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We present the only proof of Pierre Fermat by descente infinie that is knownto exist today. As the text of its Latin original requires active mathematicalinterpretation, it is more a proof sketch than a proper mathematical proof. Wediscuss descente infinie from the mathematical, logical, historical,linguistic, and refined logic-historical points of view. We provide therequired preliminaries from number theory and develop a self-contained proof ina modern form, which nevertheless is intended to follow Fermat's ideas closely.We then annotate an English translation of Fermat's original proof with termsfrom the modern proof. Including all important facts, we present a concise andself-contained discussion of Fermat's proof sketch, which is easily accessibleto 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 theexperts in these fields. Last but not least, this paper fills a gap regardingthe easy accessibility of the subject.

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