Let $x=512=2^9$, $y=675=3^3 5^2$, $z=720=2^4 3^2 5^1$. Then the problem asks us to prove that $x^3+y^3+z^3$ is not prime. Since $2z^2=3xy$, we have that $x^3+y^3+z^3 = x^3+y^3+(-z)^3 -3xy(-z)$, which we can factor as $(x+y-z)(x^2+zx-xy+zy+z^2+y^2)$. Thus, $x^3+y^3+z^3 = (x+y-z)(x^2+zx-xy+zy+z^2+y^2) = 467\cdot1745209$.

Nice solution!!

Where did the motivation for the factorization of $x^3+y^3+z^3 = x^3+y^3+(-z)^3 -3xy(-z)$ to $(x+y-z)(x^2+zx-xy+zy+z^2+y^2)$ come from?

Python54 wrote: Where did the motivation for the factorization of $x^3+y^3+z^3 = x^3+y^3+(-z)^3 -3xy(-z)$ to $(x+y-z)(x^2+zx-xy+zy+z^2+y^2)$ come from? It is well known identity