Equivalence of Fermat's Last Theorem and Beal's Conjecture

Abstract

It is proved in this paper that (1){ \bf Fermat's Last Theorem:} If $\pi$ is an odd prime, there are no relatively prime solutions $x, y, z$ to the equation $z^\pi=x^\pi+y^\pi,$ and (2) { \bf Beal's Conjecture :} The equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive integers $x, y, z$ with $\mu, \xi, \nu$ odd primes at least $3$. It is proved that these two statements are equivalent.