Fermat 420
From: ben ito (benito20044_at_yahoo-dot-com.no-spam.invalid)
Date: 11/21/04
- Next message: Stephen J. Herschkorn: "Re: Set Theory"
- Previous message: Jason Pawloski: "Re: Book recommendations in these areas."
- Messages sorted by: [ date ] [ thread ]
Date: 21 Nov 2004 15:28:42 -0600
1564.140 in reply to 1564.139
Fermat's Last Theorem
Ben Ito
11-21-04
I will show that Fermat's n=4 and Wiles' proofs are invalid then prove
Fermat's last theorem using the law of cosine.
l. Introduction
Fermat's last theorem states that
X^n + Y^n = Z^n, (equ 1)
when n>2 does not form integer solutions of X, Y and Z.
2. Fermat's n=4 Proof
Fermat's n=4 proof is described. Fermat uses the integer solution
equations of n=2,
X = 2uv, Y = u^2 - v^2, and Z = u^2 + v^2 (equ 2a,b,c),
to derive,
X'^2 = 2uv, Y'^2 = u^2 - v^2, and Z = u^2 + v^2 (equ 3a,b,c),
(Shanks, p.141). Equations 3a,b,c are used to prove that n=4 does not
form integer solutions. Fermat's proof is only proving equations
3a,b,c do not form integer solutions. Proving n=4 using only
equations 3a,b,c violates logic. There are an infinite number of
integer combinations of X and Y that are not proven in Fermat's n=4
proof; Fermat is only proving that the equations 3a,b,c do not form
solutions; therefore, Fermat's n=4 proof is incomplete and therefore,
invalid.
3. Wiles' Proof
Wiles' proof of Fermat's last theorem uses Fermat's elliptic curve.
The elliptic curve equation is derived using the integer solution
equations of n=2 (Osserman, p.21),
X = 2uv, Y = u^2 - v^2, and Z = u^2 + v^2 (equ 4),
Therefore, the elliptic curve is only valid for n=2. Wiles' proof of
Fermat last theorem is using Fermat's elliptic curve to prove n>2;
therefore, Wiles' proof using elliptic curves is invalid since the
elliptic curve is only valid for n=2.
4. Proof
I will prove Fermat's last theorem. The variables X, Y and Z represent
the sides of a triangle. When n>2, the following equation describes
the length of Z,
Z = (X^n + Y^n)^(1/n). (equ 5)
The length of Z is represented with the law of cosine.
Z^2 = X^2 + Y^2 - 2XYcos(A). (equ 6)
therefore,
Z = [X^2 + Y^2 - 2XYcos(A)]^(1/2) (equ 7)
Equating equations 5 and 7,
Z = (X^n + Y^n)^(1/n) = [X^2 + Y^2 - 2XYcos(A)]^(1/2) (equ 8)
Since X=Y=Z =/ 0, the minimum and maximum angle of A is,
0 < A< 60 degrees (equ 9)
I will prove that
X^2 + Y^2 - 2XYcos(A) (equ 10)
of equation 8 always forms a non-integer by showing that 2XYcos(A)
always forms a non-integer using the cosine expansion,
cos(A) = 1 - (A^2)/2! + (A^4)/4! - (A^6)/6! + ............. . (equ
11).
Using equation 11 in equation 10, the value of 2XYcos(A) is always a
non-integer; therefore, equation 10 forms a non-integer which proves
that equation 8 can never form a integer value of Z which complete
the proof of Fermat's Last Theorem.
5. Conclusion
Fermat's n=4 proof is a deception that implies that equations 3a,b,c
represent all integers; however, 3a,b,c does not include all integers
of X and Y; therefore, Fermat's n=4 proof is incomplete and therefore
invalid.
Fermat's elliptic curves are derived using the integer solution
equations of n=2; therefore, an elliptic curve can not be used to
prove Fermat's last theorem when n>2 since the elliptic curve is
only valid when n=2.
I will prove Fermat's last theorem by showing that the law of cosine
never forms a integer value of Z when n>2.
6. References
Robert Osserman. Fermat's Last Theorem (a supplement to the video).
MSRI. 1994
Daniel Shanks. Solved and Unsolved Problems in Number Theory. Chelsea
Pub. 1985.
*-----------------------*
Posted at:
www.GroupSrv.com
*-----------------------*
- Next message: Stephen J. Herschkorn: "Re: Set Theory"
- Previous message: Jason Pawloski: "Re: Book recommendations in these areas."
- Messages sorted by: [ date ] [ thread ]
Relevant Pages
|
