Show $4x^3 + y^3 = 792,864,313,578,917,724,246$ has no solution for $x, y \in \mathbb{Z}$.

Question

I think it involves something about looking at the last digits of the number and/or modular arithmetic but I don't remember how to do this. Help?

Answer 1

Assume there is such a solution $(x,y)$. Since $4x^3$ and the right hands side are even, we conclude that $y$ is even. Then the left hand side is a multiple of $4$, but the right hand side is not - contradiction!

Answer 2

You want the sum to be even. That means that $y$ must be even. So say $y = 2m$, then $$ 4x^3 + y^3 = 4x^3 + 8m^3 = 4(x^3 + 2m^3). $$ (It is not hard to see that $2\cdot 39643215678945886212\color{red}3 = 792864313578917724246$)