Power Fibonacci sequences in quadratic integer modulo m

Paul Ryan A. Longhas, Cyryn Jade L. Prendol, Jenelyn F. Bantilan, and Larra L. De Leon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 1, Pages 133–145
DOI: 10.7546/nntdm.2025.31.1.133-145
Full paper (PDF, 246 Kb)

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Authors and affiliations

Paul Ryan A. Longhas
Department of Mathematics and Statistics, Polytechnic University of the Philippines
Manila 1008, Philippines

Cyryn Jade L. Prendol
Department of Mathematics and Statistics, Polytechnic University of the Philippines
Manila 1008, Philippines

Jenelyn F. Bantilan
Department of Mathematics and Statistics, Polytechnic University of the Philippines
Manila 1008, Philippines

Larra L. De Leon
Department of Mathematics and Statistics, Polytechnic University of the Philippines
Manila 1008, Philippines

Abstract

The power Fibonacci sequence in \mathbb{Z}_m[\sqrt{\delta}] is defined as a Fibonacci sequence \linebreak F_n = F_{n-1} + F_{n-2} where F_0 = 1 and F_1 = a, such that a\in \mathbb{Z}_m[\sqrt{\delta}] and F_n\equiv a^n \:(\mathrm{mod}\: m), for all n\in \mathbb{N}\cup \{0\}. In this paper, we investigated the existence of power Fibonacci sequences in \mathbb{Z}_m[\sqrt{\delta}], and the number of power Fibonacci sequences in \mathbb{Z}_m[\sqrt{\delta}] for a given m, where \delta is a square-free integer. Furthermore, we determined explicitly all power Fibonacci sequences in \mathbb{Z}_{p^k}[\sqrt{\delta}], where p is a prime number.

Keywords

  • Power Fibonacci sequences
  • Fibonacci sequence
  • Legendre symbol
  • Quadratic equation
  • Square-free integer

2020 Mathematics Subject Classification

  • 11B39
  • 11B50

References

  1. Hull, R. (1932). The numbers of solutions of congruences involving only kth powers. Transactions of the American Mathematical Society, 34(4), 908–937.
  2. Ide, J., & Renault, M. S. (2012). Power Fibonacci sequences. The Fibonacci Quarterly, 50(2), 175–179.

Manuscript history

  • Received: 5 May 2024
  • Revised: 11 April 2025
  • Accepted: 24 April 2025
  • Online First: 25 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

Longhas, P. R. A., Prendol, C. J. L., Bantilan, J. F., & De Leon, L. L. (2025). Power Fibonacci sequences in quadratic integer modulo m. Notes on Number Theory and Discrete Mathematics, 31(1), 133-145, DOI: 10.7546/nntdm.2025.31.1.133-145.

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