Let $g$ be a positive integer and $S$ be the set of prime divisors of $g$. Does there exist a $g$ such that $g$ is a primitive root modulo all primes not in $S$?
I was wondering about this, but it seemed hard to prove or disprove this because there are infinitely many primes. The condition that $g$ is taken modulo $p$ which is not a prime divisor of $g$ means that $g$ is invertible modulo $p$. How can solve this question?