In Ian Stewart's book on Algebraic Number Theory, he gives the following proof that the solutions to $y^2 + 2 = x^3$ is $x=3, y = \pm 5$:
I'm confused about the step "since $c^2 + 2d^2 | 4y^2$ and $c^2 + 2d^2 | 8$ then $c^2 + 2d^2 | 4$". How is this implied?
Also, he lists several sets of valid solutions, like $c = \pm 2, d = 0$, but somehow none of these correspond to any "proper factors" of the two: why?