$\DeclareMathOperator\ord{ord}$This is related to a question in the MO post, [Does there exist a prime $p$ such that $\left|\frac{\mathrm{ord\_{p}}(a)}{\mathrm{ord\_{p}}(b)}-c\right|<\gamma$ for some small constant $\gamma$?](https://mathoverflow.net/q/376040/160943)


Let $p$ be a prime and $\ord\_{p}(a)$ be the least positive integer $d$ such that $p\mid a^{d}-1$.


If $a$ and $b$ are two coprime natural numbers greater than
1, then does there exist a prime $p$
such that $ \frac{\ord\_{p}(a)}{\ord\_{p}(b)}>1$?


**Edit:**
Some progress on this problem has been made recently [here.](https://arxiv.org/abs/2102.00370)



Does there exist a prime $p$ such that $ \frac{\operatorname{ord}_{p}(a)}{\operatorname{ord}_{p}(b)}>1?$
This question might be difficult, but heuristically it should be true. Essentially the same heuristic that yields the Artin primitive root conjecture, there should be for any such $a$ and $b$ be infinitely many primes $p$ where $a$ is a primitive root mod $p$, and $b$ is not.


Another heuristic approach is that if $a$ and $b$ are relatively prime, there should be infinitely many primes $p$ where $\frac{p-1}{2}$ is prime, $a$ is a quadratic non-residue mod $p$, and $b$ is a quadratic residue mod $p$. So if $p$ is large enough, one must have that the order of $a$ is $p-1$, and the order of $b$ is $\frac{p-1}{2}$.


