There is an abundant literature, and even here on MO no shortage of questions, on the question of the smallest prime primitivee root modulo $q$ (where $q$ is a prime, or more gener …

I have had this question on my mind for two decades. We know, after Heath-Brown, that one out (say) of 3, 5, 7 is a primitive root mod p for infinitely many primes p. We just don't …

Let $n\in\mathbb{Z}^+$ and $\alpha,\beta $ be two generators of the cyclic group $\left(\frac{\mathbb{Z}}{(2^n - 1)\mathbb{Z}},+\right)$. Question: What are known theorems regardi …

For $p$ a prime number, let $G(p)$ be the least prime $q$ such that $q$ is a primitive root mod $p$, that is $q$ generates the multiplicative group $(\mathbb Z/p\mathbb Z$)* . …

It appears (from computer experiments) that if $p$ is a prime such that 2 generates the multiplicative group $\mathbb{F}_p^\times$ of the corresponding finite field $\mathbb{F}_p$ …

Artin conjectured that if $a$ is an integer which is not a square and not $-1$ then $a$ is a primitive root for infinitely many primes. This conjecture has not been resolved, but …

If Artin's conjecture on primitive roots is true, then 2 generates $(\mathbb{Z}/p\mathbb{Z})^\times$ for infinitely many primes $p$. Can one at least show that $(\mathbb{Z}/p\mathb …

Given a positive integer $n > 1$, is it true that there exists infinitely many primes $p$ such that $n$ is a primitive root modulo $p$.