Primitive roots 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 $(\mathbb{Z}/p\mathbb{Z})^\times$ is generated by 2 and 3 for infinitely many primes $p$?
 A: I think the best approximation is due to Heath-Brown: for infinitely many primes $p$, one of $2$, $3$, $5$ is a primitive root mod $p$. Actually, this result works with any three primes in place of $2$, $3$, $5$.
A: This is an interesting question. More generally people have considered the following. Let $\Gamma$ be a subgroup of $\mathbf{Q}^*$ generated by $r$ primes. What can one say about 
$$
  N_\Gamma(X)=|\{p < X : \Gamma \bmod p \textrm{ generates } (\mathbf{Z}/p
    \mathbf{Z})^{\times}\}|.
$$
There is also a natural elliptic analogue. Thus let $E/\mathbf{Q}$ be an elliptic curve and let $\Gamma\subset E(\mathbf{Q})$ be a subgroup of rank $r$. Then we can consider
$$
  N_\Gamma(X)=|\{p < X : \Gamma \bmod p \textrm{ generates } E(\mathbf{Z}/p
    \mathbf{Z})\}|.
$$
Gupta and Murty give a number of results, both conditional and unconditional, in their paper Primitive points on elliptic curves, Compositio Math. 58 (1986), 13–44. For example, if $r\ge6$ and $E$ has complex multiplication, then they prove unconditionally that $N_\Gamma(X)\gg X/(\log X)^2$. In the non-CM case, assuming the GRH, they prove that if $r\ge18$, then $N_\Gamma(X)\gg X/(\log X)$.
It would be interesting to investigate similar questions on higher dimensional algebraic groups, either abelian varieties of dimension $\ge2$, or even on $(\mathbf{Q})^{\times}\times(\mathbf{Q})^{\times}\times\cdots\times(\mathbf{Q})^{\times}$.
