Some new results on the largest cycle consisting of quadratic residues

Prabin Das and Pinkimani Goswami
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 4, Pages 843–850
DOI: 10.7546/nntdm.2024.30.4.843-850
Full paper (PDF, 216 Kb)

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Authors and affiliations

Prabin Das
Department of Mathematics, University of Science and Technology Meghalaya
Baridua, India

Pinkimani Goswami
Department of Mathematics, University of Science and Technology Meghalaya
Baridua, India

Abstract

The length of the largest cycle consisting of quadratic residues of a positive integer n is denoted by L(n). In this paper, we have obtained a formula for finding L(p), where p is a prime. Also, we attempt to characterize a prime number p in terms of the largest cycle consisting of quadratic residues of p.

Keywords

  • Quadratic residues
  • Fermat primes
  • Mersenne prime
  • Largest cycle
  • Legendre symbol

2020 Mathematics Subject Classification

  • 11A07

References

  1. Burton, D. M. (2012). Elementary Number Theory. (7th ed.). TATA McGraw-Hill Edition.
  2. Xu, H., (2016). The largest cycles consist by the quadratic residues and Fermat primes. Preprint. arXiv:1601.06509v2[math.NT] 27 Jan 2016.
  3. Somer, L., & Křížek, M. (2004). On a connection of number theory with graph theory. Czechoslovak Mathematical Journal, 54(129), 465–485.

Manuscript history

  • Received: 14 October 2023
  • Revised: 25 November 2024
  • Accepted: 10 December 2024
  • Online First: 11 December 2024

Copyright information

Ⓒ 2024 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

Das, P., & Goswami, P. (2024). Some new results on the largest cycle consisting of quadratic residues. Notes on Number Theory and Discrete Mathematics, 30(4), 843-850, DOI: 10.7546/nntdm.2024.30.4.843-850.

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