# Integer solutions of $ax^3 + bx^2 + cx - y^2 = k$

The Mordell curve $$y^2=x^3+k$$ is known to have finite many solutions for every integer $k\ne 0$.

How can I verify if an equation like $$ax^3 + bx^2 + cx - y^2 = k$$ with $a,b,c \in \mathbb{Z}$ have at least one solution for all $k > 0?$ Otherwise, for which values of $k$ we have solutions?

More specifically, for which $k$ we can find solutions to $12x^3 + 36x^2 + 24x - y^2 = k?$ For example, if $k =279,$ $x = 19$ and $y = 309.$

I tried to solve it putting the equation in the Weierstrass form, and using the Mordell-Weil theorem to find possible integer solutions, but this idea doesn't work...

• The Mordell curve has only finite many integer points, true, but it is not at all trivial to find all the solutions in general. The known upper bound, depending on $k$, is far too large for brute force calculation. – Peter Mar 3 '18 at 13:59