# SEMI-PRIMITIVE ROOT MODULO n

• Lee, Ki-Suk (Department of Mathematics Education, Korea National University of Education) ;
• Kwon, Mi-Yeon (Department of Mathematics, University of Wisconsin-Platteville) ;
• Kang, Min-Kyung (Department of Mathematics Education, Korea National University of Education) ;
• Shin, Gi-Cheol (Department of Mathematics Education, Korea National University of Education)
• Accepted : 2011.03.25
• Published : 2011.06.25
• 101 23

#### Abstract

Consider a multiplicative group of integers modulo n, denoted by $\mathbb{Z}_n^*$. Any element $a{\in}\mathbb{Z}_n^*$ n is said to be a semi-primitive root if the order of a modulo n is $\phi$(n)/2, where $\phi$(n) is the Euler phi-function. In this paper, we classify the multiplicative groups of integers having semi-primitive roots and give interesting properties of such groups.

#### Keywords

Multiplicative group of integers modulo n;primitive roots;semi-primitive roots

#### References

1. M. Abramowitz, I. A. Stegunl, Handbook of Mathematical Functions, Dover Publication, New York, 1964.
2. C. F. Gauss; A. A. Clarke (translator into English), Disquisitiones Arithemeticae, Springer, New York, 1986.
3. H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhauser, Boston, 1994.
4. H. E. Rose, A Course in Number Theory, Oxford University Press Inc., New York, 1994
5. J. K. Strayer, Elementary Number Theory, Waveland Press, Inc., 2002.

#### Cited by

1. MULTIPLICATIVE GROUPS OF INTEGERS WITH SEMI-PRIMITIVE ROOTS MODULO n vol.28, pp.1, 2013, https://doi.org/10.4134/CKMS.2013.28.1.071