Let $p$ be a prime number. Is there a prime number $q$ such that $p$ is a primitive root of $q$?



Gupta and R. Murta showed that there were infinitely many primes $p$ for which there are infinitely many $q$. HeathBrown generalized this to show that there are at most two prime numbers $p$ for which there are not infinitely many $q$ that they are primitive roots modulo $q$. HeathBrown, D. R.(4OXM) Artin's conjecture for primitive roots. Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 145, 27–38. 11A07 (11N13 11N35) Gupta, Rajiv(1IASP); Murty, M. Ram(3MGL) A remark on Artin's conjecture. Invent. Math. 78 (1984), no. 1, 127–130. 11A07 

