A Self-Contained and Easily Accessible Discussion of the Method of Descente Infinie and Fermat's Only Explicitly Known Proof by Descente Infinie

TL;DR

This paper provides a clear, self-contained modern presentation and analysis of Fermat's only known proof by descente infinie, making it accessible to both mathematicians and historians.

Contribution

It offers the first comprehensive, accessible modern reconstruction of Fermat's proof by descente infinie, including annotations and historical context.

Findings

Clarifies Fermat's proof sketch for laymen and experts

Provides a modern, self-contained proof with annotations

Enhances understanding of the Method of Descente Infinie

Abstract

We present the only proof of Pierre Fermat by descente infinie that is known to exist today. As the text of its Latin original requires active mathematical interpretation, it is more a proof sketch than a proper mathematical proof. We discuss descente infinie from the mathematical, logical, historical, linguistic, and refined logic-historical points of view. We provide the required preliminaries from number theory and develop a self-contained proof in a modern form, which nevertheless is intended to follow Fermat's ideas closely. We then annotate an English translation of Fermat's original proof with terms from the modern proof. Including all important facts, we present a concise and self-contained discussion of Fermat's proof sketch, which is easily accessible to laymen in number theory as well as to laymen in the history of mathematics, and which provides new clarification of the Method of Descente Infinie to the experts in these fields. Last but not least, this paper fills a gap regarding the easy accessibility of the subject.