty of Science, Pamukkale University
20160, Kınıklı, Denizli, Türkiye

Abstract

The analytical study of the Pell number and the role of floor and ceiling functions into their computation is examined. Closed expressions of Pell numbers were initially derived using Binet’s formula, followed by an asymptotic behavior study of the sequence using this formula. Taking into account the decreasing trend in the term |\beta|^n = |1 - \sqrt{2}|^n for large values of n, a formula that closely approximates Pell numbers has been developed. In this context, relationships between numbers are clarified using floor and ceiling functions. The accuracy with which various theorems and lemmas mathematically prove these approximations is also included. The study also looks at limit processes with emphasis placed upon the determining influence that the ratio \alpha = 1 + \sqrt{2} has on the growth rate of the sequence.

Keywords

  • Pell numbers
  • Floor function
  • Ceiling function
  • Recurrence relation
  • Binet’s formula

2020 Mathematics Subject Classification

  • 11B39
  • 11B83
  • 11B37
  • 26A18
  • 39A10
  • 11A55

References

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  2. Dickson, L. E. (2005). History of the Theory of Numbers, Volume II: Diophantine Analysis. Dover Publications.
  3. Graham, R. L., Knuth, D. E., & Patashnik, O. (1994). Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley.
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  5. Koshy, T. (2007). Elementary Number Theory with Applications (2nd ed.). Elsevier.
  6. Koshy, T. (2014). Pell and Pell–Lucas Numbers with Applications. Springer.
  7. Koshy, T. (2017). Fibonacci and Lucas Numbers with Applications (Vol. 1, 2nd ed., pp. 147–158). John Wiley & Sons.
  8. Niven, I., Zuckerman, H. S., & Montgomery, H. L. (1991). An Introduction to the Theory of Numbers (5th ed.). John Wiley & Sons.

Manuscript history

  • Received: 28 November 2024
  • Revised: 21 May 2025
  • Accepted: 29 May 2025
  • Online First: 2 June 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

Sulan, İ., & Aşçı, M. (2025). Floor and ceiling functions for Pell numbers. Notes on Number Theory and Discrete Mathematics, 31(2), 326-334, DOI: 10.7546/nntdm.2025.31.2.326-334.

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