# on prime numbers which are primitive roots of a prime

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

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Have a look at Artin's conjecture. –  Felipe Voloch Jun 29 '13 at 12:55
This question appears to be off-topic because it is about a well-known open problem. –  Felipe Voloch Jun 29 '13 at 12:55
Ali asks if there is one (as opposed to infinitely many) such prime number $q$. This may be easier. –  Péter Komjáth Jun 29 '13 at 13:08

Gupta and R. Murta showed that there were infinitely many primes $p$ for which there are infinitely many $q$. Heath-Brown 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 $q$.