Fermat's Last Theorem for the Exponent 3

TL;DR

This paper simplifies the classical proof of Fermat's Last Theorem for exponent 3 by providing a shorter lemma proof and reducing the descent argument from two cases to one, streamlining the overall proof process.

Contribution

It offers a more concise proof of Fermat's Last Theorem for exponent 3 by simplifying key lemmas and the descent method used in previous proofs.

Findings

Shorter proof of the central lemma

Reduction of descent from two cases to one

Streamlined proof of Fermat's Last Theorem for exponent 3

Abstract

`Fermat's Last Theorem for the exponent 3 has received numerous proofs, the most common of which being either in Euler's or in Gauss' style. This latter works entirely in the ring of integers of the quadratic field generated by the square root of -3. A proof in Euler's style is based on a central lemma related to properties of the quadratic form x^2 + 3y^2, then it proceeds by descent by considering Two main cases. In the present version, the central lemma receives a short proof and the classical proof by descent in Two main cases boils down to a One-case proof.