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 $p$?
[Math] Primitive roots
nt.number-theoryprimitive-roots
Related Solutions
The answer is "yes" - the order mod p of 2 is almost always as large as the square root of p (actually you get epsilon less than this in the exponent). If you take r multiplicatively independent numbers and ask for the group they generate mod p, the exponent is r/(r + 1). This is a paper of mine, and then in a paper of the Murtys, and I think is referenced in some form by Heath-Brown (it is the less deep part of his technique - to get something serious out of it you need something like Chen's method for Goldbach).
There is a simple answer here, so someone might as well record it.
Let $n$ be a nonzero integer. If $n = -1$ or $n$ is a square then there is no prime $p > 3$ such that $n$ is a primitive root modulo $p$. There are no other obvious obstructions. (It is worth thinking for a second why we do not have to rule out $n$ being a cube, for instance: this is a nice exercise in cyclic group theory.)
There is a famous conjecture that these obvious necessary conditions are the only ones: namely Artin's Primitive Root Conjecture asserts that for any integer $n$ which is not $0$, $-1$ and not a square, there are infinitely many prime numbers $p$ such that $n$ is a primitive root modulo $p$. In fact the conjecture is more precise than this: the set of primes $p$ for which such an $n$ is a primitive root is conjectured to have positive relative density among all primes and, at least under some mild additional restrictions, this density is conjectured to be a certain specific number which is independent of $n$:
$ C = \prod_{p \text{ prime}} \left(1- \frac{1}{p(p-1)} \right)$;
this $C$ is known as Artin's constant. This conjecture was proved by C. Hooley in 1967 assuming the Generalized Riemann Hypothesis. More recently unconditional results have been given by Gupta, R. Murty and Heath-Brown which consider several numbers $n$ at a time and show that Artin's Conjecture must be true for at least one of them. But the conjecture is still open for any one fixed value of $n$.
Best Answer
I think the best approximation is due to Heath-Brown (Quart. J. Math. Oxford Ser. 37, 27-38.): for infinitely many primes p, one of 2,3,5 is a primitive root mod p.
Actually Heath-Brown's theorem works for any three primes in place of 2,3,5. You can find his paper online here (praise Google).