# A proof of Fermat's last Theorem using elementary algebra

• James E. Joseph Howard University

### Abstract

In 1995, A. Wiles [1,2] announced a proof of Fermat's Last Theorem, which is stated as follows: 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, a simpler proof of this theorem is offered. It is proved that if $$\pi$$ is an odd prime and $$x$$, $$y$$, $$z$$ are positive integers satisfying $$z^\pi=x^\pi+y^\pi,$$ then $$x$$, $$y,$$ and $$z$$ are each divisible by $$\pi$$. The proof offered by Wiles uses cyclic groups. It is the purpose of this note to offer a proof of this result, with techniques within the grasp of Fermat.