Fermat's Last Theorem

Ben Ito

10-27-04

I will solve Fermat's last theorem.

l. Introduction

I will show that Fermat (n=4) and Wiles' proofs are invalid then prove that x^n + y^Y = z^n only forms integer solutions when n>2 using a circle transformation.

2. Fermat's Proof (n=4)

Fermat's Proof when n=4 is described. The proof of n=4 using the following equations

A^2 = 2uv, B^2 = u^2 - v^2, and C^2 = u^2 + v^2, (Shanks, p, 148) which conflicts with,

A = 2uv, B = u^2 - v^2, and C = u^2 + v^2. (Shanks, p.141) that are the original equations.

Consequently, 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 = x(x - a^p)(x + b^p) (equ 3)

where

a^p + b^p = c^p (equ 4)

a,b and c are constants of equation 3. Wiles assumed that since equation 4 is similar to Fermat's equation that Fermat's last theorem can be described using elliptic curves. Wiles does not prove Fermat's last theorem since n>3 is not represented by the elliptic curve equation. There are an infinite number of high order equations that are formed when n>3 that are not represented with Wiles' ellipitic curves.

4. Ito's Proof.

I will form the Proof of Fermat's Last Theorem by showing why right triangles form integer solutions. Wiles and Fermat's proofs are base on condradictions. My prove will be base on discovering how a right triangle forms the integer solutions. I will use x,y and z to represent the side of a triangle. Let z = c where c is an integer; Fermat equation becomes the equation of a circle (n=2),

x^2 + y^2 = c^2.

In the case of the circle transformation, the hypotenus of the right triangle becomes the radius of the circle. Consequently, the transformed circle and the right triangles can be represented together to form the integer solutions by increasing the value of c. Consequently, only the right triangles form the condition where the transformation forms the triangle and the transformed structure on the same plane. Consequently, only right triangles when n=2 form the possiblility of integer solutions of Fermat's equation.

5. Conclusion

I have shown that Fermat's derivation of n=4 includes a mathematical error. I then show that Wiles proof of Fermat's Last theorem only describe lower orders of n; therefore, Wiles' proof is incomplete. I then show that only the right triangle cannot be tranformed onto a plane of the transformtion of z = c where c is an integer. Therefore, only the right triangles cannot form integer solutions of x, y and z.

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5. 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.

6. 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. Also, thanks to my Mom and Daddy.