Simple method to prove $a^3+b^3=1$ has no integer solutions if $ab\neq 0$ Any simple method to prove $a^3+b^3=1$ has no integer solutions if $ab\neq 0$ that does not involve Fermat's last theorem?

$$1 = a^3+b^3 = (a+b)(a^2-ab+b^2) = (a+b)((a+b)^2-3ab)$$
implies $a+b=\pm 1$ and so $(a+b)^2 - 3ab = 1 - 3ab \not\in \{-1,1\}$ if $ab \ne 0$
