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 $k\\geq 0$ .

单词	PropertiesOfTheIndexOfAnIntegerWithRespectToAPrimitiveRoot
释义	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 $k\\geq 0$ .
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