FERMAT'S LAST THEOREM & THE BUILDING BLOCKS OF THE UNIVERSE

Fermat’s Last Theorem

and

the Building Blocks of the Universe

©Edward R. Close, January 18. 2018

There is a deep connection between Fermat’s Last Theorem (FLT) and the geometric structure of the universe. It has been overlooked by mainstream science for nearly 400 years. The connection is simple, but has been obscured by abstract mathematical complexity and the myopic restrictions of academic specialization. It has been 353 years since Fermat died, and I think it is time for the importance of his work to be more fully recognized. This essay is an attempt to do that.

There are many observations about numbers that are easy to state and easy to understand, and yet very difficult to prove, and FLT is a prime example of this kind of statement. Fermat said that he had found a “marvelous” proof, but, because it was never found, and no one else could produce one, mathematicians considered FLT nothing more than a conjecture from 1637, when Fermat made the statement, until 1994, when British mathematician Andrew Wiles produced a 129-page proof that has been accepted by the community of professional mathematicians, allowing FLT to finally take its place as a legitimate mathematical theorem. Because of the complexity of Wiles’ proof, relying on theorems that were unknown in Fermat’s day, many mathematicians doubt that Fermat actually proved it.

Pierre de Fermat was a lawyer and judge at the Parlement of Toulouse, in France, but his real passion was mathematics. In 1637, he wrote the following statement in Latin in the margin of his personal copy of Arithmetica by Diophantus of Alexandria:

"Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.”

Which, translated into English is:

"It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

The theorem stated in somewhat simpler English is:

Two whole numbers raised to any power greater than 2, and then added together, cannot be equal to a third whole number raised to the same power.

In simple mathematical notation:
Xⁿ + Yⁿ ≠ Zⁿ if X, Y, Z, and n are integers and n ≥3.
For ease of expression in this discussion, I shall call the equation Xⁿ + Yⁿ = Zⁿ Fermat’s equation. Note that the ‘equals’ sign (=) is replaced by ≠ (is not equal to) if Fermat’s Conjecture is true. Fermat’s proofs for the case n = 3 emerged from his writings after his death, but no general proof for all n > 2 was ever found.

FLT is a statement about numbers that is easy to state and easy to understand, but it went without proof for more than 300 years, even though the best and brightest mathematicians tried to prove (or disprove) it. Professional mathematicians correctly labeled Fermat’s Last Theorem a conjecture, which is what a mathematical statement should be called until it is either proved or disproved. It is likely that every mathematician alive in the last three centuries has tried to prove FLT, because the longer it went without resolution, the more famous it became, and resolving such a puzzle would assure the one who was first to solve it great recognition and fame; and mathematicians rank right up there with, or very close to, physicists, as a group of people with well-developed egos.

Karl Friedrich Gauss, arguably one of the greatest mathematicians who ever lived, was no exception. He invented modular algebra as a tool for use to investigate Diophantine equations, which is what Fermat’s equation is. [Diophantus specialized in solving equations with whole-number (integer) solutions. So, mathematicians call equations for which only integer solutions are sought, Diophantine equations.] Gauss proved Fermat’s Conjecture for n = 3, using modular algebra and complex numbers, but failed to be able to generalize the proof to greater values of n. When asked about his interest in Fermat’s Conjecture by Astronomer Wilhelm Olbers, Gauss said:

“I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.”

Could this disdain for FLT be the statement of a big ego, unwilling to admit that he couldn’t prove it, even though he tried? Probably so. Even a great man like Gauss can be blinded by his own brilliance. But I think Gauss can be forgiven his short-sightedness, because many brilliant thinkers believe that there are mathematical statements that can never be proved or disproved, and that FLT is an abstract theorem with little or no practical application outside of abstract number theory. Modern mathematicians, on the other hand, who believe the same things, should not be let off the hook so easily, because they have Gӧdel’s Incompleteness Theorems, which sheds some light on the questions of proof and provability.

In 1931, Kurt Gӧdel proved, with two decisive theorems, that no consistent system of mathematical logic is ever complete. He proved that within any logical system, there will always be statements that, even though they may be easy to understand, cannot be proved true or false within the system in which they are defined. Some mathematicians thought that FLT was actually such a statement, just as Gauss did, even though he died 76 years before Gӧdel published his proofs.

The fact is however, that Gӧdel’s Incompleteness Theorems do not say that there are propositions that can never be proved. They only say that there are propositions that cannot be proved using only the system of logic within which they were stated. Every logical system is based on one or more a priori assumptions. A meaningful statement that cannot be proved true or false within a given logical system, may actually be provable when the a priori assumptions of the system are either added to, or changed in some fundamental way.

I believe that Pierre de Fermat, and Diophantus of Alexandria, - and perhaps a few others – at least had inklings of the importance of Fermat’s Last Theorem as it relates to the building blocks of the universe, but modern mathematicians and physicists, and scientists in general, seem to have missed the point, mainly because of narrow academic specialization, coupled with a bit of human ego and pride.

With the discovery by Max Planck, already more than 100 years ago, that the mass and energy of the physical universe are meted out in multiples of a quantum unit, it should have been obvious that if the right units of equivalence are used, then Diophantine equations will be needed to describe the combinations of elementary particles that make up physical reality. That, however, has not been the case. Current mainstream science has not yet expanded its a priori assumptions to include quantization of the basic parameters of the physical universe. When we do so, as we have done with the definition of Triadic Rotational Units of Equivalence (TRUE), the true quantum units based on the physical features of the electron, Fermat’s Last Theorem emerges as a key mathematical concept in the process of revealing the nature of reality. Please let me explain:

The general Fermat equation, Xⁿ + Yⁿ = Zⁿ with X, Y, Z, and n equal to positive integers, is a special case of the more general quantum combination expression:

When the basic unit of measurement is defined as the smallest quantum equivalence unit, all cases of this expression are Diophantine equations, and when m = 1 and n = 2, we have the equation X₁ + X₂ = Z, which describes the linear actualization of the closure of integers; for example:

1 + 2 = 3, 2 + 2 = 4, 2 + 3 = 5,…etc.

When m = 2 and n = 2, we have (X₁)² + (X₂)² = Z², for which all integral solutions are quantum actualizations of Pythagorean triples; e.g., 3² + 4² = 5², 5² + 12² = 13², 7² + 24² = 25², etc. I have derived a simple ratio formula for producing the Pythagorean triples. See Appendix A of The Book of Atma, Published in 1977.

When m = 3 and n = 2, we have (X₁)³ + (X₂)³ = Z³, for which, Fermat’s Last Theorem tells us there are no integer solutions. This means that there are no quantum actualizations of this equation, because linear values cubed are volumes, and that is why there are no combinations of two quarks forming a larger particle. Two quarks cannot combine volumetrically to form a symmetrically stable third particle. However, when m = 3 and n = 3, we have: (X₁)³ + (X₂)³ + (X₃)³ = Z³, and we find there are quantum actualizations of this equation. For example:

3³ + 4³ + 5³ = 6³, and 1³ + 6³ + 8³ = 9³, etc. 

This is why quarks combine in threes to produce the symmetrically stable particles known as protons and neutrons. This is just the first example of the importance of Fermat’s Last Theorem in understanding the quantum combinations that form the subatomic particles that make up the elements of the Periodic Table.

I proved Fermat’s Last Theorem in 1964, 327 years after Fermat’s statement, and 30 years before Wiles’ proof. I published the original proof as an appendix to in The Book of Atma in 1977.  I documented the proof and began submitting it to professional mathematicians in 1965. For that reason, I refer to it as FLT65 in my subsequent writings. (See posts on Proof of Fermat’s Last Theorem on this blog and Reality Begins with Consciousness). Since 1965, I have submitted the proof to more than fifty mathematicians, both professional and amateur. Out of the fifty plus, four have accepted it as valid, two professional mathematicians and two capable mathematicians with degrees in sciences that require familiarity with advanced mathematics; but, only one has publicly defended it. Why is this?

It must be said in passing, that the validity of the use of FLT in the application of the Triadic Dimensional Vortical Paradigm (TDVP), developed by Dr. Vernon Neppe and myself in 2011, to quantum physics is completely independent of the validity or invalidity of my 1965 proof, because FLT has long been known to be true for values of n between 3 and 9, the range of TDVP. But, if FLT65 is valid, why hasn’t it been accepted by the community of number theory mathematicians, as has Wiles’ proof? I have published what I see as the reasons, and details of this story in other posts on this website (search the blog archives for Fermat’s Last Theorem) but, my intent in this post is to explain it as briefly and clearly as I can.

The FLT65 proof relies on a very simple basic mathematical theorem known as the Division Algorithm. More specifically, it depends on a corollary of the Division Algorithm that says that one integer, call it A, is a factor of another integer, call that integer B, if, and only if, the remainder when B is divided by A is zero. For example, 9/2 = 4, with a remainder of 1, while 9/3 = 3 + 0. So, 3 is a factor of 9 but 2 is not. The requirement for a zero remainder is patently self-evident for integers, and is it proved in FLT65 for algebraic polynomials consisting of real numbers. This part of FLT65 is never questioned by skeptical reviewers.

In FLT65, the Fermat equation for n equal to a prime number greater than 2 is rewritten as Zⁿ - Yⁿ = Xⁿ and factored into two polynomials, one a first-degree binomial (a polynomial of two terms, consisting of the variable Z plus an integer constant) and the other an (n-1)-degree polynomial with n terms:

Zⁿ – Yⁿ = (Z-Y)( Z^n-1 + Z^n-2Y + Z^n-3Y² +•••+ Y^n-1) = Xⁿ

The factored form of the Fermat equation is chosen so that the (n-1)-degree polynomial must be equal to the nth power of an integer factor of X. This means that the (n-1)-degree polynomial factor must be divisible by that integer, a factor of X. None of this is disputable, and was not disputed by any of the reviewers.

The integer divisor, because its value is unknown, and because the integers are closed with respect to addition, can be represented by the variable Z minus an integer constant. When the (n-1)-degree polynomial factor of the Fermat equation is divided by the integer represented by Z minus a, where a is an integer constant, the remainder is a polynomial comprised of positive integers, and thus cannot equal zero for any integer solution of the Fermat equation. The fact that the remainder cannot equal zero for any integer solution of the Fermat equation means that, by the Division Algorithm corollary cited above, the (n-1)-degree polynomial cannot be divisible by an integer factor of X, which proves FLT for all n.

Is FLT65 a valid proof of FLT, or not? It seems that this should be a question that could be answered decisively, very quickly. But, given the 350-year history of FLT, mathematicians consider the claim of a simple proof an extraordinary claim, and, of course, an extraordinary claim requires extraordinary proof.

The question is: If FLT65 is so simple that given the Division Algorithm and its corollaries, the proof can be described in two pages, in contrast with Wile’s proof of 129 pages, given both Ribet’s theorem and a special case of the modality theorem for elliptical functions, why has FLT65 not been accepted by more than a handful of reviewers?

I have answered this question in the History of FLT65 and other discussions previously posted, but the reasons can be summarized as follows:

First, there is a cultural bias in the community of professional mathematicians against considering the possibility that anyone outside the academic mathematics community might be able to produce a valid proof of FLT. This is an endemic attitude epitomized by: “If I can’t prove it, no one can, and it’s not that important anyway.” (Reminiscent, e.g., of Gauss’ denial, and definitely reflected in Descartes’ attempts to discredit Fermat.)

I developed FLT65 while teaching secondary-school mathematics, shortly after earning my degree in mathematics. The first professional mathematician to whom I submitted FLT65, was a math professor at a Midwestern University, who also happened to be the President of the state’s Academy of Sciences at the time. He returned my proof with a brief note that said: “Your proof is invalid because, if true, it would hold for n = 2.” Of course, the case n = 2 of the Fermat equation is the Pythagorean Theorem equation, with integer solutions called the Pythagorean triples, as noted above. I was stunned. His answer was unbelievable! It revealed that he hadn’t read the proof at all. The case n = 2 was eliminated on the first page!

Most of the university mathematics professors to whom I submitted FLT65, simply ignored it. This is actually quite understandable, because they receive hundreds of half-baked proofs and mathematical ramblings from would-be mathematicians every year. Some number theory professors have form letters they send out in response to such unsolicited proofs, while others just refuse to waste their time reading purported proofs submitted by anyone unknown to them.

Second, mathematicians who tried to disprove FLT65 (there were four) were not willing to go beyond trying to disprove it, probably largely because of the endemic attitude of disbelief cited above, or because of the fear of loosing credibility in the professional community. One of these reviewers thought that the notation used in FLT65 was confusing, and suggested that if the standard notations for variables and constants, integers and rational numbers were used, the error in the reasoning would probably become clear. I rewrote FLT65, changing the notation as appropriate, and found that it made no difference, since for integer solutions of the Fermat equation, the only distinction necessary was between variables and constants, which was already done in the original FLT65, and for integer solutions, both variables and constants are integers by definition.

Finally, my submittals of FLT65 to professional mathematicians between 1965 and 2013, a period of nearly fifty years, were sporadic because of my career. Working as a systems analyst, environmental engineer, and consultant, I was involved in projects that required frequent moves from state to state, across the country, and out of the country for prolonged periods. As a result, some reviewers lost interest, and some. unfortunately, have passed away. Over the years, I have kept a file of all meaningful correspondences and attempts to disprove FLT65.

While FLT65 has failed to get support from any mathematician with much influence in the professional mathematics community, no one has been able to actually refute it. All the attempts to do so have involved one or more of the following approaches: 1) The proposition that the Division Algorithm and its corollaries may not apply to integers. 2) The production of a “counter-example”, a set of three integers which, when substituted into the polynomials on FLT65, appear to contradict the Division Algorithm corollary. 3) The argument that even though specific examples failed to disprove FLT65, they could represent a loophole in the proof.

1)    The proposition that the remainder corollary of the Division Algorithm might not apply to integer solutions of the Fermat equation, was suggested by several reviewers. However, none of them offered a general proof of this. In fact, they couldn’t because the opposite is true: The corollary applies over the field of real numbers, which includes the integers, so it applies to integer polynomials. This is stated in FLT65 and demonstrated in the proof of the Division Algorithm and its corollaries, included as the first part of FLT65.

2)    Because of their belief that FLT65 could not be valid, some reviewers tried to produce counter-examples with integer values that appeared to contradict the remainder corollary. This approach proved to be exceptionally subtle and misleading because one can indeed find integers that, when substituted into the n = 3 Fermat equation’s second-degree (n-1) factor, will produce a value that contains the divisor as an integer factor, even though the remainder is non-zero. It was easy to show, however, that the integers the reviewers chose for such examples were not solutions of the Fermat equation. For that reason, the approach was a form of misdirection. It focused the attention on a demonstration that had nothing to do with FLT. Not only that, if anyone could actually produce a counter-example, it would not only disprove FLT65, it would disprove Wiles' proof as well, because it would produce an integer solution for the Fermat equation.

3)    One reviewer, who appeared to be well-qualified to review FLT65, announced that he had disproved FLT65 with a counter-example. When shown that his example was not relevant to the Fermat equation, he admitted that his “counter-example” did not disprove FLT65, but still maintained that it revealed a loophole in the proof, because if integers could be found that produce a value for the polynomial factor of the Fermat equation that contains the divisor as an integer factor, even though the remainder is non-zero, who is to say there isn’t at least one set of such integers that would actually produce an integer solution to the FLT equation?

Position #3 gave me some pause, until I realized it could only be true if proposition #1 were true, i.e., there would have to be integer polynomials for which the Division Algorithm and its corollaries did not hold. But, the Division Algorithm and its corollaries are proved across the field of real numbers in the first part of FLT65, and integers are real numbers. To see the truth of this clearly for the Fermat equation, one only has to do the following:

Assume there is an integer solution for the Fermat equation for some integer value of n ≥ 3, and substitute the three integers of the solution into the factored Fermat equation, (Z-Y)( Z^n-1 + Z^n-2Y + Z^n-3Y² +•••+ Y^n-1) = Xⁿ. Then the integer polynomial f(Z) =  Z^n-1 + Z^n-2Y + Z^n-3Y² +•••+ Y^n-1 must contain a factor of X. (In fact, it must contain the factor raised to the nth power). And since X and Z are positive integers, Z is larger than X, and integers are closed with respect to addition, there is a positive integer a, such that the factor of X is equal to Z – a. Then the integer polynomial f(Z) divided by Z – a yields a remainder equal to:

a^n-1 + a^n-2Y + a^n-3Y² +•••+ Y^n-1, and since a and Y are positive integers, the remainder is non-zero for all values of a and Y. But, the integer polynomial f(Z) can contain Z–a, if, and only if, the remainder is zero.

If there is any lingering concern that when the integer polynomials, f(Z) and Z–a, are reduced to single integers, A and B, respectively, (as they certainly can be, if there are integer solutions for the Fermat equation, because integers are closed with respect to addition and multiplication), A might still contain B as a factor, it is dispelled by the following demonstration:

There is no question that, if there is an integer solution (X,Y,Z) of the Fermat equation, the equation can be expressed as the integer polynomials displayed above. And, as integer polynomials, Z–a divides f(Z), if and only if the remainder is zero. Therefore, if we set the remainder equal to zero and solve for a, and determine the values of X, Y and Z for each value of a, we will obtain exactly n-1 solutions for the Fermat equation. When we solve for a, however, we find that a cannot be an integer, and therefore, if two of the three X,Y,Z values for any solution are integers, then the third is a non-integer. So, solving for a, produces n-1 non-integer solutions to the Fermat equation, and one additional solution is provided by a = Z which implies X = 0, a legitimate solution of the Fermat equation. This means that we have the n solutions of the Fermat equation, and by the Fundamental Theorem of Algebra (FTA), there are no more solutions.

The Fundamental Theorem of Algebra states that every non-zero, single-variable polynomial of degree n with complex coefficients has exactly n complex roots.

For any integral solution of the Fermat equation, f(Z) is a non-zero, single variable polynomial of degree n, and the coefficients of f(Z) are real numbers, and all real numbers are complex numbers with the imaginary term equal to zero. So, there cannot be more than n solutions to the Fermat equation, and none of them are positive integer solutions with X, Y and Z equal to positive integers. 

Conclusion: All of the legitimate questions raised by reviewers of FLT over the years have been eliminated and resolved. Therefore: 

The FLT65 proof is complete and valid as it was written in 1965.

COMMENTARY AND TRIBUTE TO PIERRE DE FERMAT:

The FLT65 proof contains concepts that would indeed have been available to Fermat, even though they are probably in different form and with different notation than he would have used in 1637. I consider FLT65 to be an elegant proof, because it relies on a deep truth about the fundamental mathematical operation of division, which applies to all real numbers. I have also validated the FLT65 proof in previous written presentations using the logic of infinite descent, Fermat’s favorite method of proof. This means that Fermat definitely could have found his “marvelous” proof. 

To Pierre de Fermat I want to say:

Requiesce in pace, Pierre, tuus lumen mathematicum esse iudicavit!

A NEW LOOK AT FERMAT'S LAST THEOREM

A NEW ANALYSIS OF MY 1965 PROOF

A friend pointed out to me that I spend a lot of time and effort these days defending my 1965 proof of Fermat’s Last Theorem (FLT65) against hypotheses and propositions that are designed to show flaws in FLT65’s logic. My friend suggested that instead of being on the defensive, perhaps it is time for me to take a different approach. The only legitimate questions raised by critics seem to involve, in one way or another, questioning the legitimacy of the application of the division algorithm and corollary III to integers in FLT65. So let’s have a look at how the division algorithm is applied in FLT65.

Division is one of the four fundamental operations of mathematics, and the division algorithm describes the operation of division for polynomials (algebraic summations with multiple terms). The following are examples of algebraic polynomials:

f(z) = z³ – a³; g(y) = y² + by + c; h(x) = x⁴ – 2x² + 5x + 13; and q(X) = 12X

A general expression representing an algebraic polynomial in x, of degree n, is given by;

f(x) = axⁿ + bx^n-1 + cx^n-2 + …+ sx + k, where there can be any number of terms, and a, b, c. …s, and k are constants that can be either positive, negative or zero.

The first part of FLT65 shows that for any polynomial f(x) over the field of real numbers, there exist unique polynomials, q(x) and g(x), such that f(x)/g(x) = q(x) + r(x)/g(x). This is nothing more than a statement of the division algorithm for polynomials. I also provided proof of the three corollaries of the division algorithm used in the FLT65 proof, including corollary I, which says that if f(x) and g(x) contain a common factor, r(x) contains it also; corollary II, which says that the remainder, when f(x) is divided by z-a is f(a); and corollary III that says that a polynomial f(x) of degree greater than 1, is divisible by the polynomial x-a, if and only if, f(a)=0. FLT65 also contains proof that if FLT is true for n = p, when p is a prime number greater than 2, then it is true for all n.

In FLT65, if x, y and z are integers, then, for a comprehensive proof, it is sufficient that they are relatively prime integers. For FLT65, y is chosen as a variable that does not contain p, and f(z) is defined as the larger factor of the right-hand side of the equation below, the Fermat equation:

y^p = z^p – x^p = (z-x)(z^p-1 + z^p-2x + z^p-3x² +•••+ x^p-1)

It is shown that z–x, and f(z), as factors of y^p, can be considered to be relatively prime. Thus f(z) = A^p, and A is an integer factor of y.

It is noted in FLT65 that the division algorithm and corollaries hold when the terms and coefficients of the polynomials are integers because the integers are elements of the field of real numbers. Also note that there are no restrictions on the application of the division algorithm and its corollaries with regard to the values of the coefficients, or the number of terms in the polynomial to which they are applied. This means that the statement of the division algorithm, f(x)/g(x) = q(x) + r(x)/g(x), is true whether f(x) and g(x) are polynomials of many terms or single terms.

Applying the algorithm and corollaries to the polynomial f(z), and defining g(z) as z-a, we have:

(z^p-1 + z^p-2x + z^p-3x² +•••+ x^p-1)/(z-a) = q(z) + f(a)/(z-a). Multiplying through by z-a, we have:

(z^p-1 + z^p-2x + z^p-3x² +•••+ x^p-1) = q(z)(z-a) + f(a).

From this it is clear that the polynomial f(z) is factorable into the two polynomial factors q(z) and z-a, if f(a) = 0.

We also know that, if there is an integer solution to the equation z^p – x^p = y^p, then y^p is an integer containing the integer factor f(Z) = A^p. = (Z-a)^p. This gives us two equations:

1)         (z^p-1 + z^p-2x + z^p-3x² +•••+ x^p-1) = q(z)(z-a) + f(a), an algebraic polynomial equation and

2)         (Z^p-1 + Z^p-2X + Z^p-3X² +•••+ X^p-1) = q(Z)(Z-a) + f(a), the same equation with all integer terms. Equation #2 exists if there is an integer solution (X,Y,Z) for z^p – x^p = y^p.

For an integer solution, if one exists, equation #2 can be reduced to an equation of single integer terms:

3)         A^p = Q·A + R, which implies that the integer f(Z) = A^p is equal to the product of the two factors Q and A, if and only if R = 0.

Critics have suggested that there is no correspondence between the polynomial factors of equation #1 and the integer factors of equation #3, in which case, f(a) ≠0 does not necessarily imply that R ≠0, and the fact that the polynomial f(z) cannot contain the polynomial z-a as a factor, has no bearing on whether the integer that f(Z) reduces to can contain the integer value that (Z-a) reduces to, or not.

So the real question whose answer will resolve the disagreement about the validity of FLT65 is:

What is the nature of the relationship between the polynomial factors of equation #1 and the integer factors of equation #3?

In FLT65, Z-a is defined as A, a factor of Y^p, and f(Z) = A^p, and A is an integer if there is an integer solution. By inspection of equations #2 & 3 above, we see that there is a one-to-one relationship between the four expressions of the two equations. By direct substitution of single-integer values for X, Z and a, assuming an integer solution of the FLT equation, for any prime degree, p, the polynomial factor (Z^p-1 + Z^p-2X + Z^p-3X² +•••+ X^p-1) yields A^p, the polynomial factor q(Z) yields Q, (Z-a) yields A, and f(a) yields R. This means that R = f(a) ≠ 0, and we have a contradiction:

For there to be an integer solution to the FLT equation, Y^p =A^pB^p =f(Z)B^p → f(Z) =A^p and this in light of equation #3 implies R has to be zero. But because of the form of f(z), with all coefficients equal to unity except the constant term, which is zero, f(a), which yields R, cannot be zero. And thus the fact that f(a) ≠ 0 for equation #1, implies that R ≠ 0 in equation #3, and this contradiction, featured in FLT65, is sufficient to prove there are no integer solutions for the FLT equation.

Edward R. Close August 3, 2017

A NOTE ON MY 1965 PROOF OF FERMT'S LAST THEOREM

FERMAT'S LAST THEOREM PROVED 30 YEARS BEFORE SIR ANDREW WILES' PROOF

Over the years I have submitted my proof to more than 50 professional mathematicians. The mathematicians who have  rejected my 1965 proof have done so primarily because of the belief that there cannot be a proof of FLT using simple mathematical concepts.The few who have actually tried to refute FLT65, have attempted to support this belief with the fact that the division algorithm may or may not apply to integer constants obtained by substituting specific integer values into integer polynomials and reducing them to single integers. But there is no proof that it is true for the integer polynomials of the Fermat equation, and so three of them have resorted to demonstrations that have no relevance to actual solutions of the Fermat equation, to try to make the point that FLT65 may not be valid.

Their idea that the division algorithm might not apply to the integer polynomials of the Fermat equation factor f(Z) = Z^p-1 + Z^p-2X + Z^p-3X² + ••• + X^p-1 arises from the fact that, for given integer values of X and Z, f(Z) can be reduced to a single integer (a constant), and if that single integer is not prime, in general, one of its integer factors may or may not contain the integer equal to the integer value of Z₁ – a.

FLT65 provides a way to determine whether or not any specific single integer value of f(Z) (a polynomial factor of the Fermat equation) can contain the specific single integer value of Z –a (a polynomial factor of Y in the Fermat equation) as a factor, using the division algorithm and its three corollaries. For an integer solution of the Fermat equation, f(Z₁) must not only contain Z₁ – a, it must be equal to (Z₁ – a)^p.

The division algorithm and its corollaries, by definition, apply to all polynomials with real number variables, so they apply to polynomials of integer variables in the same way they apply to all polynomials of real numbers because integers are real numbers which, along with non-integer rational and irrational numbers, comprise the field of real numbers. Finally, an integer solution of the Fermat equation, if there is one, is simply one of the infinite number of solutions to one of the Fermat equations, and the three numbers of any solution are a set of three numbers existing in the field of real numbers. FLT65 demonstrates the fact that for the Fermat integer polynomials f(Z) and Z – a, where both polynomials must be factors of Y^p, there are no integer values of a, X and Z for which Z – a divides f(Z), because the remainder will always be non-zero.

After more than 40 years, I still have hope that more mainstream mathematicians will join the small, but growing number of mathematicians who agree that there are no fatal flaws in the logic of FLT65.

Edward R. Close, June 26, 2017

For those who are not familiar with Fermat's Last Theorem, I've pasted in a previous discussion and some relevant links below.

The Basic FLT65 Proof

The following steps summarize the logic and mathematics of FLT65. For brevity, I will not present proofs of the steps here because they are so easily proved that they can be proved by a bright high school algebra student. If these steps aren’t obvious go to http://www.erclosetphysics.com/search?q=Fermat%27s+Last+Theorem+Part+1.

STEP #1: The first step in FLT65 was to provide a rigorous proof of the division algorithm and its three corollaries. The reason I provided this proof first, even though it was well known to mathematicians, was to show that it applies to all polynomials across the field of real numbers, including integers, and to highlight the fact that the uniqueness of the dividend and remainder allows the all-inclusive “if and only if” of Corollary III. These points were pointed out in FLT65.

STEP #2: If there is an integer solution for Fermat’s equation: xⁿ + yⁿ = zⁿ, to prove or disprove it, we need only consider n as prime numbers, p >2, and x, y, and z as relatively prime positive integers. Proof of this is included in FLT65 allowing us to proceed to Step 3.

STEP #3: Fermat’s equation can be rewritten as z^p – x^p = y^p, and since all prime numbers >2 are odd, factored as follows: z^p – x^p = (z-x)( z^p-1 + z^p-2x + z^p-3x² +•••+ x^p-1) = y^p

Similarly, z^p – y^p = (z-y)( z^p-1 + z^p-2y + z^p-3y² +•••+ y^p-1) = x^p.

For the next step, and throughout this discussion, keep in mind that we have assumed that there are integer solutions to Fermat’s equation, so the approach is to determine whether this assumption leads to a contradiction. If it leads to a contradiction, FLT is proved.

STEP #4: It is easy to show by simple algebraic division that the only common factor that may be shared between the factors of the Fermat equation is the integer p, and since x, y, and z are relatively prime integers, if either x or y contains p as a factor, the other cannot. See the proofs of this in the original proof in the link above. So we can let y represent the one that does not contain p. It then follows that the two factors of the left hand side of the first equation of step 3 are relatively prime and thus are perfect p-powers of integers. Thus, by inspection of

(z-x)(z^p-1 + z^p-2x + z^p-3x² +•••+ x^p-1) = y^p, we see that we can write

(z-x)= B^p, and (z^p-1 + z^p-2x + z^p-3x² +•••+ x^p-1) = A^p, where A and B are positive integers.

STEP #5: If there is an integer solution, then x and y are specific integers X₁ and Y₁, and the p-1 polynomial in z, f(z)=(z^p-1 + z^p-2X₁ + z^p-3X₁² +•••+ X₁ ^p-1) = A^p, and B^pA^p = Y₁^p. That is, the two factors must be perfect p-powers, integers raised to the pth power.

STEP #6: In a positive integer solution, z >Y₁ >A, and by closure of integers, there is a positive integer a, such that A= (z – a), and by corollary II of the division algorithm, when f(z)=(z^p-1+ z^p-2X₁ + z^p-3X₁² +•••+ X₁ ^p-1) is divided by Z – a, the remainder is equal to f(a) = a^p-1+ a^p-2X₁ + a^p-3X₁² +•••+ X₁ ^p-1.

STEP #7: Corollary III of the division algorithm says that f(z) is divisible by z –a if, and only if, f(a) = 0. But, since f(a) is the sum of p positive integers, it can never equal zero. Thus by assuming there is an integer solution of z^p – x^p = y^p, we have produced a contradiction proving Fermat’s Last Theorem.

Discussion

Note that the case n = 4 is not addressed in this proof. It was overlooked in FLT65, but this was not a problem because there were several known proofs for n = 4, including one by Fermat himself.

So FLT65 is effectively a complete and valid proof of FLT; but approximately 90% of the mathematicians to whom the proof was submitted over the years did not respond at all. This is because, before Sir Andrew Wiles’ proof was accepted, professional mathematicians received hundreds of supposed proofs of Fermat’s last theorem per year.

If you’ve ever taught mathematics and had to evaluate proofs developed by students, you know it can often be very challenging and time consuming, and attempts at proofs by amateur mathematicians are usually filled with all kinds of errors. In addition, because in more than 300 years, so many first-rate mathematicians had tried to prove or disprove FLT and failed, most mathematicians consider reviewing such ‘proofs’ a waste of time.

I’m sure that this was the reason the first mathematician to whom I sent it rejected it. The reason he gave, however, was that, if FLT65 were true, it would also apply to the case n = 2. [When n = 2, we have z² – x² = y², which does have integer solutions known as the Pythagorean triples, e.g. 3,4,5]. Of course by giving this reason for rejecting FLT65, he revealed the fact that he hadn’t read it, because it is clear to anyone with basic math skills reading the first page that the method of proof of FLT65 doesn’t apply to the case n = 2.

Of the 10% who did respond, most gave the opinion that there had to be a mistake somewhere, but failed to point one out, or provide any mathematical argument supporting their opinion. Of the remaining recipients of FLT65, only a few provided any sort of mathematical demonstration supporting their opinions. Those responses are presented in the article accessed by the links provided above. Those arguments were all easily refuted. However, one of those demonstrations, actually offered with different numerical values by three reviewers, is worth mentioning here because it is a classic example of inadvertent misdirection, and it also shows how tricky a proof of FLT can be.

The argument they put forth was that the division algorithm and corollaries certainly apply to algebraic polynomials, but they may not necessarily apply to the integers obtained when, for specific integer values of z, a and X₁, the algebraic polynomials z-a, f(z) and f(a) are reduced to single integer values. This is an interesting conjecture, but none of the reviewers attempted to prove or disprove it, instead they offered what they thought were counterexamples to FLT65 for n = 3. They selected integer values of z, a, and X₁ that, when substituted into f(z) and f(a), produced an integer value for f(z) that contained the integer z-a as a factor, even though f(a) did not equal zero, appearing to violate corollary III of the division algorithm.

It is worth contemplating this argument a little more deeply for a moment, because by doing so, we expose the fact that such a demonstration is not actually a counterexample, but is in fact, an inadvertent misdirection, shifting attention away from the fact that z must be part of an integer solution to the Fermat equation. It is not hard to find positive integer values for z, a, and X₁ such that f(z) is divisible by z-a, and of course f(a)= a^p-1+ a^p-2X₁ + a^p-3X₁² +•••+ X₁ ^p-1 will still be non-zero because all the terms are positive integers. But because the values of z and a selected have no relation to the Fermat equation, these demonstrations have no bearing on the logic of FLT65.

FLT65 started with the assumption that there is an integer solution for the Fermat equation. This means that for a numerical example to be relevant, z must be part of an integer solution of Fermat’s equation. The issue is not whether you can find integer values for z, a, and X₁ that will make f(z) divisible by z-a; the relevant point here is that, if there is an integer solution, the value of z must satisfy Fermat’s equation. Then, because f(z) and z-a are both polynomials in z, the algorithm and corollaries apply, and the remainder must equal zero for f(z) to equal a perfect p-power, A^p, if the assumption of an integer solution is true. But, of course for Fermat’s equation, f(a) cannot equal zero. -- End of story!

I think these reviewers were so intent on trying to find a way to disprove FLT65, which they were convinced from the beginning could not be valid, that they were blinded to the fact that, if their ‘counterexamples’ were valid, they would actually have provided integer solutions for z^p – x^p = y^p, directly disproving FLT, and thereby also disproving Andrew Wiles’ proof.

So after fifty years, FLT65 still has not been refuted. Those who tried have failed, but only a few besides myself have accepted it as valid, and two of them have since passed away. Many of the mathematicians who have reviewed it believe it cannot be valid, and two even claimed to have refuted it, but their arguments were easily disproved. See the details in the links provided above. 

After fifty years, I would like to have closure; so anyone out there who believes the proof is faulty or incomplete is challenged to provide irrefutable mathematical proof that I can understand supporting that belief. If you can prove to me that FLT65 is wrong, I will acknowledge you proof and send you a check for $100.

Unlike Sir Andrew Wiles’ proof of FLT, which is hundreds of pages long, drawing on a very sophisticated knowledge and understanding of elliptic functions and modular algebra, FLT65 is a relatively simple proof relying only on basic mathematical principles. I believe that Pierre de Fermat will rest easier when FLT65 is recognized as valid, because it proves that he could have proved his famous theorem with mathematics available in 1637. If FLT65 is is at last recognized as correct, I, and poor Fermat will have closure.