Vol.:(0123456789) Foundations of Science https://doi.org/10.1007/s10699-019-09642-3 1 3 Historical and Foundational Details on the Method of Infnite Descent: Every Prime Number of the Form 4n + 1 is the Sum of Two Squares Paolo Bussotti 1  · Rafaele Pisano 2,3,4 © Springer Nature B.V. 2020 Abstract Pierre de Fermat (1601/7–1665) 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 difcult theorems thanks to a method of his own invention: the infnite descent (Fermat 1891–1922, II, pp. 431–436). He wrote of numerous applications of this procedure. Unfortu- nately, 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: (1) it follows all the elements of which Fermat wrote in his outline; (2) it represents a good introduc- tion 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. Keywords Fermat · Infnite descent · Number theory · Foundations of mathematics · Relationship logic-mathematics 1 Introduction The Infnite Descent is a mathematical method used in the theory of numbers. It is based on the third excluded principle and relies on the fact that the natural numbers are a well-ordered set. For example, given a certain known equation, typically a Diophantine one, while looking for its solutions, fnally one can arrive to claim that it has no solution supposing that it has solutions and showing that under this hypothesis, an infnite descent in integers might be con- structed. This is absurd. Hence, the equation has no solution. Nowadays, this mathematical * Rafaele Pisano rafaele.pisano@univ-lille.fr Paolo Bussotti paolo.bussotti@uniud.it 1 Udine University, Udine, Italy 2 IEMN, Lille University, Lille, France 3 Afliated to (2015–2019) School/SCFS, Sydney University, Sydney, Australia 4 Afliated to CPNSS, LSE, London, UK