Fermat’s Last Theorem
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:
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.
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)( Zn-1 + Zn-2Y + Zn-3Y2 +•••+ Yn-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)( Zn-1 + Zn-2Y + Zn-3Y2 +•••+ Yn-1) = Xn. Then the integer polynomial f(Z) = Zn-1 + Zn-2Y + Zn-3Y2 +•••+ Yn-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:
an-1 + an-2Y + an-3Y2 +•••+ Yn-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:
Requiesce in pace, Pierre, tuus lumen mathematicum esse iudicavit!
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:
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