For questions about primitive roots in modular arithmetic, index calculus, and applications in cryptography. For questions about primitive roots of unity, use the (roots-of-unity) tag instead.
If $n$ is a positive integer, a primitive root modulo $n$ is an integer whose multiplicative order modulo $n$ is equal to $\varphi(n)$, Euler's totient function evaluated in $n$.
A primitive root modulo $n$ is often identified with its corresponding element of $\mathbb Z/n\mathbb Z$. With this identification, a number is a primitive root modulo $n$ if and only if it is a generator of the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$, in which case this group is cyclic.
A primitive root modulo $n$ exists if and only if $n$ is equal to $2$, $4$, $p^k$ or $2p^k$ for some odd prime $p$ and some positive integer $k$.
For questions about primitive roots of unity, consider using the roots-of-unity tag.