# Some questions concerning the generators of cyclic groups

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?