Method of Infinite Descent and 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 using the famous mathematical tool of Fermat called the method of infinite decent. However, Euler did not establish in full the key lemma required in the proof. Since then, several authors have published proofs for the cubic exponent but Euler's proof may have been supposed to be the simplest. Paulo 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 in his recent article [3, Eng.Transl.] claims that he may have reconstructed Euler’s proof for the key lemma. However, none of these proofs is short nor easy to understand compared to the simplicity of the theorem and the method of infinite decent The main objective of this paper is to provide a simple, short and independent proof for the theorem using the method of infinite decent. It is assumed that the equation 3 3 3 z  y  x , (x, y) 1 has non trivial integer solutions for (x, y, z) and their parametric representation [5] is obtained with one necessary condition that must be satisfied by the parameters. Using this necessary condition, an analytical proof of the theorem is given using the method contradiction. The proof is based on the method of finding roots of a cubic formulated by Tartagalia and Cardan [4], which is very much older than Fermat’s last theorem.