**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**Y**.*=*^{n}*Z*^{n }- X^{n}*Briefly, it does this as follows*:
·
The right-hand side of the
equation

**Y**can be factored into two polynomials:*=*^{p}*Z*^{p }– X^{p}

*Z*= (Z – X)(Z^{p }– X^{p}^{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**, a perfect^{p}**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**, the remainder***unique**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***+Xa*^{p-1}*+…+X*^{p-2}^{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,**; so the fact that*IF, AND ONLY IF*, f(A) = 0**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**since for an integer solution of the Fermat equation,***unavoidable contradiction***Z – a***must divide***f(Z)**because**f(Z) = (Z – a)**, a perfect^{p}**p**-power of an integer.
·

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