I have the equation $p=qb^4-a^2$, where $p<q$ are odd primes and $a\in\mathbb{Z}$ and $b\in\mathbb{N}$.
Is it either provable or is there a conjecture for the existence of infinite many solutions $(a,b,p,q)$?
What I know so far:
I found in Bordellès Book "Arithmetic Tales. Advanced Edition" (page 295) that according to Polignac's conjecture (which isn't proved), there are infinitely many examples per any even difference of both primes for the slightly different case: $q=pb^4+a^2$.
I would already be very happy to know if there is any theorem or conjecture that says there are infinitely many solutions for my current case $p=qb^4-a^2$.