A simple and short proof of Fermat’s last theorem for n = 3

Abstract

The first proof of Fermat’s last theorem for the exponent n  3 was given by Leonard Euler. However, Euler did not establish in full the key lemma required in the proof [1]. Since then, several authors have published proof for the cubic exponent but Euler’s proof may have been supposed to be the simplest. Ribenboim [1] claims that he has patched up Euler’s proof and Edwards [2] also has given a proof of the critical and key lemma of Euler’s proof using the ring of complex numbers. Recently, Macys [3] in his article, claims that he may have reconstructed Euler are proof by providing an elementary proof for the key lemma. However, in this authors’ point of view, none of these proofs is short nor easy to understand compared to the simplicity of the wording and the meaning of the theorem.The main objective of this paper is, therefore, to provide a simple and short proof for the theorem. It is assumed that the equation , ( , ) 1 3 3 3 z  y  x x y  has non-trivial integer solutions for (x, y, z) . Parametric solution of x, y, z and a necessary condition that must be satisfied by the parameters can be obtained using elementary mathematics. The necessary condition is obtained and the theorem is proved by showing that this necessary condition is never satisfied.