I was wondering whether one can prove for some fixed integer powers of fixed primes $p_1^a$ and $p_2^b$ that there exists some prime $p_3$ which has a primitive root g, such that:
$p_1^a - p_2^b g \equiv 0 \pmod{p_3}$
That is:
$\frac{p_1^a}{p_2^b} \equiv g \pmod{p_3}$
I was thinking that for $p_3 = 2p_4 + 1$ there should be a high likelihood of finding such a generator since $\varphi(p_3) = 2p_4$ in that case, thus limiting the possible sub-groups -- but I thought this seems like a problem for which there probably exists established theorems hence the question.
Thanks a lot!