The subtle mistake is caused by reviewers ignoring the uniqueness requirement for the application of corollary III of the division algorithm. Ignoring the uniqueness requirement leads to the erroneous conclusion that FLT65 is wrong as follows:
The reviewer reasons: “FLT65 correctly shows that when f(Z), an algebraic polynomial of degree p-1 in Z, is divided by Z – a, a factor of Y1, the remainder is f(a) = an-1+X1an-2+…+X1n-1. And FLT65 also correctly states that f(a) cannot equal zero because a and X1 are positive integers if the x, y and z of the Fermat equation are positive integers, but corollary III says f(Z) is divisible by Z – a, if, and only if, f(a) = 0. And we know that this is true for algebraic polynomials, but we can find integers, a, X and Z, such that the integer value of f(Z) is divisible by the integer value of Z – a, even though f(a) is not equal to zero. Therefore, FLT65 is wrong!”
This reasoning sounds quite convincing until you realize that the uniqueness requirement has been ignored. What corollary III actually says, as quoted in FLT65, is that “if q(Z) and r(Z) are unique, f(a) cannot contain Z - a, and so it follows that a polynomial, f(Z), of degree greater than 1, is divisible by Z - a IF AND ONLY IF, f(a) = 0. “
The quotient and remainder are unique only when the remainder has the minimum value of either zero, or less than Z – a, and the author has shown above that for the examples, and for any integers a, X and Z, the unique remainder and quotient can only occur when a is a non-integer.
The proof of corollary III depends on the uniqueness of the quotient and remainder with division of f(Z) by Z – a for any specific integer value of x = X1. The uniqueness requirement for application of the corollary is clearly stated in FLT65, and the complete proof is attached in Appendix A. Note that the use of X as the variable in the proof of the division algorithm and corollaries in FLT65 is arbitrary and follows the generic practice of using X for the variable unless other letters or symbols are needed for clarity. The variable Z is used in these discussions to conform to its usage in the FLT equation.
In the examples mistakenly thought to be counterexamples disproving FLT65, the quotient and remainder only become unique when f(a) reaches its minimum value and the quotient reaches its maximum in the repeated division of the remainder (which must contain Z – a if there are integer solutions for the FLT equation) by Z – a = ai-1 – ai. When the unique minimum remainder and maximum quotient for a given X1 are reached, and the value of Z – a is identical with the integer factor obtained by substituting the unique values of ai-1 and ai, for Z and a, corollary III applies and f(a) ≠ 0 implies f(Z) cannot contain Z – a or its integer value, as a factor, and FLT is proved. See Appendix E for complete details.This contradiction proving FLT is made obvious by the repetitive division of the remainder by Z - a, a process of infinite descent, which was a favorite method used by Fermat.
Thus, Pierre de Fermat’s claim that he had found a ‘marvelous’ proof, may be vindicated, as the FLT65 concepts were certainly available to Fermat in 1637.
The only significant objections to FLT65, raised by mathematicians who have reviewed it, are based on the hypothesis that the division algorithm which applies to algebraic polynomials, might not apply to integer polynomials formed by integer solutions of the Fermat equation, if there are any.
Some reviewers have jumped to the conclusion that this concern reveals a fatal flaw in FLT65, because one can find integer values for X, a and Z that produce values of Z – a that divide f(Z), even though the remainder f(a) is not zero, seeming to contradict corollary III of the division algorithm. However, this is a subtle misdirection of logic made by reviewers who have ignored the uniqueness requirement for application of corollary III. As stated in the proof of Corollaries II and III in FLT65: if q(Z) and r(Z) are unique, f(a) cannot contain Z – a, and so it follows that a polynomial f(Z) of degree greater than one is divisible by Z –a, IF AND ONLY IF, f(a) = 0.
So we see that corollary III is not actually violated by the reviewers’ examples. The integers chosen in such examples do not produce valid counterexamples because, as shown above, they do not produce unique values for q(Z) and F(a). The fact that integers are closed with respect to addition applied to the integer equation f(Z)/(Z – a) = q(Z)+ f(a)/(Z – a) implies that because f(Z) contains (Z – a)n, f(a) must also contain Z – a, but when q(Z) and F(a) are unique, as they must be for an integer solution to the Fermat equation, we have an unavoidable contradiction proving that, if X and Y are integers, a cannot be an integer, implying that Z cannot be an integer, proving FLT.
The hypothesis that the division algorithm might not apply to integer polynomials is falsified in two independent ways:
(1.) the division algorithm expresses a basic mathematical relationship that applies to all polynomials composed of elements of the field of real numbers, including the integers. This means that integer polynomials derived from integer solutions of the FLT equation must conform to the division algorithm and its corollaries.
(2.) The author demonstrates in Appendix E by the method of infinite descent that the argument that f(z) may be divisible by Z – a for some integer values of Z, is simply irrelevant. The attempted counterexamples fail because, if there are integers that satisfy the Fermat equation, they must also satisfy the mathematical relationship between the integer polynomials formed by factoring the Fermat equation. The resulting factors are algebraic polynomials, and thus the division algorithm applies to them. As integer polynomials, they comprise a subset of the real number polynomials formed by the p real number solutions of the Fermat equation.
This means that f(z)g(z) =Ap(z – a)p = Yp, a condition that is necessary if there are integer solutions, can only be true if z = a, which produces the trivial solution with Y = 0, or if a is a non-integer. (See Appendix F.) If a is a non-integer, Y, which contains Z – a, cannot be an integer, producing an unresolvable contradiction. So, the fact that f(a) ≠ 0 when f(Z) is divided by Z - a, actually proves that the equation xn + yn = zn, where n is a prime > 2, has no solution in integers, as stated in FLT65.
The author’s position has been, and remains, that FLT65, his 1965 proof, is conclusive and complete. He maintains this position for two simple reasons:
(1.) The division algorithm and corollaries applied in the proof are valid for polynomials over the entire field  of real numbers, which includes the integers, and
(2.) Application of the division algorithm to the FLT equation produces a unique remainder that allows us to determine the numerical type of the third variable when two of them are assumed to be integers (Appendix F), and except for one trivial solution, that type is non-integer.
 In mathematics, a field is a group of ordered quantitative elements that contains a multiplicative inverse for every nonzero element. As such it is an arithmetical structure within which the operations of addition, subtraction, multiplication, and division are consistent and valid, and it supports an algebraic structure within which the same basic operations are valid. Geometrically, a field is a dimensional domain within which arithmetically and algebraically defined elements can be described and located. The three most basic mathematical fields are the field of real numbers, the field of rational numbers, which include integers, and the field of complex numbers.
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