FLT65: FERMAT'S LAST THEOREM PART 4

FERMAT'S LAST THEOREM PART 4

NOTE: This part contains some original mathematics. The theorems stated and proved in this section are used in the explanation that follows, but If math is not your thing, you can read the text to get the gist of it and skip over the math. The references cited will be posted with the last part of this paper.

A NOTE ON THE NATURE OF MATHEMATICAL LOGIC AND PROOF

If FLT65 is a valid proof of Fermat’s Last Theorem, why hasn’t it been recognized in all these years? There are a number of reasons, and they are discussed below, but an important underlying reason is the lack of consensus among reviewers concerning exactly what constitutes mathematical logic and proof. We must, therefore, start by explaining what is meant by those terms in this presentation:

Mathematical logic starts with

(1.) the inductive reasoning process of identifying self-evident axioms underlying the logical patterns of reality, like distinctions of subject and object, separation and unity, enumeration and combination, difference and equivalence, extension and content, dependent and independent variables, functional relationships, and

(2.) formalizing the fundamental operations of substitution, addition, subtraction, multiplication and division. Based on such axiomatic concepts and fundamental operations, the logical structures of mathematical systems are built up by processes of deductive reasoning.

 (3.) Mathematical proof consists of using mathematical operations and substitutions of equivalents to determine whether a given statement or proposed theorem is consistent with accepted mathematical logic and the axioms underlying that system.

This sounds like a boring, rigidly structured affair that might be completed in short order. But that is, in fact, not the case.

The history of FLT65 provides a good example of how mathematics is not the ‘cut and dried’ subject that most people seem to think it is. One reviewer of FLT65, a highly intelligent man with deep interests in science and mathematics and significant math skills— but not a professional mathematician —said, after discussing FLT65 for some time with  the author, something to the effect that, “of all things, a mathematical statement like a proposed proof of FLT is something we should absolutely be able to come to an agreement about in short order, because it’s either true or it’s not.” This reflects an erroneous belief about mathematics that was common, even among professional mathematicians, including the great German mathematician David Hilbert, before Kurt Gödel published his proofs of two theorems now known as Gӧdel’s incompleteness theorems (GIT) (1929 – 1931) ^[6]. Gӧdel’s incompleteness theorems are very relevant to FLT and FLT65 because they explain, in part, why some reviewers have not understood FLT65. Additionally, after Gӧdel’s work became widely known, many mathematicians suspected that FLT might actually be an unprovable statement.

Gӧdel’s work was a bombshell that changed mathematics forever: It destroyed Hilbert’s dream of establishing all mathematical logic upon a finite, consistent foundation of axioms, from which all statements could be proved either true, false or meaningless by deductive reasoning. Finding such an all-encompassing algorithmic system was known as the ‘entscheidungsproblem’ (decision problem) ^[7]. Gödel proved, however, that no system of mathematical logic is ever complete. This means that there can be no complete entscheidung algorithm for all mathematical statements. It means that there can be logically legitimate questions posed within any given axiomatic mathematical system that are meaningful, but unprovable within that system.

In the introduction to his breakthrough paper, Gödel said: “As is well known, the development of mathematics towards greater certainty has led to the formalization of much of it such that you can complete proofs by mechanically following a few basic rules. The most comprehensive current formal axiomatic systems are the system of the Whitehead-Russell Principia Mathematica on the one hand, and the Zermelo-Fraenkel axiom-system of set theory on the other hand. These two systems are so thoroughly developed that one can formalize in them all methods of proof currently in use in mathematics, i.e. you can reduce the methods of constructing proofs to a few axioms and deduction rules. Therefore, it seems plausible to [as Hilbert did][1] that these deduction rules are sufficient to decide all mathematical questions expressible in those systems. We will show that this is not true, but that there are even relatively easy problems in the theory of ordinary whole numbers that cannot be decided from existing axioms.”^[8]

 A mathematical statement that has not been proved is properly called a conjecture. With a conjecture, the question to be decided is whether the conjecture is true or not. After Gödel, some mathematicians believed that Fermat’s conjecture (FLT), a legitimate mathematical question, might be an unprovable statement. This idea, however, was based on a misunderstanding of the incompleteness theorems. The incompleteness theorems do not, as one might suppose, imply that questions that cannot be decided within the axiomatic system they are framed in, are necessarily ultimately unprovable. Gӧdel’s theorems did not prove that there are undecidable questions, they proved that there are some questions for which the mathematical system within which one is operating may be inadequate. The system of mathematical logic within which a conjecture is stated may have to be expanded with one or more new axioms and new definitions and theorems before the statement can be proved to be either true or false.

In order for a reader to understand the applicability of the “Division Algorithm” and its three corollaries as they are used in FLT65, it may be necessary to expand the logic of the mathematical system from which the reviewer is operating to include the following definitions, axiomatic principle and theorems:

Concepts of importance in FLT65

1A   Mutually Co-Dependent Variables.

Variables are mutually co-dependent if they are related only by the form of the algebraic expression or equation they appear in, with no designated difference between them in terms of dependence or independence. For example, in the equation zⁿ - xⁿ = yⁿ, the variables x, y and z are mutually co-dependent. If we designate one of them as the dependent variable and assign values to the others so that we can set up an equation in terms of functions of the dependent variable, the variables are no longer mutually co-dependent. For example, if x and y are replaced by specific types of variables or constants, like X and Y for integer variables, or specific values, X₁ and Y₁, z becomes dependent upon their values and we say the equation is of the nth degree in z.[2]

1B. The Principle of Designation of Dependent and Independent Variables:

Given a polynomial f(x, y, z, … ) a function of several variables, over the field of real numbers, any of the otherwise mutually co-dependent variables, x, y, z, …, may be designated as dependent and/or independent, as appropriate to make the polynomial representative of meaningful information as part of an equation or descriptive statement.

This statement expresses an inherent fundamental feature of algebraic expressions consistent with the axioms of a system of mathematical logic. It is a principle rather than a theorem because, while it depends upon the validity of the axioms and fundamental operations of mathematics, it requires no multi-step proof by deductive reasoning from those axioms. This operational principle allows us to use polynomials to set up potentially solvable equations for application to real problems. This principle is relevant to FLT65 and this discussion. This is so because at several points, the designations of dependent and independent variables are changed as needed to express functional relationships consistent with application of the division algorithm. It was used without comment in FLT65, and its use is made explicit here for more clarity.

2A. Parallel Polynomials:

Parallel polynomials are polynomials of the same degree and the same algebraic form, i.e. with the same number of terms and the same coefficients, but with different variables. For example, a₁xⁿ + a₂x^n-1 + a₃x^n-2 + … + a_n+1 and a₁z ⁿ + a₂z^n-1 + a₃z^n-2 + … + a_n+1 are parallel polynomials, as are Az² + Bzx + Cx² and Ax² + Byx + Cy².

2B. The Theorem of Parallel Factorization:

If a given polynomial, f(z), of degree n > 1, defined over the field of real numbers, is divisible by z –a, then the parallel polynomial f(a) is also divisible by z – a.

This theorem is proved indirectly in FLT65, and is proved formally here for the first time, as far as the author knows. It could also be considered to be a corollary to the division algorithm. Note that the choice of symbols used to represent variables in proofs is arbitrary: The validity of axioms and theorems do not depend upon the symbols used—the use of any consistent set of symbols would produce the same theorems. However, the symbols a, x, y, and z were chosen to correspond to the symbols used throughout this paper to facilitate understanding and application of the theorems to FLT65. In the generalized two-variable algebraic polynomial forms used in the following proofs, x and z were chosen for the same reason.

Proof: Consider the generalized n-degree polynomial:

f(z, x) =Azⁿ +Bz^n-1x +Cz^n-2x² +…+Mxⁿ. …………………………………………………………..       1

By the Axiom of Unrestricted Designation of Variables, we may designate x = X₁, to obtain the single-variable polynomial

f(z) =Azⁿ +Bz^n-1X₁ +Cz^n-2X₁² + … + MX₁ⁿ  ……………………………………………………….  2

If we divide f(z) by z – a, by Corollary II of the Division Algorithm, the remainder,

f(a) = Aaⁿ +Ba^n-1X₁ +Ca^n-2X₁² +…+ MX₁ⁿ. ……………………………………………….  3

Then, f(z) – f(a) = Azⁿ - Aaⁿ + Bz^n-1X₁ - Ba^n-1X₁² + Cz^n-2X₁ ² - Ca^n-2X₁² + …+ LzX₁^n-1 - LaX₁^n-1 + Mxⁿ - MX₁ⁿ = A(zⁿ - aⁿ) + B X₁(z^n-1 - aⁿ) +… + 0 = A(z – a)(z^n-1 + z^n-2a +…+ a^n-1) +BX₁(z – a)(z^n-2 + z^n-3a +…+ a^n-2) +…+ LX₁^n-1(z - a) →f(z) – f(a) = (z – a) {A(z^n-1 + z^n-2a + … + a^n-1) + B X₁(z^n-2 + z^n-3a + … + a^n-2) +…+ LX₁^n-1}.  ………………………………….. 4

Then, since f(z) is divisible by z – a, and f(z) – f(a) = (z – a) {A(z^n-1 + … + a^n-1)+B X₁(z^n-2 + … + a^n-2) +…+ L X₁^n-1}, by Corollary I of the Division Algorithm, f(a) is also divisible by z – a, Q.E.D. ………………………………………………………………………………  5

Use of this theorem shortens the discussion of FLT and FLT65 considerably, as the reader will see in the discussions that follow. Also, proof of the companion theorem for integers given below is important because it removes some of the mystery expressed by reviewers concerning the application of theorems that are true for algebraic polynomials to the subgroup involving integers, namely integer polynomials.

2C. The Theorem of Parallel Factorization for Polynomials of Integer Variables:

 If a given polynomial, f(Z), defined over the ring of integers is divisible by Z –a, the parallel polynomial f(a) defined over the ring of integers is also divisible by Z – a.

This can be inferred from the proof of the theorem of parallel factorization for polynomials over the field on real numbers and the fact that the integers are a subset of the set of real numbers. Or, stated more formally: It follows from the fact that all of the elements of the ring of integers are contained within the set of elements of the field of real numbers. However, importantly, it can be proved independent of the proof for polynomials over the field of real numbers as follows:

Proof:

Given the general polynomial in two variables:

f(z,x) =Azⁿ +Bz^n-1x +Cz^n-2x² +…+Mxⁿ . ………………………………………………. 1

By the Axiom of Unrestricted Designation of Variables, we may designate x = X₁ a specific integer and z = Z, an integer variable, in the general polynomial to obtain f(Z) =AZⁿ +BZ^n-1X₁ +CZ^n-2X₁² +…+MX₁ⁿ. ……………………………………………. 2

And then, when we divide f(Z) by Z – a, by Corollary II of the Division Algorithm, the remainder is the parallel polynomial

f(a) = Aaⁿ +Ba^n-1X₁ +Ca^n-2X₁² +…+MX₁ⁿ. ……………………………………………… 3

 Then, f(Z)–f(a) = AZⁿ - Aaⁿ + BZ^n-1X₁ - Ba^n-1X₁² + CZ^n-2X₁ ² - Ca^n-2X₁² + …+ LZX₁^n-1 - LaX₁^n-1 +Mxⁿ - Mxⁿ =A(Zⁿ - aⁿ) + Bx(Z^n-1- aⁿ) +…+ 0 =A(Z – a)(Z^n-1 +Z^n-2a + …+ a^n-1) +BX₁(Z – a)(Z^n-2 +Z^n-3a +…+ a^n-2) +…+LX₁^n-1(Z - a) → f(Z) – f(a) = (Z – a){A(Z^n-1 + Z^n-2a + … + a^n-1) +BX₁(Z^n-2 + Z^n-3a + … + a^n-2) +…+ LZ^n-1} ………………………………………… 4

Then, If f(Z) is divisible by Z – a, and f(Z) – f(a) = (Z – a){A(Z^n-1 +  … + a^n-1) + BX₁(z^n-2 + … + a^n-2) +…+ LX₁^n-1}, then, by Corollary I of the Division Algorithm, f(a) is also divisible by Z – a, Q.E.D. …………………………………………..………………………………… 5

By incorporating the operational principle, definitions and theorems presented above into the axiomatic system of mathematical logic and applying them to FLT, we will show how  the FLT65 demonstration that f(a), the remainder obtained by dividing f(Z) by Z – a can never equal zero [i.e. f(a) ≠ 0], provides an unavoidable contradiction proving Fermat’s Last Theorem.

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[1] A comment added by the author of this paper.

[2] See Appendix D for detailed notational definitions.