$$p^2-1 = kq^3 - kq$$ For a given $k$ where $p$ and $q$ are prime, is there something that can be told about this Diophantine equation, for example, whether it has finite solutions or if there exists solutions they cannot be bigger than a certain number.
In general certain properties of the numbers that satisfy this Diophantine equation.
I've ran code to test it out and observed that values of q are generally around the value of k (around the same magnitude).
Update: I ran the code for $5 < k < 1000$ and $q < 100000$ and there was only one instance of $k$ where $q/k > 2$ which was 995 and $q/k$ had a value of 5.82 with $q$ being 5741