primitive rootPrimitiveRoot2006-07-11 11:12:482008-01-25 16:29:53Definition% this is the default PlanetMath preamble. as your knowledge
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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 \emph{primitive root} is a generator of this group of units when it is cyclic.
Equivalently, one can define the integer $r$ to be a {\em 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 $70(\ln p)^2$.
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Wikipedia, ``Primitive root modulo n''
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