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

- Lee, K., M. Kwon, M. K. Kang, & G. Shin (2011) Semi-primitive root modulo n, Honam Math. J., 33(2), 181–186.
- 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

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.