fermat (revised 10-31-04)
From: ben ito (benito20044_at_yahoo-dot-com.no-spam.invalid)
Date: 10/31/04
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Date: 31 Oct 2004 15:28:59 -0600
Nate,
There are an infinite number of probems with Fermat's n=4 proof.
Heres another one.
Ben
ben ito <benito2444@yahoo.com> wrote:
Fermat's Last Theorem
Ben Ito
10-31-04
I will solve Fermat's last theorem.
l. Introduction
I will show that Fermat's (n=4) and Wiles' proofs are invalid then
prove that Fermat's equation
x^n + y^n = z^n (equ 1)
only forms integer solutions when n>2 using a transformation.
2. Fermat's Proof (n=4)
The following equations are used to describe the integer solutions of
Fermat's equation (n=2),
A = 2uv, B = u^2 - v^2, and C = u^2 + v^2 (equ 2)
(Shanks, p.141). The varibales A, B and C represent integer solutions
of Fermat's equation (equ l). Using u = 2 and v = 1, in equation 2, A
= 4, B = 3 and C = 5 which are integer solutions when n=2.
Fermat's proof for n=4 is described. Fermat implies that by proving
that,
A^4 + B^4 = C^2 (equ 3).
does not form integer solutions also proves that
A^4 + B^4 = C^4 (equ 4)
does not form integer solutions. Fermat uses the following equations
to prove that equation 3 does not form interger solutions,
A^2 = 2uv, B^2 = u^2 - v^2, and C = u^2 + v^2 (equ 5),
Fermat is using the equations that describe the integer solutions of
n=2 (equ 2) to prove that equation 3 does not form integer solutions;
however, the equation derived from n=2
x^2 + y^2 = z^2 (equ 6)
is completely different form equation 3; therefore, Fermat is
violating logic by implying that the equations that describe the
integer solutions of n=2 (equ 2) can be used to prove that equation 3
does not form integer solutions. Therefore, Fermat's proof for n=4 is
incomplete and therefore invalid.
3. Wiles Proof
Wiles proof of Fermat's Last Theorem is based on the elliptic curve
equation,
y^2 = ax^3 + bx + c (equ 7)
where
a^p + b^p = c^p (equ 8)
Wiles assumed that since equation 8 is similar to equation l that
Fermat's Last Theorem can be described using elliptic curves;
however, Fermat's equation is not dependent on an ellilptic curve;
therefore, equation 8 is not Fermat's equation (equ l) as implied by
Wiles. In addition, there are an infinite number of higher order
equations of x and y, when n>3, that are not represented with
Wiles' ellipitic curves. Wiles is implying that equation 8 represent
the higher orders of x and y using elliptics curves; however, Wiles'
ellipitic curve only represent lower orders of x and y; therefore,
Wiles' proof of Fermat's Last Theorem is incomplete and therefore
invalid. Wiles ignores that Fermat's equation is not dependent on an
ellipitc curve and that elliptic curves do not represent higher
orders of x and y.
4. Ito's Proof.
I will form the Proof of Fermat's Last Theorem by showing that only
right triangles form integer solutions. I will use x, y and z to
represent the sides of a right triangle when n=2. Using a
transformation, let z = c (integer), Fermat equation becomes the
equation of a circle (n=2),
x^2 + y^2 = c^2. (equ 9)
In the circle transformation, the hypotenus of the right triangle
becomes the radius of the circle. Consequently, a circle of radius r
and the right triangle with a hypotenus z can be represented together
on the x-y plane which forms the primary alignment. Only n=2 forms the
x, y and z lengths on the x-y plane which allows for the possible
formation of integer solution of Fermat's equation when n=2. The
equation describe with Fermat's equation when n>2 never forms a
transformed structure (z=c) that forms the x, y and z lengths on the
x-y plane; therefore, only n=2 of Fermat's equation forms the integer
solutions.
5. Conclusion
I have shown that Fermat's derivation of n=4 is base on questionable
logic. I then show that Wiles' proof of Fermat's Last theorem only
describe lower orders of x and y with ellipitic curves; therefore,
Wiles' proof is incomplete and therefore invalid. I then show that
the transformation (z=c) forms the x, y and z lengths on the x-y
plane only when n=2;consequently, only n=2 forms the condition where
the integer solutions can be formed.
*************************
...
6. References
Robert Osserman. Fermat's Last Theorem (a supplement to the video).
MSRI Berkeley. 1994
Marilyn vos savant. The World's Most Famoous Math Problem. St Martin's
Press. 1993
Daniel Shanks. Solved and Unsolved Problems in Number Theory. Chelsea
Pub. 1985.
7. Acknownlegment
Special thanks to Rudi, Nate, Peter, 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
and UCLA mathematics Dept.
*-----------------------*
Posted at:
www.GroupSrv.com
*-----------------------*
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