1564.1

Fermat's Last Theorem

Ben Ito

10-24-04

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 the circle transformations.

2. Fermat's Proof (n=4)

Fermat's Proof when n=4 is described. Fermat's proof is based on the following statement, "Finally, if there any solutions of

x^4 + y^4 = z^4 (equ 1)

in whole numbers, then we could let c =z^2 to get x^4 + y^4 = c^2, and applying the above argument woud lead to a contradiction. We are forced to conclude that no such solution is possible, which proves Fermat's Last Theorem for the case n=4." (Osserman, p. 22).

Fermat is assuming that c = z^2 contradicts Pythagorean's equation; however, when c = z^2 is used in x^4 + y^4 =c ^2,

x^4 + y^4 = z^4,(equ 2)

which eliminates the contradiction; therefore, Fermat's justification that no integer solution exist for n=4 is invalid.

3. Wiles' Proof

Wiles implies that Diophantine equations x^n + Y^n = z^n can be translated to described a set of elliptic curves. "These curves represent the surface of a torus, an object shaped like a smooth doughnut." (Savant, p. 30). However, the equation for an elliptic curve is

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

Elliptic curve are only valid for n=3; therefore, Wiles' assumption that he has solved Fermat's Last Theorem using elliptic curves is invalid.

4. B. Ito's Proof.

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