# 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.### References

H. Edwards, {it Fermat's Last Theorem:A Genetic Introduction

to Algebraic Number Theory/}, Springer-Verlag, New York, (1977).

A. Wiles, {it Modular ellipic eurves and Fermat's Last

Theorem/}, Ann. Math. 141 (1995), 443-551.

A. Wiles and R. Taylor, {it Ring-theoretic properties of

certain Heche algebras/}, Ann. Math. 141 (1995), 553-573.

*Journal of Progressive Research in Mathematics*,

*10*(1), 1446-1447. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/931

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