The Alternative Form of Fermat's Equation

TL;DR

This paper introduces an alternative form of Fermat's equation that simplifies proofs of the first case of Fermat's Last Theorem for specific exponents, highlighting a more elementary and efficient method.

Contribution

It presents a new form of Fermat's equation that allows simpler proofs for certain exponents, demonstrating an effective alternative to classical approaches.

Findings

Simplified proofs for exponents 3, 5, 7, 11, 13

Method avoids complex calculations of traditional proofs

Illustrates the effectiveness of the alternative form

Abstract

An alternative form of Fermats equation[1] is proposed. It represents a portion of the identity that includes three terms of Fermats original equation. This alternative form permits an elementary and compact proof of the first case of Fermats Theorem (FT) for a number of specific exponents. Proofs are given for exponents n equal to 3, 5, 7,11 and 13. All these cases have already been proven using the original Fermats equation, not to mention the fact that a complete proof of FT was given by A. Wiles [2]. In view of this, the results presented here carry a purely methodological interest. They illustrate the effectiveness and simplicity of the method,compared with the well-known classical approach. An alternative form of the equation permits use of the criterion of the incompatibility of its terms, avoiding the labor-intensive and sophisticated calculations associated with traditional approach.