Is $x^3 + y^3 = z^3$ possible?

Question

Is $x^3 + y^3 = z^3$ possible when $x$, $y$ and $z$ are integers? If not, how to prove that they are not possible?

(I am a grade 10 student so please answer in a simple way)

Answer 1

By Fermat's Last Theorem (Andrew Wiles proved it), if for $x,y,z\in\mathbb Z$ $$x^3+y^3=z^3,$$ then $$xyz=0.$$

This implies that at least one of $x,y,z$ has to be $0$.

So, you'll know there are infinitely many solutions such as $(0,t,t),(t,0,t),(t,-t,0)$ for any $t\in\mathbb Z$.