Prove that there does not exist an integer solution for the diophantine equation $x^5 - y^2 = 4$.
It's obvious that $x$ and $y$ are of the same parity. We can also claim that if $x$ is odd, then it is $1 \pmod 4$. Also, if $x$ and $y$ are even, then $y \equiv 2 \pmod 4\text{ since } x^5 \text{ is a multiple of } 32$ and if $y$ were a multiple of $4$ then it would be at a distance of at least $16$ from $x^5$ or be equal to it. These are my observations.
How should I proceed with the proof? Please give hints.
Hint: Consider working modulo $11$.