The question is

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> Find all primes $(p,q)$ such that $pq$ divides $p^3 + q^3 +1$.
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My attempt:

This reduces to finding primes $p,q$ such that p divides $q^3+1=(q+1)(q^2-q+1)$ and q divides $p^3+1=(p+1)(p^2-p+1)$.

Now we have 4 cases: If p divides $q+1$ and q divides $p+1$, then $\{p,q\}=\{2,3\}$.

I don't know how to deal with the other cases. For example, what happens if p divides $q+1$ and q divides $p^2-p+1$?