Are all Mordell equations $y^2=x^3+k$, for any integer $k$, solvable

Question

Are all Mordell equations $y^2=x^3+k$, for any integer $k$, solvable? Not that there are solutions $x,y$ for every $k$, but that you can determine for every $k$ if there are solutions, and if there are, what those solutions are. That is also considering that you have a finite amount of time to solve them.

Answer 1

In fact, there are algorithms for finding all integer solutions to any elliptic curve over the rationals.

See, for example, the paper Solving elliptic diophantine equations: the general cubic case.

In particular, the authors mention explicit upper bounds for integer solutions, so in principle we can find all integer solutions just by checking a finite list. There are more practical algorithms out there, of course, but this gives an affirmative answer to your question.

Answer 2

Theorem by Baker (see here, page $20$): If $y^2=x^3+k$ for some $x,y,k\in\Bbb Z, k\neq 0$, then $$\max\{|x|,|y|\}\le \left(e^{10^{10}}\right)^{|k|^{10\, 000}}$$

Practically, it's not solvable for all $k\in\Bbb Z_{\neq 0}$ because of computer power/time constraints. Mordell's equation is fully solved when $0<|k|\le 10^4$ and for some $10^4<|k|\le 10^5$, but not when $|k|>10^5$. See this paper from $1998$ (i.e. written only $17$ years ago).