TITLE: Least prime in Artin's primitive root conjecture
QUESTION [6 upvotes]: Let $a$ be an integer which is neither a square nor $-1$. Artin's conjecture states that there are infinitely many primes $p$ for which $a$ is a primitive root modulo $p$. My question is whether there is anything on the literature for
(1) the (conjectural) size of the smallest such prime $p$.
(2) Conditional or unconditional upper or lower bounds.
Regarding (2), one can chase down the implied constants in Hooley's paper (Hooley, Christopher (1967). "On Artin's conjecture." J. Reine Angew. Math. 225, 209-220) to show that his asymptotic must kick in after $x\geq x_0=|a|^{C\log \log 3|a|}$ for some absolute $C>0$. Thus, under GRH the least prime is at most  $$|a|^{C\log \log 3|a|}.$$

REPLY [8 votes]: One would guess much smaller.  If p=2q+1 where p and q are prime, then
a is a primitive root mod p IFF (a/p)=-1 and p does not divide a+1.
One would guess from the usual heuristics that there is such a prime p once one has tested about c log |a| loglog |a| of them. Therefore we would expect to find such a p which is << log |a| (log log |a|)^3.  For most a, one of the first 100 Sophie Germain primes will work; indeed I very much doubt you can compute an example where they will not.