properties of the index of an integer with respect to a primitive root

Definition.

Let $m>1$ be an integer such that the integer $g$ is a primitive root for $m$ . Suppose $a$ is another integer relatively prime to $g$ . The index of $a$ (to base $g$ ) is the smallest positive integer $n$ such that $g^{n}\equiv a\mod m$ , and it is denoted by $\operatorname{ind}a$ or $\operatorname{ind}_{g}a$ .

Proposition.

Suppose $g$ is a primitive root of $m$ .

1.

$\operatorname{ind}1\equiv 0\mod\phi(m)$ ; $\operatorname{ind}g\equiv 1\mod\phi(m)$ , where $\phi$ is the Euler phi function.
2.

$a\equiv b\mod m$ if and only if $\operatorname{ind}a\equiv\operatorname{ind}b\mod\phi(m)$ .
3.

$\operatorname{ind}(ab)\equiv\operatorname{ind}a+\operatorname{ind}b\mod\phi(m)$ .
4.

$\operatorname{ind}a^{k}\equiv k\operatorname{ind}a\mod\phi(m)$ for any $k\geq 0$ .

Title	properties of the index of an integer with respect to a primitive root
Canonical name	PropertiesOfTheIndexOfAnIntegerWithRespectToAPrimitiveRoot
Date of creation	2013-03-22 16:20:52
Last modified on	2013-03-22 16:20:52
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	4
Author	alozano (2414)
Entry type	Theorem
Classification	msc 11-00