SOME DIOPHANTUS-FERMAT DOUBLE EQUATIONS EQUIVALENT TO FREY’S ELLIPTIC CURVE

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.