Fermat's Last Theorem
Ben Ito
11-4-04 I will show that Fermat's n=4 proof assumes that the integer solutions of n=2 represent all the possible integer solutions yet n=2 and n=4 form completely different equations; therefore, the solutions of n=2 cannot represent all the integer possibilities of n=4. I will then show that Fermat's derivation of the elliptic curves equation is based on equations that describe a right triangle; consequently, Wiles' proof of Fermat's Last Theorem is invalid since only n=2 can be represented with an elliptic curve. I will then prove that only a right triangle when n=2 forms the alignment and consistency that allows for the formation of integer solutions.
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 variables 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 integer solutions,
A'^2 = 2uv, B'^2 = u^2 - v^2, and C' = u^2 + v^2 (equ 5),
Fermat is using the equations (equ 5) that describe the integer solutions of n=2 to prove that equation 3 does not form integer solutions; however, the equation derived from n=2
A^2 + B^2 = C^2 (equ 6)
is completely different form equation 3;
A^4 + B^4 = C^2 (equ 7).
Using C = constant, equation 6 forms the equation of a circle; however, using C = constant in equation 7, a circle equation is not formed; therefore, equation 6 and equation 7 are completely different equations.
Fermat is implying that by proving that integer solutions of n=2 do not form solutions to n=4 he has proven n=4 does not form integer solutions; however, n=2 solutions do not include all possible integer combinations of A, B and C. Equation 5 only represents the integer solutions of n=2, not all possible integer combinations that Fermat is implying is represented. The proof for n=4 must include all possible integer combinations of A', B', and C'; however, Fermat's n=4 proof does not included all possibilities of A', B' and C'. Example, A=23, B=24 and C=25 are not included in Fermat's n=4 proof. There are an infinite number of integer combinations that are not included in Fermat's n=4 proof. Fermat is implying that n=2 and n=4 are similar equations and that there is a relationship between the n=2 and n=4 equations; therefore, Fermat uses the solutions of n=2 to prove that n=4 doesn't form solutions; however, the equations formed by n=2 and n=4 are completely different equations; therefore, using the integer solution of n=2 to prove Fermat's n=4 is an incomplete proof.
In addition, the equations that are used by Fermat (equ 5) form variables A', B' and C' where at least one or more of the variables are non-integers; square rooting both sides of equation 5 forms the following equation,
A' = (2uv)^(1/2), B' = (u^2 - v^2)^(1/2), and C' = (u^2 + v^2)^(1/2) (equ 8),
Using u = 2 and v = 1, in equation 8,
A=2 , B = 3^(1/2) and C = 5^(1/2).(equ 9)
Using equation 8, at least one of the variables of the set A', B' and C' forms a non-integer. Another example, using u = 8 and v = 3, in equation 8
A = 48^(1/2), B = 55^(1/2) and C = 73^(1/2).(equ 10)
All of the sets of A', B' and C' that are derived using equation 8 form non-integer sets. It is questionable how Fermat uses non-integers to prove that n=4 does not form integer solutions. Equations 6 and 7 are completely different equations; therefore, using equations 8 with equation 7 will always form non-integer solutions. Fermat must prove that all possible integer combinations of A' B' and C' do not from integer solutions.
Fermat is justifying the non-existence of integer solutions by proving that a single group of solutions (equ 8), do not form integer solutions then concluding that the proof includes all possible integer combination. Fermat proof does not prove that all possible integer combinations of A', B' and C' do not form integer solutions.Consequently, Fermat's n=4 proof is incomplete and therefore invalid.
3. Wiles Proof
Wiles' Proof of Fermat's Last Theorem originates from Fermat's elliptic curves.
"Fermat counter example leads to elliptic curves." (Ribet, video)
Fermat implied that elliptic curves may be where the possible proof of his equation could be obtained. Fermat's elliptic curves is the origin of Taniyama-Shimura Conjecture and Wiles proofs. The derivation of Fermat's elliptic curves are described in the following:
"Fermat was to show that if a right triangle has whole-number sides, then its area cannot be a perfect square. If the sides are a, b, c, then the area is ab/2. The statement then is that the pair of equations,
a^2 + b^2 = c^2 (equ 11)
and
ab = 2d^2 (equ 12)
have no whole number solutions a, b, c, d. Since every solution of the first equation is of the form
a = k(m^2 - n^2), b = k2mn, c = k(m^2 + n^2),(equ 13)
the second equation takes the form:
2d^2 = ab = k^2(m^2 - n^2)2mn = 2k^2mn(m^2 - n^2), (equ 14)
or
d^2 = k^2(m^3n - mn^3).(equ 15)
We now let
x = m/n, y = (dn^2)/k.(equ 16)
Then
x^3 - x = (m^3)/(n^2) - m/n = (m^3n - mn^3)/n^4 = (n^4d^2)/k^2 = y^2 (equ 17)
Proving that there are no whole number solutions a, b , c, d to the original equations is the same as proving that there is no solution x, y to the elliptic equation
y^2 = x^3 - x, (equ 18)
where x and y are fractions m/n, r/s with whole numbers m,n,r,s." (Osserman, p. 21-2).
However, Fermat's elliptic curve derivation is based on right triangle equations 11, 12 and 13; therefore, Fermat's elliptic curve derivation is based on n=2. Consequently, the elliptic curve derivation is only valid for n=2; therefore, Wiles proof of Fermat's Last Theorem is invalid since only n=2 can be represented with Fermat's elliptic curves.
Wiles proof of Fermat's Last Theorem is based on the elliptic curve equation (Poorten, p. 196-7),
y^2 = x(x - a^n)(x + b^n) (equ 19).
"Casually in the middle of a conversation this friend told me that Ken Ribet had proved a link between Taniyama-Shimura and Fermat's Last Theorem." (Nova, 2000).
"This paper has appeared in Acta Arith. 79 (1997), no. 1, 7-16, and so the dvi version has been removed. We discuss the equation a^p + 2^n b^p + c^p =0 in which a, b, and c are non-zero relatively prime integers, p is an odd prime number, and n is a positive integer." (Ribet p. 7-6)
Wiles implies that Ribets' equation is Fermat's equation
a^n + b^n = c^n (equ 19).
Wiles does not derive equation 19 from equation 18; Wiles is implying the existence of equation 19.
"Ribet and Wiles studied this curve under the assumption that there exist a nonzero integer c such that a^n + b^n = c^n." (Ribenboim, p. 247).
It's questionable how Wiles derive Fermat's equation from Ribet's equation then basing his entire proof on an equation 19. The variables of Fermat's equation are in the x, y variables of the ellipitic curve equation; therefore, Freys, Ribet, Wiles and Taniyama-Shimura assumption that the constant of the elliptic curve equation (equ 11), a and b, represent Fermat's equation is invalid. In addition, Fermat's elliptic curve equation is derive using the integer solution equations of n=2 (equ 13); therefore, n>2 cannot be represent with the elliptic curves. Consequently, Wiles proof of Fermat's Last Theorem is invalid.
4. Ito's Proof.
I will form the Proof of Fermat's Last Theorem by showing that only 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 the equation of a circle (n=2),
x^2 + y^2 = c^2. (equ 21)
In the circle transformation, the hypotenuse of the right triangle becomes the radius of the circle. Consequently, a circle of radius r and the right triangle with a hypotenuse z can be represented together on the x-y plane which forms the primary alignment. Only n=2 forms the x, y and z lengths on the x-y plane (primary alignment) which allows for the possible formation of integer solution of Fermat's equation when n=2. The equation describe with Fermat's equation when n>2 never forms the primary alignment where the transformed structure (z=c) forms the x, y and z triangle on the x-y plane;
The secondary alignment occurs when the value of c is increased. Example, when c = 3, equation 9 does not form the secondary alignment required to form integer solutions; however, when z is increased to c = 5, the secondary alignment is formed and integer solutions occurs at x=3 and x=4. As c increases, the secondary alignment shift in and out of phase. At c = 13, the secondary alignment is formed at x=12 and x=5.
The secondary alignment has consistency. For each solution value of c, two symmetric solutions are formed, example, c=13, x=12 and x=5. The reason that two symmetric solutions are formed is because of the consistency of the triangles formed within the circle; the end of the hypotenuse or tip of the radius traces the structure of a circle; therefore, the angle of the triangle and the variable x can be represented with a sinusoidal equation. Consequently, the primary, secondary alignments and the consistency of the secondary alignment form the integer solutions of n = 2. Only n=2 forms the alignment and consistency that is required to form integer solutions.
therefore, only n=2 of Fermat's equation forms the integer solutions.
5. Conclusion
I have shown that Fermat's derivation of n=4 is base on questionable logic since Fermat uses non-integer solutions of A, B and C to prove that Fermat's n=4 equation does not from integer solution.
Wiles' proof of Fermat's Last theorem is based on the implied assumption that
a^n + b^n = c^n(equ 15)
describes an elliptic curve. Wiles does not derive equation 15; Wiles assumes that equation 15 and the elliptic curve equation are related and represent all possible integer solutions which is not logically possible; therefore, Wiles' proof of Fermat's Last Theorem is incomplete and therefore invalid.
I will solve Fermat's Last Theorem by proving that only n=2 forms integer solutions by showing that the transformation when z=c where c is an integer forms the conditions that allow for the formation of the integer solutions of x, y and z. These conditions only occur when n=2; therefore, only n=2 forms integer solutions which completes the proof of Fermat's Last Theorem.
6. References
Robert Osserman. Fermat's Last Theorem (a supplement to the video). MSRI Berkeley. 1994
Marilyn vos savant. The World's Most Famous Math Problem. St Martin's Press. 1993
Daniel Shanks. Solved and Unsolved Problems in Number Theory. Chelsea Pub. 1985.
A. J. Van Der Poorten. Notes on Fermat's Last Theorem. John Wiley. 1996
7. Acknowledgment
Special thanks to Rudi, Nate, Peter, Jakob, Joesph, Ken, 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, ASU, Rutgers and UCLA mathematics Dept.
Followup postings for this message are closed.