The equation $y^2 = x^3 + k$ for $k = (4n-1)^3 - 4m^2$, with $m, n \in \mathbb{N}$ and no prime number that p is congruent to 1 modulo 4 divids m, doesn't have any answer and its proof can be obtained by using quadratic reciprocity law.
Do you know answers of this equation for two or three different values of $k$? In addition, do you know any reference about that?
This is a famous class of elliptic curves, called Mordell's equation, or sometimes Mordell-Bachet equation. See also here, or here for some discussions on MSE. For a specific example with $k=2000000$ see also here. A further reference is this article by Keith Conrad.