Abstract

In this work I demonstrate that a possible origin of the Frey elliptic curve derives from an appropriate use of the double equations of Diophantus-Fermat and from an isomorphism: a birational application between the double equations and an elliptic curve.

From this origin I deduce a Fundamental Theorem which allows an exact reformulation of Fermat's Last Theorem.

A complete proof of this Theorem, consisting of a system of homogeneous ternary quadratic Diophantine equations, is certainly possible also through methods known and discovered by Fermat,in order to solve his extraordinary equation. 

Keywords and Phrases

Fermat's Last Theorem, Arithmetic algebraic geometry, Diophantine geometry.

A.M.S. subject classification

11D41 (primary), 11G05 (secondary).

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