Historical and Foundational Details on the Method of Infinite Descent: Every Prime Number of the Form 4n + 1 is the Sum of Two Squares

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 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: 1) it follows all the elements of which Fermat wrote in his outline; 2) 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 theoretical context.