General Pythagorean Theorem And Frey's Elliptical Equation
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 - ap)(x + bp) can be derived from Fermat's equality cp = ap + bp with p>2 and c,a,b,p integers.  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.
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.. Here, consider FEE which can be rewritten as
x(x2 - apbp) = (ap - bp)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 - apbp) X2 = (ap - bp)x2 Y2 = y2 (4)
from which I conclude that, since Z2,X2,Y2are positive numbers
x>0 x2>apbp ap>bp x,a,b,p integers (5)
The two last inequalities (5) say that x2>a2p which, since a = x and b=y in FE, is impossible when p>0. This means equality zp =xp + yp 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).
Copyright © 2008 James Constant