Primitive root

A unit $g$ modulo $N$ is a primitive root when its multiplicative order is $\phi (N)$.

Let $p$ be an odd prime and let $b$ be primitive modulo $p$. Then $b$ is primitive modulo every $p^m$ if and only if it is primitive modulo $p^2$. In that case

An integer $n$ has a primitive root exactly whenwhere $p$ is an odd prime and $a\ge 1$. When primitive roots exist, their number is $\phi (\phi (n))$.