Show that $x^3 + y^3 + z^3 = 4$ has no solutions?

Question

As the title says, I've been asked to show that the equation $x^3+y^3+z^3 = 4$ has no solutions for any integers $x$, $y$, and $z$.

I'm a bit stuck on where to start, so I'd appreciate any help or hints. Thanks!

Answer 1

For any $x\in\mathbb{Z}$, $\;x^3\!\pmod{9}$ can be only $0$ or $\pm 1$, hence the sum of three cubes cannot be a number of the form $9k\pm 4$.

There is a well-known conjecture stating that every integer number $\not\equiv\pm 4\pmod{9}$ can be expressed as the sum of three cubes, and every integer number $\equiv 4\pmod{9}$ can be expressed as the sum of four cubes.