We know, from Bertand's postulate that there is always a prime number between $n$ and $2n$, for $n > 3$. I'm wondering if there is a similar conjecture for the number $2$ being a principal root of unity modulo a prime $p$, with maximum multiplicative order.
To make this more clear, I'm wondering if we can say something like $2$ is a principal root of unity, with maximum multiplicative order, modulo some prime $p$, with $n \le p \le \alpha \cdot n$, for all $n$ and some constant $\alpha$ that is preferably small. Does this $\alpha << \infty$ exist?
According to Artin's conjecture on primitive roots, there are infinitely many primes for which $2$ is a primitive root, and the asymptotic density of these in the primes is Artin's constant, approximately 0.3739558136. However, the conjecture is still open.
See also OEIS sequence A001122.