PREMISE

&nbsp;

This work contains two papers: the first entitled "Euler’s double equations equivalent to Fermat’s Last
Theorem" presents a marvellous "Eulerian" proof &nbsp;of &nbsp;Fermat’s
Last Theorem, which&nbsp; could have entered
in a not very narrow margin, i.e. in only a few pages (less than 8).

The second instead, entitled "The
origin of the Frey elliptic curve in a too narrow margin" &nbsp;provides a proof, which is not elementary (25
pages): It is in various ways articulated and sometimes the author use facts
with are proven later, but it is still addressed in an appropriate manner. 

This proof is however conditioned by presence of a right triangle (very
often used by Fermat in his elusive digressions on natural numbers) or more
precisely from a Pythagorean equation, which has a role decisive in the
reconstruction of the lost proof.

Regarding the first paper, following an
analogous and almost identical approach to that of A. Wiles, I tried to
translate the aforementioned bond into a possible proof of Fermat's Theorem. 

More precisely, through the aid of a Diophantine equation of second
degree, homogeneous and ternary, solved not directly, but as a consequence of
the resolution of the double Euler equations that originated it, I was able to
obtain the following result: the intersection of the infinite solutions of
Euler's double equations gives rise to an empty set and this only by exploiting
a &nbsp;well known &nbsp;Legendre Theorem, which concerns the
properties of all the Diophantine equations of the second degree, homogeneous
and ternary.

Consequently, the double equations of Euler are discordant, just as in
the case of exponent equal to 4, solved by Fermat himself (in this case, in the
double equations of Euler, defined by the expressions (0) of my work, attached,
is just assume that m = -n = 1). 

&nbsp;

I report that the “
Journal of Analysis &amp; Number Theory” has made this paper available online
at 

&nbsp;

http://www.naturalspublishing.com/ContIss.asp?IssID=1779

The lost proof of Fermat's Last Theorem PDF

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