Enumeration of cyclic vertices and components over the congruence a^{11} \equiv b \pmod n

Sanjay Kumar Thakur, Pinkimani Goswami and Gautam Chandra Ray
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 3, Pages 525–537
DOI: 10.7546/nntdm.2023.29.3.525-537
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Authors and affiliations

Sanjay Kumar Thakur
Department of Mathematics, Science College Kokrajhar
Assam, India

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

Gautam Chandra Ray
Department of Mathematics, CIT Kokrajhar
Assam, India

Abstract

For each positive integer n, we assign a digraph \Gamma(n,11) whose set of vertices is Z_n=\lbrace 0,1,2, \ldots, n-1\rbrace and there exists exactly one directed edge from the vertex a to the vertex b iff a^{11}\equiv b \pmod n. Using the ideas of modular arithmetic, cyclic vertices are presented and established for n=3^k in the digraph \Gamma(n,11). Also, the number of cycles and the number of components in the digraph \Gamma(n,11) is presented for n=3^k,7^k with the help of Carmichael’s lambda function. It is proved that for k\geq 1, the number of components in the digraph \Gamma(3^k,11) is (2k+1) and for k>2 the digraph \Gamma(3^k,11) has (k-1) non-isomorphic cycles of length greater than 1, whereas the number of components of the digraph \Gamma(7^k,11) is (8k-3).

Keywords

  • Digraph
  • Fixed point
  • Power digraph
  • Carmichael λ-function
  • Cycles
  • Components

2020 Mathematics Subject Classification

  • 05C20
  • 11A07
  • 11A15

References

  1. Blanton Jr., E., Hurd, S., & McCranie, J. (1992). On a digraph defined by square modulo n. The Fibonacci Quarterly, 34, 322–334.
  2. Bryant, S. (1967). Groups, graphs and Fermat’s last theorem. The American Mathematical Monthly, 74(2), 152–156.
  3. Carmichael, R. D. (1910). Note on a new number function. Bulletin of American
    Mathematical Society, 16(5), 232–238.
  4. Goswami, P., Thakur, S. K., & Ray, G. C. (2023). The structure of power digraph connected with the congruence a11 ≡ b (mod n). Proyecciones (Antofagasta), 42(2), 457–477.
  5. Harary, F. (1969). Graph Theory. Addison-Wesley Publication Company, London.
  6. Křížek, M., Luca, F., & Somer, L. (2001). 17 Lectures on Fermat Numbers: From Number Theory to Geometry. CMS Books in Mathematics, vol. 9, Springer-Verlag, New York.
  7. Křížek, M., & Somer, L. (2004). Sophie Germain little suns. Mathematica Slovaca, 54(5), 433–442.
  8. Lucheta, C., Miller, E. & Reiter, C. (1996). Digraphs from Powers modulo p. The Fibonacci Quarterly, 34, 226–239.
  9. Mateen, M. H., & Mahmood, M. K. (2019). Power digraphs associated with the congruence xk ≡ y (mod n). Punjab University Journal of Mathematics, 51, 93–102.
  10. Rahmati, M. (2017). Some digraphs attached with congruence x5 ≡ y (mod n). Journal of Mathematical Extension, 11(1), 47–56.
  11. Rogers, T. D. (1996). The graph of the square mapping on the prime fields. Discrete Mathematics, 148, 317–324.
  12. Skowronek-Kaziow, J., & Gora, Z. (2009). Properties of digraphs connected with some congruence relations. Czechoslovak Mathematical Journal, 59, 39–49.
  13. Somer, L., & Křížek, M. (2004). On a connection of number theory with graph theory. Czechoslovak Mathematical Journal, 54, 465–485.
  14. Somer, L., & Křížek, M. (2009). On symmetric digraphs of the congruence xk ≡ y (mod n). Discrete Mathematics, 309, 1999–2009.
  15. Somer, L., & Křížek, M. (2011). The structure of digraphs associated with the congruence xk ≡ y (mod n). Czechoslovak Mathematical Journal, 61, 337–358.
  16. Szalay, L. (1992). Egy diszkrét iteráció a számelméletben. Szombathelyi Berzsenyi Dániel Tanárképző Főiskola Tudományos Közleményei, 8, 71–91 (Hungarian).

Manuscript history

  • Received: 25 August 2022
  • Revised: 3 March 2023
  • Accepted: 20 July 2023
  • Online First: 26 July 2023

Copyright information

Ⓒ 2023 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

Thakur, S. K., Goswami, P., & Ray, G. C. (2023). Enumeration of cyclic vertices and components over the congruence a^{11} \equiv b \pmod n. Notes on Number Theory and Discrete Mathematics, 29(3), 525-537, DOI: 10.7546/nntdm.2023.29.3.525-537.

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