$m^2+2017=n^3$ has no solutions

Question

Show that $m^2+2017=n^3$ has no solutions for positive integers $m,n$.

I'm having trouble tackling this one, especially since $\mathbb{Z}[\sqrt{-2017}]$ isn't a UFD. We can write the equation as $m^2+45^2=n^3+8$ or $m^2+17^2=n^3-12^3$, but I can't do much with either.

Answer 1

Your claim is false.

$$81060^2 + 2017 = 1873^3$$