On the number of semi-primitive roots modulo n

Pinkimani Goswami and Madan Mohan Singh
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 21, 2015, Number 4, Pages 48—55
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Authors and affiliations

Pinkimani Goswami
Department of Mathematics, North-Eastren Hill University
Permanent Campus, Shillong–793022, Maghalaya, India

Madan Mohan Singh
Department of Basic Sciences and Social Sciences, North-Eastern Hill University
Permanent Campus, Shillong–793022, Maghalaya, India

Abstract

Consider the multiplicative group of integers modulo n, denoted by ℤ*n. An element a ∈ ℤ*n is said to be a semi-primitive root modulo n if the order of a is φ (n)/2, where φ(n) is the Euler’s phi-function. In this paper, we’ll discuss on the number of semi-primitive roots of non-cyclic group ℤ*n and study the relation between S(n) and K(n), where S(n) is the set of all semi-primitive roots of non-cyclic group ℤ*n and K(n) is the set of all quadratic non-residues modulo n.

Keywords

  • Multiplicative group of integers modulo n
  • Primitive roots
  • Semi-primitive roots
  • Quadratic non-residues
  • Fermat primes

AMS Classification

  • 11A07

References

  1. Lee, K., M. Kwon, M. K. Kang, & G. Shin (2011) Semi-primitive root modulo n, Honam Math. J., 33(2), 181–186.
  2. Lee, K., M. Kwon, & G. Shin (2013) Multiplicative groups of integers with semi-primitive roots modulo n, Commum. Korean Math. Soc., 28(1), 71–77.

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Cite this paper

APA

Goswami, P., & Singh, M. M. (2015). On the number of semi-primitive roots modulo n. Notes on Number Theory and Discrete Mathematics, 21(4), 48-55.

Chicago

Goswami, Pinkimani, and Madan Mohan Singh. “On the Number of Semi-primitive Roots Modulo n.” Notes on Number Theory and Discrete Mathematics 21, no. 4 (2015): 48-55.

MLA

Goswami, Pinkimani, and Madan Mohan Singh. “On the Number of Semi-primitive Roots Modulo n.” Notes on Number Theory and Discrete Mathematics 21.4 (2015): 48-5. Print.

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