Considering $\mod 11$ we get that $x^5 \equiv 0,1,-1 \pmod {11}$, and $y^2 \equiv 0,1,3,4,5,9$, so $y^2 + 4 \equiv 2,4,5,7,8,9$, which none is equal to $0,1$ or $10$. So there are no solutions. The total number of solutions is $0$.

1998 Balkan MO p4