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
Full paper (PDF, 95 Kb)

Details

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.

Related papers