Proofs of Fermat's Last Theorem\\and Beal's Conjecture

Abstract

If $\pi$ is an odd prime and $x, y, z,$ are relatively prime positive integers, then $z^\pi\not=x^\pi+y^\pi.$ In this note, an elegant simple proof of this theorem is given  that if $\pi$ is an odd prime and $x, y, z$ are positive integers satisfying $z^\pi=x^\pi+y^\pi,$ then  $x, y, z,$ are each divisible by $2:$ (Beal\rq{}s conjecture) The equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive integers $x, y, z, $ with $\xi, \mu, \nu$ primes at least $3.$ is also proved; that is $x, y, z $ are all even.