# Generalization of Artin conjecture to square of primes [closed]

Artin's conjecture on primitive roots states that a given integer a that is neither a perfect square nor −1 is a primitive root modulo infinitely many primes p. Can we expect that it is also a primitive root modulo infinitely many squares of primes?

Yes. While being a primitive root (mod $$p$$) does not guarantee being a primitive root (mod $$p^2$$), it makes it extremely likely. (The proportion of exceptions for a given prime $$p$$ is $$1/p$$.) Therefore we expect that for the vast majority of primes $$p$$ for which $$a$$ is a primitive root, $$a$$ is also a primitive root (mod $$p^2$$). Almost equivalently, we expect there to be very few primes $$p$$ such that $$p^2\mid(a^p-a)$$. However, all of these statements are beyond our current ability to prove, I believe.
• This seems like a perfectly fine answer for a perfectly fine question. I'm confused why it is closed. The weaker question as to whether $2^{p-1} \not\equiv 1 \mod p^2$ for infinitely many $p$ still remains open. – user760870 Jul 12 at 16:56