properties of the index of an integer with respect to a primitive root
Definition.
Let $m\mathrm{>}\mathrm{1}$ 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 ${g}^{n}\mathrm{\equiv}a\mathrm{mod}m$, and it is denoted by $\mathrm{ind}\mathit{}a$ or ${\mathrm{ind}}_{g}\mathit{}a$.
Proposition.
Suppose $g$ is a primitive root of $m$.

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

2.
$a\equiv bmodm$ if and only if $\mathrm{ind}a\equiv \mathrm{ind}bmod\varphi (m)$.

3.
$\mathrm{ind}(ab)\equiv \mathrm{ind}a+\mathrm{ind}bmod\varphi (m)$.

4.
$\mathrm{ind}{a}^{k}\equiv k\mathrm{ind}amod\varphi (m)$ for any $k\ge 0$.
Title  properties of the index of an integer with respect to a primitive root 

Canonical name  PropertiesOfTheIndexOfAnIntegerWithRespectToAPrimitiveRoot 
Date of creation  20130322 16:20:52 
Last modified on  20130322 16:20:52 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  4 
Author  alozano (2414) 
Entry type  Theorem 
Classification  msc 1100 