General Pythagorean Theorem And Frey's Elliptical Equation

James Constant

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Wile's proof of Fermat's Last Theorem (FLT) is suspect because it rests on Frey's elliptic equation (FEE) which Rivet proves is not modular and, therefore, it and its association with Fermat's equality (FE) do not exist. Frey claims his elliptical equation y² =x(x - a^p)(x + b^p) can be derived from  Fermat's equality c^p = a^p + b^p with p>2 and c,a,b,p integers. [1] In what follows I introduce a General Pythagorean Theorem (GPT) which can be used to study any equation involving parameters x,y. I will then study FEE will prove that Wile's proof of FLT is false because

1.        FEE does not reduce to and cannot be derived from FLT or FE;

2.        FEE is fatally flawed because it disproves the arithmetic rule of addition;

3.        FEE is fatally flawed because it disproves the Pythagorean Theorem (PT);

4.        Only FEE is relevant for proving FLT in Wile's proof thereof; and

5.        Ribet's and Wile's theorems on the modularity of elliptical equations are not relevant for proving FLT in Wile's proof thereof.

I begin by writing the PT


                z² =x² + y²              z,x,y   all numbers             (1)

in which numbers  z,x,y are  rational,  irrational, positive  or  negative and numbers  z²,x²,y² are positive numbers.

Next, I write the GPT

                Z²=X²+Y²            Z=Z(z,x,y),X=X(z,x,y),Y=Y(z,x,y) all numbers    (2)

in which numbers Z,X,Y are  rational,  irrational , positive  or  negative and numbers  Z²,X²,Y²are positive numbers. 

The GPT equation (2) can be used to evaluate any equation with numbers z,x,y.  Elsewhere, I use the GPT equation (2) to analyze FE and find that it disproves PT and the arithmetic rule of addition.[2]. Here, consider FEE which can be rewritten as

               x(x² - a^pb^p) = (a^p - b^p)x²+ y²            x,y,a,b,p integers       (3)

Equation (3) shows that FEE cannot be reduced to or derived from FLT or FE (QED 1).  However, FEE equation (3) can now be identified with the GPT equation (2)

                Z²= x(x² - a^pb^p)        X² =  (a^{p -} b^p)x²        Y² =  y²                 (4)

from which I conclude that, since  Z²,X²,Y²are positive numbers

                x>0        x²>a^pb^p      a^p>b^p              x,a,b,p  integers              (5)

or           x<0        x²<a^pb^p      a^p>b^p              x,a,b,p  integers               (6)


The two last inequalities (5) say that x²>a^2p which, since a = x and b=y in FE, is impossible when p>0. This means equality z^p =x^p + y^p is impossible when p = 1,2,3, . . Note:  I  disregard  inequalities (6)since p cannot be less than 1.

Thus with p=1 inequalities (5) disprove the arithmetic rule of addition (QED 2); with p=2, inequalities (5) disprove the PT equation (1) (QED 3); and with p=3, inequalities (5) disprove FE and thus prove FLT (QED 4 and QED 5). In view of the last two inequalities, the first inequality says that x in PT can never be a negative value, which disproves  the PT equation (1) (QED 3).


Therefore, since Frey's elliptical equation is not deriveable from FE, is fatally flawed and since Ribet's and Wile's modularity theorems are not relevant for proving Fermat's last theorem in Wile's proof thereof, Wile's proof of Fermat's Last theorem is false. 

 

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1. S. Singh, Fermast's Last Theorem, pp 216-222, Fourth Estate, London 1998

2.  http://www.coolissues.com/mathematics/Wile'sproofofFLT.html

        Copyright © 2008 James Constant

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