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 $p$?

I think the best approximation is due to HeathBrown (Quart. J. Math. Oxford Ser. 37, 2738.): for infinitely many primes p, one of 2,3,5 is a primitive root mod p. Actually HeathBrown's theorem works for any three primes in place of 2,3,5. You can find his paper online here (praise Google). 


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 nonCM 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 $(\mathbf{Q})^{\times}\times(\mathbf{Q})^{\times}\times\cdots\times(\mathbf{Q})^{\times}$. 


This is just a comment, but I'm unable to do it, so I post it as an answer: $2^{11} \equiv 3^{11} \equiv 11 \, \mod 23$. And a similar situation happens with other primes: $47, 71, 73, 97, 167, 191, 193, 239, 241, 263, 307, 311, 313, ...$ although not always with $ord_{p}(2)=ord_{p}(3)=\frac{p1}{2}$. 

