Fermat 420
From: ben ito (benito20044_at_yahoo-dot-com.no-spam.invalid)
Date: 11/28/04
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Date: 27 Nov 2004 19:29:40 -0600
UNDATE 11-27-04
Fermat's Last Theorem
Ben Ito
11-27-04
I will show that Fermat's n=4 and Wiles' proofs are invalid then prove
Fermat's last theorem using algebra.
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 that are derive from equations
2a,b,c are used to prove that n=4 does not form integer solutions.
Fermat's proof is only proving equations 3a,b,c that are derived from
the integer solution equation of n=2 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. The elliptic curve y^2 = x^3 -
x is derived from the n=2 equations; therefore, I question how Wiles
could use the ellipitc curves to prove n>2 since once you start
using n>2 than the elliptic curves would have to be derived using
n>2. However, elliptic curves are unique to n=2. Example, n=3
does not form an elliptic curve. How does Wiles' justifies using
elliptic curves, if n=3 does not form an elliptic curve?
4. Proof
I will prove Fermat's last theorem. Using n=3 in equation 1,
X^3 + Y^3 = Z^3, (equ 5)
let X and Y represent all integer combinations of X and Y.
Squaring both sides of equation 5,
(X^3 + Y^3)^2 = Z^6, (equ 6)
then taking the cube root of equation 6,
(X^3 + Y^3)^(2/3) = Z^2.(equ 7)
First, squaring the integer value of,
(X^3 + Y^3)^2, (equ 8)
equation 8 always forms an integer value. Cube rooting the integer
value of equation 8,
(X^3 + Y^3)^(2/3), (equ 9)
equation 9 always forms a non-integer.
Next, squaring rooting the non-integer value of equation 9 to derive Z
(equ 7),
[(X^3 + Y^3)^(2/3)]^(1/2) = Z, (equ 10)
Equation 10 always forms a non-integer value of Z. Consequently, for
n=3, I have proven that Z never forms an integer, for all integer
combinations of X and Y. This method can be used to prove n>2
(odd) and a similar method can be used to prove n>2 (even).
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've proved Fermat's last theorem by showing that n>2 for all
integer combinations of X and Y never forms an integer value of Z
using algebra.
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.
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