Solutions to $x^2 + y^3 = z^2$

Question

The question I was given: Find all solutions to $x^2 + y^3 = z^2$ in which x,y, and z are pairwise relatively prime and y is even. What I have so far: If $y$ is even then the equation becomes $x^2+8*b^3=z^2$ where b is the integer $y/2$ and $x$ and $z$ must be odd. I don't know if I should do anything mod 8 or how to approach this problem.

Answer 1

Above equation shown below;

$x^2 + y^3 = z^2$

Since variable $(y)$ is required to be even & comment

given by "Lulu" gives rational solution of $(x,z)$ instead of integer solution,

another solution is given below:

$(x,y,z)= ((2k^2-k),(2k),(2k^2+k))$

For $k=2$ we get $(x,y,z)=(6,4,10)$