Fermat's Last Theorem
I will solve Fermat's last theorem.
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 (Savant, p. 22), (Osserman, p. 22). Fermat equation is represented with
x^n + y^n = z^n. (equ 1)
where n= 1,2,3,4,5,6,7,8,9,.........
Fermat uses a transformation from n=1,2,3,4,5... to n = 4,8,12,20...... 4k where k=1,2,3,4,5,6,7........
Consequently, x-->X, y-->Y and z-->Z. The variables x,y,z asociated with n=1,2,3,4... and the variables X,Y,Z are assiociated with n= 4,8,12,20.....
"If n= 4k, then y^n + y^n = z^n is impossible; X^4 + Y^4 = Z^4 in which
X=x^k, Y=y^k, and Z = z^k." (Savant, p. 22). (equ 2)
However, the transformation from n=1,2,3,... to n=4,8,12,20, equation 2 becomes part of X^4 + Y^4 + Z^4; therefore, equations 2 are valid. Fermat's proof is based on the assumption that equations 2 are not possible.
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)
The largest power of an elliptic curves is three; therefore, to represent all values of n>3 violates logic. How can an elliptic curves represent n=20 which forms power of 20.
Consequently, Wiles' Proof of Fermat's equation base on ellilptic curves is invalid.
4. B. Ito's Proof.