Sequences in finite fields yielding divisors of Mersenne, Fermat and Lehmer numbers, I

A. M. S. Ramasamy
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 1, Pages 116–140
DOI: 10.7546/nntdm.2024.30.1.116-140
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Authors and affiliations

A. M. S. Ramasamy
Department of Mathematics, Pondicherry University
Pondicherry – 605014, India

Abstract

The aim of this work is to present a method using the cyclic sequences \{M_k\},\{\theta_{t,k}\} and \{\psi_{t,k} \} in the finite fields \mathbb{F}_\rho, with \rho a prime, that yield divisors of Mersenne, Fermat and Lehmer numbers. The transformations \tau_t and \sigma_t are introduced which lead to the proof of the cyclic nature of the sequences \{\theta_{t,k}\} and \{\psi_{t,k}\}. Results on the roots of the H(x)-polynomials in \mathbb{F}_\rho form the central theme of the study.

Keywords

  • Satellite polynomials
  • M-cycle
  • Background prime
  • The transformations \tau_t and \sigma_t
  • Symmetric and skew-symmetric properties
  • Pivotal elements
  • Euler’s totient function

2020 Mathematics Subject Classification

  • 11A51
  • 11B50
  • 11C08
  • 11T06

References

  1. Brillhart, J. (1964). On the factors of certain Mersenne numbers II. Mathematics of Computation, 18, 87–92.
  2. Brillhart, J., & Johnson, G. D. (1960). On the factors of certain Mersenne numbers. Mathematics of Computation, 14, 365–369.
  3. Hardy, G. H., & Wright, E. M. (1971). An Introduction to the Theory of Numbers. (4th ed.). The English Language Book Society.
  4. Kang, S. W. (1989). On the primality of the Mersenne number Mp. Journal of the Korean Mathematical Society, 26(1), 75–82.
  5. Kravitz, S. (1961). Divisors of Mersenne numbers 10, 000 < p < 15, 000. Mathematics of Computation, 15, 292–293.
  6. Leyendekkers, J. V., & Shannon, A. G. (2005). Fermat and Mersenne numbers. Notes on Number Theory and Discrete Mathematics, 11(4), 17–24.
  7. Mohanty, S. P., & Ramasamy, A. M. S. (1985). The characteristic number of two simultaneous Pell’s equations and its application. Bulletin of the Belgian Mathematical Society – Simon Stevin, 59(2), 203–214.
  8. Ramasamy, A. M. S. (2006). Generalized version of the characteristic number of two simultaneous Pell’s equations. The Rocky Mountain Journal of Mathematics, 36(2), 699–720.
  9. Ribenboim, P. (1996). The New Book of Prime Number Records. Springer–Verlag.

Manuscript history

  • Received: 25 August 2023
  • Revised: 4 March 2024
  • Accepted: 6 March 2024
  • Online First: 7 March 2024

Copyright information

Ⓒ 2024 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

Ramasamy, A. M. S. (2024). Sequences in finite fields yielding divisors of Mersenne, Fermat and Lehmer numbers, I. Notes on Number Theory and Discrete Mathematics, 30(1), 116-140, DOI: 10.7546/nntdm.2024.30.1.116-140.

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