Re: Fermat's Last Theorem
From: shedar (nobody_at_nonesuch.com)
Date: 11/03/04
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Date: Wed, 03 Nov 2004 22:15:13 GMT
See: news:2ughm2F28gr7nU1@uni-berlin.de.
"ben ito" <benito20044@yahoo-dot-com.no-spam.invalid> wrote in message
news:41892382_1@Usenet.com...
> Fermat's Last Theorem
> Ben Ito
> 11-03-04
>
> 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). Fermat uses the following equations to prove that
> n=4 does not form integer solutions,
>
> A^2 = 2uv, B^2 = u^2 - v^2, and C = u^2 + v^2 (equ 3),
>
>
> Fermat is violating logic by implying that equations that describe the
> integer solutions of n=2 (equ 2) can be used to prove that n=4 does
> not form integer solutions; however, n=2 and n=4 form completely
> different equations; therefore, the integer solution equations, of
> n=2, cannot be used to prove n=4. Fermat's proof for n=4 is invalid.
>
> 3. Wiles Proof
>
> 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 4)
>
> where
>
> a^n + b^n = c^n (equ 5).
>
> Frey does not derive equation 5 from equation 4; Frey and Wiles are
> implying the existence of equation 5.
>
> "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).
>
> It's questionable that Wiles can assume the existence of equation 5
> without deriving it then basing his entire proof on an implied
> equation. Fermat's equation is not dependent on an elliptic curve.
> Using n=2, in equation 4, the following equation is formed,
>
> y^2 = x(x - a^2)(x + b^2) (equ 6)
>
> equation 6 is not Pythagoreans equations; therefore, equation 6 is
> unrelated to Fermat's equation. Consequently, Wiles' proof of
> Fermat's Last Theorem is invalid.
>
> 4. Ito's Proof.
>
> I will prove Fermat's Last Theorem. I will use x, y and z to represent
> the sides of a triangle. When n=2, using a transformation by letting
> z = c (integer), Fermat's equation becomes the equation of a circle,
>
>
> x^2 + y^2 = c^2. (equ 7)
>
> In the transformation, equation 7 and Fermat's equation (n=2) can be
> represented simultaneously on the xy plane. Consequently, the
> equation of a circle of radius r (equ 7) and the right triangles with
> a hypotenuse z (n=2) can be represented on the x-y plane. Only n=2
> forms the x, y and z lengths on the x-y plane which allows for the
> possible formation of integer solution. When n>2, the
> transformed equations (z=c) can not be represented on the x-y plane
> with Fermat's equation; therefore, only n=2 of Fermat's equation
> forms the integer solutions.
>
>
> 5. Conclusion
>
> Fermat is using the integer solutions of n=2 to prove that n=4 does
> not form integer solutions; however, the equations of n=2 and n=4 are
> complete different equations; therefore, the solutions of n=2 cannot
> be used in Fermat's n=4 proof; consequently, Fermat's prove is
> invalid.
>
> Wiles' proof of Fermat's Last theorem is based on the implied
> assumption that
>
> a^n + b^n = c^n (equ 8)
>
> and the elliptic curve equation
>
> y^2 = x(x - a^2)(x + b^2) (equ 9)
>
> are related. However, equations 8 and 9 are completely different
> equations; therefore, Wiles' proof using elliptic curves to prove
> Fermat's Last Theorem is invalid since equation 8 cannot be derived
> from equation 9; using n=2 in equation 9, Pythagoreans' equation is
> not formed. Consequently, Wiles' proof of Fermats' theorem is
> invalid.
>
> I will solve Fermat's Last Theorem by showing that when z = c, where c
> is an integer, a circle equation and right triangle equation can be
> represented on the x-y plane simultaneously. Only n=2 forms the
> circle equation and Fermat equation (n=2) on the x-y plane. When
> n>2 the equations formed never can be represented simultaneously
> on the x-y plane. The representation of the transformed structure
> (z=c) and Fermat's equation allow for the possibility of integer
> solution to form. Therefore, only n=2 forms integer solutions of
> Fermat's equation.
>
> 6. References
>
> Daniel Shanks. Solved and Unsolved Problems in Number Theory. Chelsea
> Pub. 1985.
>
> Paulo Ribenboim. Fermat's Last Theorem for Amateurs. Springer.
> 1999.
>
> A. J. Van Der Poorten. Notes on Fermat's Last Theorem. John Wiley.
> 1996
>
> *-----------------------*
> Posted at:
> www.GroupSrv.com
> *-----------------------*
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