Recently I carried out an elementary proof of the following assertion, which is a special case of Fermat's last theorem:
If $p$ is an odd prime and $x, y, z > 0$ are integers such that $(x, y) = 1$ and $z-y \mid x$ and $z-y \neq 1,$ then $x^{p} + y^{p} \neq z^{p}$
I am wondering, has there already been any proof in literature?
This result is an immediate consequence of the "Relations of Barlow". Without loss of generality we can assume that $p$ is not a factor of $x$. Then these relations state that there are coprime integers $r$ and $s$ such that $$x=rs,z-y=s^p$$ Your condition that $z-y$ divides $x$ would mean that $s^p$ divides $rs$ and is clearly impossible.