FERMAT'S LAST THEOREM FLT65 PART SIX

FERMAT' LAST THEOREM FLT65 PART SIX

FLT65 provides a straight-forward proof using the Division Algorithm and its corollaries to show that for n equal to p, any prime greater than 2, there are no purely integer solutions for Yⁿ = Zⁿ - Xⁿ. Briefly, it does this as follows:

·       The right-hand side of the equation Y^p = Z^p – X^p can be factored into two polynomials:

                       Z^p – X^p = (Z – X)(Z^p-1+XZ^p-2+…+X^p-1)                                                  (1.)

·       These factors can be considered to be co-prime (not containing common factors) for reasons that are detailed in FLT65 and Appendix C.

·       Since f(Z) and g(Z) are co-prime and their product is equal to Y^p, a perfect p power integer, both factors must be equal to perfect p-powers of co-prime integers if all three variables, X, Y and Z, are to be integers.

·       The proof continues by dividing the p - 1 degree polynomial, designated by f(Z), by another first degree polynomial, Z – a, where a is defined by f(Z) = (Z – a)^p: 

f(Z)/(Z – a) = q(Z) + r(Z)/(Z – a)                                                      (2.)

and for X and Z equal to specific integers, a becomes the variable: f(Z)/(Z – a) = q(a) + r(a)/(Z – a),  Note that for every specific value of Z, there is an a,  but for only one specific integer value of Z, when q(a) is maximum and r(a) is minimum, are they unique.

·       The division algorithm and its corollaries apply to all algebraic polynomials in real number variables, including the algebraic polynomial factor of the FLT equation, f(Z) = Z^p-1+XZ^p-2+…+X^p-1.

·       And since integer solutions of Y^p = Z^p – X^p, if any exist, form a subset of the real number solutions of the equation, integer polynomials formed with them must also conform to the Division Algorithm and its corollaries.                                                              

·       By corollary II of the division algorithm, the polynomial remainder when f(Z) is divided by Z – a is f(a), and by Corollary III of the Division Algorithm, if q(a) and f(a) are unique, the remainder must equal zero if f(Z) is divisible by Z - a.                            

·       But, when f(Z) is divided by Z – a, the remainder, r(a) =f(a) = a^p-1+Xa^p-2+…+X^p-1 cannot equal zero because if X and a are equal to integers, f(a) is always positive.

·       For only one specific pair of integer values of a and Z, the quotient and remainder, q(a) and r(a), are unique, and corollary III says that, if the quotient and remainder are unique, a polynomial f(Z) is divisible by Z – a, IF, AND ONLY IF, f(A) = 0; so the fact that f(a) ≠ 0 implies that the polynomial factors, f(Z) and Z – a, are co-prime for those specific integer values of a and Z. This is an unavoidable contradiction since for an integer solution of the Fermat equation, Z – a must divide f(Z) because f(Z) = (Z – a)^p, a perfect p-power of an integer.                

·       Thus, if a, X and Z are integers, f(a) ≠ 0 proves FLT.