Least primitive root: Difference between revisions

Latest revision as of 22:54, 29 May 2010

Definition

Suppose $n$ is a natural number such that the multiplicative group modulo $n$ is cyclic (i.e., $n$ is either a prime number, or $4$ , or a power of an odd prime, or twice the power of an odd prime). The least primitive root or smallest primitive root modulo $n$ is the smallest natural number $a$ such that $a$ is a primitive root modulo $n$ .

Particular cases

Here, we only list $n$ equal to a prime or the square of a prime. For something that is a higher power of an odd prime, the smallest primitive root modulo the square of the prime works.

$n$	Smallest primitive root modulo $n$
2	1
3	2
4	3
5	2
7	3
11	2
13	2
17	3

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 ==Definition==
-Suppose <math>n</math> is a [[natural number]] such that the multiplicative group modulo <math>n</math> is cyclic (i.e., <math>n</math> is either a [[prime number]], or <math>4</math>, or a power of an odd prime, or twice the power of an odd prime). The '''smallest primitive root''' modulo <math>n</math> is the smallest [[natural number]] <math>a</math> such that <math>a</math> is a [[defining ingredient::primitive root]] modulo <math>n</math>.
+Suppose <math>n</math> is a [[natural number]] such that the multiplicative group modulo <math>n</math> is cyclic (i.e., <math>n</math> is either a [[prime number]], or <math>4</math>, or a power of an odd prime, or twice the power of an odd prime). The '''least primitive root''' or '''smallest primitive root''' modulo <math>n</math> is the smallest [[natural number]] <math>a</math> such that <math>a</math> is a [[defining ingredient::primitive root]] modulo <math>n</math>.
 ==Particular cases==