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
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Related: Solve $(x+1)^n-x^n=p^m$ in positive integers and Solve $n^s-(n-1)^t=1$ for $n,s,t>0$ – Bart Michels Jul 03 '15 at 10:07
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I have edited the question with an equivalent equation. – user26486 Jul 04 '15 at 08:00
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We may assume that $n=4$, or $n=p>2$ is prime. Then see this post. – Dietrich Burde Oct 23 '22 at 18:20