Let $g(p)$ be the least positive primitive root of the prime $p$, the primitive roots being the generators of the cyclic group $\mathbb{Z}_{p-1}$. These are the values for the first prime numbers:
$$g(3) = 2$$
$$g(5) = 2$$
$$g(7) = 3$$
$$g(11) = 2$$
$$g(13) = 2$$
$$g(17) = 3$$
$$g(19) = 2$$
$$g(23) = 5$$
$$g(29) = 2$$
$$g(31) = 3$$
$$g(37) = 2$$
$$g(41) = 6$$
I've learned that it's hard to calculate $g(p)$ for arbitrary $p$.
Main questions
(1) Is there a known connection between $g(p)$ and some other number theoretic functions?
E.g. $g(p)=$ some number theoretic function $f$ of the number $\phi(p-1)$(2) Is there a known connection between the property $g(p)=2$ and some other number theoretic properties?
E.g. $g(p)=2$ iff the number $\phi(p-1)$ has some number theoretic property $P$
Side questions
(3) What's the asymptotic probability that $g(p) = 2$?
(4) Are there values $g(p)$ cannot take?
What about $g(p) = 4$ and higher powers of $2$? What about $g(p) = 9$? See OEIS.(5) How hard is $g(p)$ exactly?
Is it possibly NP hard?
