  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).
