|
Given any positive integer $n$ , the group of units $U(\mathbb{Z}/n\mathbb{Z})$ of the ring $\mathbb{Z}/n\mathbb{Z}$ is a cyclic group iff $n$ is 4, $p^m$ or $2p^m$ for any odd positive prime $p$ and any non-negative integer $m$ . A primitive root is a generator of this group of units when it is cyclic.
Equivalently, one can define the integer $r$ to be a primitive root modulo $n$ , if the numbers $r^0,\,r^1,\,\ldots,\,r^{n-2}$ form a reduced residue system modulo $n$ .
For example, 2 is a primitive root modulo 5, since $1,\; 2,\; 2^2 = 4,\; 2^3 = 8 \equiv 3 \pmod{5}$ are all with 5 coprime positive integers less than 5.
The generalized Riemann hypothesis implies that every prime number $p$ has a primitive root below $70(\ln p)^2$ .
- Wikipedia, ``Primitive root modulo n''
|
