Foundations of Science 25 (3):671-702 (2020)

Raffaele Pisano
Université des Sciences et Technologies de Lille
Pierre de Fermat is known as the inventor of modern number theory. He invented–improved many methods useful in this discipline. Fermat often claimed to have proved his most difficult theorems thanks to a method of his own invention: the infinite descent. He wrote of numerous applications of this procedure. Unfortunately, he left only one almost complete demonstration and an outline of another demonstration. The outline concerns the theorem that every prime number of the form 4n + 1 is the sum of two squares. In this paper, we analyse a recent proof of this theorem. It is interesting because: it follows all the elements of which Fermat wrote in his outline; it represents a good introduction to all logical nuances and mathematical variants concerning this method of which Fermat spoke. The assertions by Fermat will also be framed inside their historical context. Therefore, the aims of this paper are related to the history of mathematics and to the logic of proof-methods.
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DOI 10.1007/s10699-019-09642-3
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References found in this work BETA

Newton’s Philosophiae Naturalis Principia Mathematica "Jesuit" Edition: The Tenor of a Huge Work.Raffaele Pisano & Paolo Bussotti - 2014 - Rendiconti Accademia Dei Lincei Matematica E Applicazioni 25 (4):413-444.
Descente Infinie + Deduction.Claus-Peter Wirth - 2004 - Logic Journal of the IGPL 12 (1):1-96.

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Finite Arithmetic with Infinite Descent.Yvon Gauthier - 1989 - Dialectica 43 (4):329-337.
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