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2013•Communications of the Korean Mathematical SocietyOpen access

MULTIPLICATIVE GROUPS OF INTEGERS WITH SEMI-PRIMITIVE ROOTS MODULO n

Ki-Suk Lee, Miyeon Kwon, Gicheol Shin

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Abstract

Consider a multiplicative group of integers modulo $n$ , denoted by $\mathbb{Z}_n^*$ . Any element $a{\in}\mathbb{Z}_n^*$ 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 discuss some interesting properties of the multiplicative groups of integers possessing semi-primitive roots and give its applications to solving certain congruences.

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Consider a multiplicative group of integers modulo $n$ , denoted by $\mathbb{Z}_n^*$ . Any element $a{\in}\mathbb{Z}_n^*$ 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 discuss some interesting properties of the multiplicative groups of integers possessing semi-primitive roots and give its applications to solving certain congruences.

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Available abstract

Consider a multiplicative group of integers modulo $n$ , denoted by $\mathbb{Z}_n^*$ . Any element $a{\in}\mathbb{Z}_n^*$ 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 discuss some interesting properties of the multiplicative groups of integers possessing semi-primitive roots and give its applications to solving certain congruences.

Key concepts: Mathematics, Modulo, Primitive root modulo n, Multiplicative function, Multiplicative group, Congruence relation, Combinatorics, Primitive element

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