Deduce that there are infinitely many primes p such that 2 i

Deduce that there are infinitely many primes p such that 2 is not a primitive root mod p.

Solution

Let a be a primitive root of p.

Then a+kp is a primitive root of p for every k.

By Dirichlet\'s Theorem on primes in arithmetic progression, there are infinitely many primes of the form a+kp.

Remark: The problem of whether for any prime p>2

there is a primitive roots a with 2 a (p1) is open. I believe that the best unconditional result is that for large enough p, the least prime primitive root of p is <pk, for a constant k3. Under reasonable but unproved hypotheses (versions of GRH) one can bring this down to a power of (log p).