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.
$$p^3-q^5=(p+q)^2=p^2+2pq+q^2$$ So, $p^3-q^5\ge 0$, so $p^3\ge q^5$, so $p>q$. $$p^2(p-1)=p^3-p^2=q^5+2pq+q^2=q(q^4+2p+q)$$
Hence, $p^2$ divides $q^4+2p+q$ (as it can't divide q) and so $q$ divides (p-1).
So $p^2\le q^4+2p+q$, so $p\le q^2+1$. We can then verify easily that $q=2$ has no solution for $p$.
But $p$ is prime, and $q^2+1$ and $q^2$ are not so : $$q<p\le q^2-1$$
So let $p=aq+b$ with $1\le a< q$ and $1\le b<q$.
We have $$q^5=p^3-(p+q)^2=(b^3-b^2)+q.(\dots)$$ Hence $q$ divides $b^2(b-1)$, so $b=1$
Hence $p=aq+1$ with $1\le a< q$
We have $$q^5=p^3-(p+q)^2=q.(3a-2(a+1))+q^2.(\dots)$$
Hence, $q$ divides $a-2$, so $a=2$.
So $p=2q+1$
$$q^5=q^2.(8q+12-9)$$ $$0=q^3-8q-3=(q-3).(q^2+3q+1)$$
The only positive integer solution is $q=3$ and so $p=7$.