Every prime number has a primitive root. Actually, for every prime number p a considerable percentage of the numbers from 2 to p-1 are primitive roots (with the exception p = 2 which is the only prime that has 1 as a primitive root).
Following a discussion how to find primitive roots, I thought it would be a good idea to start checking whether the small primes (2, 3, 5, 7, 11, ... ) are primitive roots because they seem to be more than average likely to be primitive roots. That obviously raises the question whether every prime p other than 2 actually has a primitive root in the range from 2 to p-1 that is prime.
Googling didn't find any answer. There doesn't seem to be an obvious answer, for example p = 41 has the primitive root 6 but neither 2 nor 3 are primitive roots (2^20 = 3^8 = 1 modulo 41). There should always be a prime primitive root because of the sheer numbers of primitive roots (p = 271 with smallest prime primitive root 43 seems quite exceptional), but a proof would be nice.