Fermat's Last Theorem

Ben Ito

10-28-04

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),

However, A and B of equations 2 and 5 are not equal,

A^2 =/ A and B^2 =/ B. (equ 6)

Consequently, Fermat is improperly using equation 2; therefore, 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,

y^2 = ax^3 + bx + c (equ 7)

where

a^p + b^p = c^p (equ

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 uses a deception by 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 why 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 an 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 and y plane which forms the primary alignment. Only when n=2 forms the primary that is required in forming the integer solution of Fermat's equation.

5. Conclusion

I have shown that Fermat's derivation of n=4 is base on a mathematical error. 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. I then show that only n=2 forms the primary alignment that forms the of integer solutions of Fermat's equation. Consequently, only n=2 forms the condition where the integer solutions can be formed.

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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