The lost proof of Fermat's last theorem

Abstract: This work contains two papers: the first published in 2022 and entitled "On the nature of some Euler's double equations equivalent to Fermat's last theorem" provides a marvellous proof through the so-called discordant forms of appropriate Euler's double equations, which could have entered in a not very narrow margin and the second instead published in 2024 and entitled "Some Diophantus-Fermat double equations equivalent to Frey's elliptic curve" provides the possible proof, which Fermat has not published in detail, but which uses the characteristic of all right-angled triangles with sides equal to whole numbers, or the famous Pythagorean identity. Some explanations in session(III) are provided: the first makes evident the nature of the "proof a' la Fermat" and the subsequent sessions clarify the direct and interesting connection of the two elementary proofs and it is necessary if you want to understand how two different elementary proofs of Fermat's Last Theorem are possible. It must be observed that those proofs must in no way be interpreted as a sort of absurd revenge of elementary number theory over more modern analytic and algebraic treatments.The author himself has added a section in which he connects his concepts with some of those used by Wiles in his complex demonstration.This implies that, to a certain extent, Wiles' demonstration inspired the author of those works. Ultimately in this paper we will illustrate how only thanks to some of Euler's discoveries was it possible to shed light on the so-called too narrow margin never written by Fermat. For this reason we will also provide some details on an article that was the real inspiration for achieving these results (see Last Conclusions).