Does $x^3+k=y^2$ has infinite solution when $k \in \{1,2\}$?

Question

I am looking for results regarding power Diophantine equation $x^3+k=y^2$ when $k \in \{1,2\}$? Does it have infinite integer solution?

If possible plz comment on What happens when $k \in \mathbb{N}$. Please provide reference (any kind of online document is okay).

Edit: I have been told, $(x^3 - 1) + 2 = y^2$ has finitely many solutions, $(x^3 - 1) + 1 = y^2$ has infinitely many solutions, where can I find about these solution?

Answer 1

The second one is trivial.

$(x^3 - 1) + 1 = y^2 $ is the same as $x^3 = y^2 $ which has the infinite class of solutions $x = n^2, y=n^3$ so that both sides are $n^6$.