# 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. 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. 2.

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

3. 3.

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

4. 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 PropertiesOfTheIndexOfAnIntegerWithRespectToAPrimitiveRoot 2013-03-22 16:20:52 2013-03-22 16:20:52 alozano (2414) alozano (2414) 4 alozano (2414) Theorem msc 11-00