fermat 420

From: ben ito (benito20044_at_yahoo-dot-com.no-spam.invalid)
Date: 12/01/04 

Date: 1 Dec 2004 00:28:51 -0600

Fermat's Last Theorem
Ben Ito
11-30-04

I will show that Fermat's n=4 is invalid. Fermat's n=4 proof is a is
based on a deception that changes variables to imply that the n=2 and
n=4 equations are identical equations.However, the n=2 and n=4
equations are completely different equations; therefore, the integer
solution equations of n=2 cannot be used to prove n=4 as Fermat has
done.

I will then show that Wiles proof of Fermat's last theorem is also
invalid. Wiles uses Fermat's elliptical curves. Fermat's derivation
of the elliptic curve equation is derived using the integer solutions
of n=2 in an area equation of a right triangle; however, the area
equation of a right triangle is completely different form the n=2
right triangle equation; therefore, Fermat's derivation is invalid
since two completely different equations are assumed to be equal in
Fermat's derivation of the elliptic curve equation.

I will prove Fermat's last theorem using mathematical quantitative
analysis. The n=1, n=2 and n=3 solutions and near solutions are
described; I will then show a pattern using mathematical quantitative
analysis to asess the effects of the numerical combinations that
describe the solutions and near solutions of n=3. From these
assesments I will show that only the n=1 and n=2 form integer
solutions using the patterns formed by the distribution of the
integers with there respective powers.

l. Introduction

Fermat went to Orléans where he studied law at the University. He
received a degree in civil law and purchased the offices of
councillor at the parliament in Toulouse. In 1631 Fermat was a lawyer
and government official in Toulouse. In his spare time Fermat did
mathematics. Fermat attempted to prove that n=4 does not form integer
solutions.

Fermat's n=4 proof depend on showing that, if three integral values of
X, Y, Z can be found which satisfy the equation, then it will be
possible to find three other and smaller integers which also satisfy
it. We show that the equation must be satisfied by three values which
obviously do not satisfy it. It would seem that this method is
inapplicable in proving n = 4.

Fermat's last theorem states that

X^n + Y^n = Z^n, (equ 2)

when n>2 does not form integer solutions of X, Y and Z. However,
Fermat did not leave a proof. Form Fermat's n=4 and the elllipitic
curve derivation, it is apparent that he did not have a proof. In
all of his attempts at forming a proof Fermat always used the integer
solution of n=2; therefore, Fermat did not understanding the basic
principle of the problem that there are an infinite number of
possible integer solution combinations that cannot be represented
with a single equations; therefore, it is impossible to prove n=4
must less n>2. I will show that Fermat's n=4 and Wiles' proofs are
invalid then prove Fermat's last theorem using mathematical
quantitative analysis.

2. Fermat's n=4 Proof

Fermat's n=4 proof is described. Fermat implies that by proving that,

X^4 + Y^4 = Z^2 (equ 3).

does not form integer solutions also proves that

X^4 + Y^4 = Z^4 (equ 4)

does not form integer solutions (Shanks, p. 144). However, using n=4
in equation 1 forms equation 3. Fermat uses equation 2 to prove n=4;
however. equations 3 and 4 are completely different equation therefore
using equation 3 to prove n=4 violates logic.

Fermat uses the integer solution equations of n=2,

X = 2uv, Y = u^2 - v^2, and Z = u^2 + v^2 (equ 5a,b,c),
 
to derive,
 
X^2 = 2uv, Y^2 = u^2 - v^2, and Z = u^2 + v^2 (equ 6a,b,c),
(Shanks, p.141) that are used to prove that n=4 does not form integer
solutions; however, Fermat squares the right sides of equations
5a,b,c without square the left side which violates algebra. Using
equations 5a,b and 6a,b,

X^2 = X and Y^2 = Y. (equ 7a,b)

equations 6a,b,c violates algebra. Using equations 7a,b in equation 3
forms

X^2 + Y^2 = Z^2 (equ 8)

However, this is implying equations 7a,b are valid; this is the key
step in Fermat's contradiction method (Osserman, p. 18). The
variables X, Y and Z are the same for all n's;

X=X, Y=Y and Z=Z (equ 9a,b,c)

when n>2.

therefore, equations 7a,b are invalid. Therefore, Fermat's
contradiction method that is used to prove n=4 is invalid.

In addition, Fermat is using equations (equ 6a,b,c) that are derived
from equation 5a,b,c to prove n=4; however,

X^2 + Y^2 = Z^2 (equ 9)

completely different from,

X^4 + Y^4 = Z^2 (equ 10).

Using Z = 6, equation 9 forms a circle of radius 6,

X^2 + Y^2 = 36.(equ 11)

Using Z = 6 in equation 10,

X^4 + Y^4 = 36,(equ 12)

equation 12 does not form a circle; therefore, equation 12 is
completely different from equation 11. The integer solutions of n=2
cannot be used to prove n=4 since equations 11 and 12 are completely
different equations. Therefore, equations 6a,b,c cannot be used to
prove n=4. Fermat proof is implying that the equations of n=2 and
n=4 are equal using equations 7a,b. Therefore, Fermat's n=4 proof is
invalid.

In addition, Fermat's n=4 proof does not testing all integer
combinations of X and Y. Fermat's proof is only proving equations
5a,b,c, that are derived from n=2, do not form integer solutions.
However, there are an infinite number of integer combinations that
are not tested in Fermat's n=4 proof; therefore, Fermat's n=4 proof
in incomplete and invalid.

Fermat is justifying the non-existence of integer solutions of n=4
using a contradiction method where a single group of equations (equ
5a,b,c) are used; however, Fermat's n=4 proof does not include all
possible integer combinations of X, Y and Z; therefore, Fermat's n=4
proof in incomplete and therefore, invalid.

3. Wiles' Proof

Wiles' proof of FLT uses Fermat's elliptic curve. The derivation of
Fermat's elliptic curve equation is described. Fermat states that a
right triangle has integer sides, X and Y;

XY = 2d^2 (equ 12)

the area described with 2d^2, the value of d cannot be an integer.
Fermat then uses the integer solution equations of n=2,

X = (m^2 - n^2), Y = 2mn, Z = (m^2 + n^2),(equ 13)

in equation 12, the second equation takes the form:

2d^2 = XY = (m^2 - n^2) x 2mn, (equ 14)

or

d^2 = (nm^3 - mn^3).(equ 15)

We now let

X = m/n, Y = (d/n^2).(equ 16)

Then

X^3 - X = (m^3)/(n^3) - m/n = (nm^3 - mn^3)/n^4 = (d^2)/n^4 = Y^2 (equ
17)

which forms the elliptic curve equation,

Y^2 = X^3 - X, (equ 17)

However, equation 12 is not equal to equation 6. The n=2 equation is
not the area equation; therefore, Fermat violates logic by using the
integer solution equation in the area equation to derive the elliptic
curve equation. Fermat is using the same method used in his n=4 proof;
therefore, the derivation of the Fermat's elliptic curve equation is
invalid. Fermat derives the elliptic curve equation by manipulating
the n=2 integer solution equations 13; however, the integer solutions
equations of n=2 is only valid for the n=2 equation; therefore,
Fermat's derivation of the elliptic curve is invalid.

Wiles proof of Fermat's Last Theorem is based on the elliptic curve
equation (Poorten, p. 196-7),

Y^2 = X(X - a^n)(X + b^n) (equ 18).

Wiles implies that equation 18 forms Fermat's equation

a^n + b^n = c^n (equ 19).

Wiles assumes the existence of equation 19.

"Ribet and Wiles studied this curve under the assumption that there
exist a nonzero integer c such that a^n + b^n = c^n." (Ribenboim, p.
247).

All the the books that describe Wiles' proof and Taniyama-Shimura
ignore the original derivation of the elliptic curve. Wiles is hiding
the fact that the elliptic curve was derived from Fermat's right
triangle area derivation (equ 12-17).

"Fermat counter example leads to elliptic curves." (Ribet, video)

The problem with Fermat last theorem is that n>2 forms an infinite
number of completely different equations, example,

X^3 + Y^3 = Z^3, X^4 + Y^4 = Z^4, X^5 + X^5 = Z^5..........(equ 20)

Therefore, once one equation is proven a completely different proof is
required; therefore, Wiles implies that the ellliptic curve's X and Y
variables are not altered when n increases. Wiles solve the proof by
using the constant a and b of the elliptic curve which violates logic
since when n increases the X and Y variables change. Wiles is assuming
that the X and Y variables powers remain that of an elliptic curve as
n increases which violates logic since every n forms a completely
different equation.

In addition, Wiles was a professor at Princeton when he published his
paper in the Annals of Mathematics, May 1995, which is published by
Princeton University Press. Wiles' was one of the editors of the
Annal of Mathematics in which he published his paper which violates
ethics.

4. Proof of FLT.

I will prove Fermat's last equation using the pattern formed by the
solutions formed by n=1 and n=2, and the near solutions of n=3 and
n=4. Using n=1 in equation 1,

X + Y = Z (equ 21)

All integers of equation 21 form solutions. An integer solution set
occurs at (1,2,3).

The integer solutions of n=2 are described with the following
equations,

X = 2uv, Y = u^2 - v^2 and Z = u^2 + v^2 (equ 22)

Not all integers form integer solutions for n=2. An integer solution
set occurs at (3,4,5).

The n=1 and n=2 solutions show a pattern where integer solutions are
formed when a solution set is forms close to the origin.

In addition, when n=1, the strength of the solution set formation is
maximum since all integers form solutions; when n=2 not all integers
form solutions; therefore, the strength of the solution formation is
weakening. When n=3,

6^3 + 8^3 = 728 where 9^3 = 729 (equ 23)

9^3 + 10^3 = 1729 where 12^3 = 1728 (equ 24)

an integer solution set does not form close to the origin for
n=3;however, equations 23 and 24 are off by one. The solution set is
weakening as n increases; at n=4

7^4 + 8^4 = 6497 where 9^4 = 6561(equ 25)

consequently, since n=3 does not from solutions near the origin, the
integer sequence ends at n = 3. Therefore, only n=1 and n=2 form
integer solutions.

5. Conclusion
I have shown that Fermat's derivation of n=4 is base on deception
since Fermat uses non-integer solutions of A', B' and C to prove that
Fermat's n=4 equation does not from integer solution. In addition,
Fermat is using the integer solution of n=2 to prove n=4 which
violates logic since n=2 and n=4 form completely different equations.
Fermat's n=4 proof does not testing all integers. Fermat's proof is
only proving equations 5a,b,c do not form integer solutions. It is
impossible to prove n=4, using Fermat's method, since it would
require an infinite number of equations; therefore, Fermat's proof of
n=4 is invalid.

Fermat's derivation of the elliptic curve equation is invalid since
Fermat uses the integer solution equations of n=2 in the area
equation to derive the elliptic curve equation however, the n=2
equations is a completely different equation than the area equation
of a right triangle'; therefore, the derivation of Fermat's elliptic
curve equation is invalid. Wiles uses the elliptic curve equation to
prove Fermat's last theorem by using the constants of the elliptic
curve equation to prove n>2; however, n>2 forms an infinite
number of equation that cannot be described with an elliptic curve.
When n>2 each value of n most form is own elliptic curve
therefore, it would require an infinite number of equation to prove
n>2 ; therefore, Wiles' proof the Fermat's last theorem is
invalid.

I will prove FLT by showing a pattern forms from the n=1 and n=2
solution sets. All of the integer of n=1 form solutions; when n=2 the
number of solutions in a finite range is less then that of n=2;
therefore, the strength of the solutions is decreasing as n
increases. When n=3 the solution sequence ends and solutions are
only formed when n = 1 and n = 2 .

6. References

Robert Osserman. Fermat's Last Theorem (a supplement to the video).
MSRI Berkeley. 1994

Marilyn vos savant. The World's Most Famous Math Problem. St Martin's
Press. 1993

Daniel Shanks. Solved and Unsolved Problems in Number Theory. Chelsea
Pub. 1985.

A. J. Van Der Poorten. Notes on Fermat's Last Theorem. John Wiley.
1996

7. Acknowledgment

Special thanks to Rudi, Nate, Peter, Jakob, Joesph, Stephen Hawkings
forum, Best Science forum, and About Physics forum, HSU, CSUS, CR,
SCC, USC, Hiram Johnson HS Sacramento (Mrs Larson), UCD, Stanford,
MIT, Harvard, ASU, Rutgers and UCLA mathematics Dept.

Memory of Yasser Arafat 11-11-04

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