(Russia) Diophantine equation involving prime numbers

(Russia) Diophantine equation involving prime numbers

Find all pairs of prime nummbers $p,q$ such that $p^3 - q^5 = (p+q)^2$.
It's obvious that $p>q$ and $q=2$ doesn't work, then both $p,q$ are odd.
Assuming $p = q + 2k$ we conclude, by the equation, that $k|q^3 - q - 4$
because $\gcd(k,q)=1$ (else $p$ is not prime) and $k=1$ has no solution.
I also tried to use some modules, but I couldn't.