Solve $x^n+z^n=(x+1)^n$ for $n\ge 3$ without FLT

Question

Is there a way to prove that for $x,z,n \in \mathbb{Z}$, $x > 0$, $z > 0$, $n > 2$, the equation

$$ x ^ n + z ^ n = (x + 1) ^ n $$

has no solution, without using Fermat's Last Theorem?

Best regards