The Cartesian method and Fermat's Last Theorem

TL;DR

This paper revisits Fermat's Last Theorem using a historical approach, analyzing Fermat's original 1637 ideas and employing a Cartesian method to demonstrate that the theorem holds, emphasizing the necessity of irrational numbers.

Contribution

The paper introduces a novel historical and philosophical perspective to prove Fermat's Last Theorem, linking it to 17th-century mathematical knowledge and the Cartesian coordinate system.

Findings

Fermat's Last Theorem is confirmed as valid.

Pythagorean triples imply the need for irrational numbers.

Historical analysis supports the theorem's validity.

Abstract

Fermat's Last Theorem is proved by using the philosophical and mathematical knowledge of 1637 when the French mathematician Pierre de Fermat claimed to have a truly marvelous proof of his conjecture. Our approach consists of setting three variables of Fermat's equation as integers and then evaluating whether the remaining variable can be an integer as well. Pythagorean triples play a fundamental role in claiming that at least an irrational number is needed to satisfy Fermat's equation. As a result, we confirm that Fermat's Last Theorem is valid.