TY - JOUR AU - James Joseph PY - 2016/12/01 Y2 - 2024/03/29 TI - Proofs of Fermat's Last Theorem\\and Beal's Conjecture JF - Journal of Progressive Research in Mathematics JA - JPRM VL - 10 IS - 1 SE - Articles DO - UR - http://scitecresearch.com/journals/index.php/jprm/article/view/931 AB - If $\pi$ is an odd prime and $x, y, z,$ are relatively prime positive integers, then $z^\pi ot=x^\pi+y^\pi.$ In this note, an elegant simple proof of this theorem is givenĀ  that if $\pi$ is an odd prime and $x, y, z$ are positive integers satisfying $z^\pi=x^\pi+y^\pi,$ thenĀ  $x, y, z,$ are each divisible by $2:$ (Beal\rq{}s conjecture) The equation $z^\xi=x^\mu+y^ u$ has no solution in relatively prime positive integers $x, y, z, $ with $\xi, \mu, u$ primes at least $3.$ is also proved; that is $x, y, z $ are all even. ER -