Sunday, January 21, 2018



Be grateful, and honor Mother Earth,
The special place that gave you birth,
Where meeting of the Yin and Yang,
With Earth and Water, Fire and Stone
Aligned so that the heavens sang,
And brought you forth in blood and bone:
To become the soul you are today.

Align once more with joy and mirth,
For you are all of Heaven’s worth,
Brought forth with tears of joy and pain,
Into a life, where you, and you alone,
Must portray a vital role, and sing again,
Your unique, angelic voice to hone,
To let God know you passed this way.

The purpose is to wake the soul
From its hypnotic dream,
To expand, and become the Whole,
To know Infinity Supreme.
So bless that little spot of Earth,
That gave to you this universe
 In which to grow and sing your song.

Align yourself, that you may blend,
From the beginning until the end,
Into the measured symphony,
Becoming the joy and ecstasy
That you are truly meant to be;
Before you hear the final gong.

Edward R. Close, January 21, 2018

Saturday, January 20, 2018



I’m currently reading Dr. Gary E. Schwartz’s new book SUPER SYNCHRONICITY, Where Science and Spirit Meet, a book which is, in my opinion, a very important contribution to human thought, science and our understanding of reality in general. In Chapter 5, Dr. Schwartz talks about the synchronous occurrences involving ravens in the life of Dr. Diane Hennacy Powell, and in his life just before and after he met Dr. Powell. The story is an excellent example of how what seems like a unique random occurrence can trigger sequences of seemingly unrelated events that have one thing in common …in this case, ravens.

At the first meeting of the Academy for the Advancement of Postmaterialist Sciences, at Canyon Ranch near Tucson, Arizona in August 2017, before I had read Super Synchronicities, I was seated next to Dr. Powell. She and I had never met before.  It turns out that Dr. Powell and I are both alumni of Johns Hopkins University in Baltimore Maryland. At this meeting, including the founders of the Academy, there were 12 members seated in a circle. As we went around the circle, introducing ourselves and expressing our interest in the hypothesis that consciousness may be fundamental, or even primary in workings of the universe, the raven connection, detailed in Super Synchronicity, was mentioned. It struck a resonating chord with me immediately.

In February and March of  2010, I was part of an expedition to the Middle East to film a documentary about the Frankincense Trail, an ancient caravan trail that stretches from what is present-day Oman, at the southern end of the Arabian Peninsula, to the Holy Land, Syria and Egypt. We filmed in Egypt and Jordan. While filming in the desert south of Cairo in Egypt, we were able to have the Great Pyramid on the Giza Plateau to ourselves for a few hours one day. After visiting the Sphinx, we walked up the hill to the great Pyramid. And, after taking a few pictures with friends in the expedition, I looked up at the great Pyramid. To my surprise, there were three large black birds circling the apex of the pyramid. They were ravens.

This may not have been too strange, but I had never seen ravens, or any other birds circling like that in any pictures, videos or films of the pyramids, and a very unusual experience that occurred in the “Queen’s Chamber” a short time later, convinced me that seeing those birds was no accident. Were they a warning? Or an important portent? You decide.

Shortly after a group of twenty of us had entered the Queen’s Chamber, the lights were turned off. We were told ahead of time that this would be happen. It was done, they said, so that we could experience the complete darkness in the heart of the Great Pyramid. As an experienced spelunker (I had explored hundreds of caves) I knew very well what absolute Darkness was like. Someone began chanting Om, and soon, it sounded as if everyone had joined in. The sound swelled and reverberated sharply from the bare stone walls of the Queen’s Chamber. It was very pleasant. But then, a few seconds before the lights were turned back on, something happened that I will never forget: 

A spinning ball of light came out of the southeast corner of the chamber and struck me on the forehead. I immediately lost all sense of balance, and began to feel the turning of the earth on its axis, the orbiting of the earth around the sun, and a third rotation: that of the galaxy whirling in the blackness of space, and then, sick with extreme vertigo, I lost consciousness”.

Two men who happened to be on either side of me, seeing that I was falling, grabbed me by the arms, preventing me from falling headlong onto the stone floor, and helped me to the entrance of the passageway between the Queen’s Chamber and the Grand Gallery, and one of them went for help. My body was carried out of the Pyramid by our Egyptian guide Fergany, and one of his assistants. When we reached the juncture between the low passageway into the Queen’s Chamber and the Grand Hall leading up to the :”King’s Chamber, we were met by Josef, the youngest son (I think about 7 or 8 years old at the time) of Gary and Mary Young, the organizers of the expedition, who placed his hand on my chest and prayed for my recovery. I threw up, and passed out again,

I was in and out of bodily awareness several times before they deposited my body on the stone just outside the entrance known as the “Robber’s Tunnel” created by force around 820 AD. What happened between periods of sickened consciousness was remarkable: One time, I seemed to be out of my body, far above the Great Pyramid, as if I were seeing it from a point near the peak, hovering in mid-air. But, the bus parking lot had disappeared, and there were trees and green grass all around, and the pyramid was blazing white with a gold capstone! Another time, I was seeing the faces of many people, coming and going, speaking to me in a mixture of languages, interspersed with lights and symbols of many kinds. And, at one point, there were bird symbols, and I remembered seeing the ravens.

Most of the members of the expedition, with the possible exception of the filming crew and the Egyptian guides, were practitioners of traditional and alternative health care modalities, and several of them came to my aid, for which I am eternally grateful. A doctor and a nurse accompanied me back to the hotel, were I had to be transported to my room on a golf cart, because I still could not walk.

The next day, we flew to Amman, Jordan and then traveled by bus to the modern city of Petra, located at the entrance to the narrow canyon leading into the ancient city of Petra, about 150 miles south of Amman. I was able to walk, but each time I closed my eyes, I would experience vertigo and begin to see more faces and hear voices. Fearing that I would become violently ill again, I ate nothing on the trip and struggled all day to keep my eyes open. I told no one what I was experiencing.

One of the men who kept me from falling in the Pyramid, sought me out in Petra and asked me if I had seen the flash of light that occurred just before I passed out. I had told no one about the experience of the light, or anything else, except to explain that I definitely had not suffered from claustrophobia or heat exhaustion, which was what most of the people in the group theorized. I had felt fine before and after entering the Pyramid, and all of the time up until I suddenly passed out. Later, another member of the group also mentioned seeing the ball of light hit me. So, I knew that the experience was not totally subjective.

As a scientist, I had to ask: What was the meaning of this strange experience? There were aspects of the experience that were very hard to explain or understand. Why had this happened to me? Had I traveled back in time several thousand years, and seen the Pyramid as it existed then? Was I seeing the faces and hearing the voices of the people who built the Pyramid? Did anyone else have such an experience?

I did a survey of the expedition members after we got home, and found that while a few had unusual experiences, no one else reported experiencing anything similar to what I went through. While I had been excited about going into the Pyramid, I certainly had not expected anything like that, and would not want to experience it again, because it was very unpleasant. Was it just a freak discharge of static electricity, and I just happened to be in the wrong place at the wrong time? Was it a meaningless random event?

As a physicist and mathematician, I have long known that there is no such thing as a random event. We exist in a mathematically ordered universe, and an event that seems to be random only appears so because we are not seeing the whole picture. Statistical analysis and probability theory are designed to enable us to handle things that appear to be random, in a logical manner, but scientifically, they are no more than theories, and theories that are built on very shaky grounds at that!

But what about things like flipping coins, opening a book without thinking about it and reading a random passage? No, sorry, not random. Just because you are not thinking about what you are doing, doesn’t mean the universe isn’t. What about random number generators? A sequence of numbers produced by a program is not random, because the program was created by someone thinking about how to avoid the value of number n+1 being related to number n or n+2.

What about numbers generated by a natural process like radioactive decay? No, they’re not truly random either, they are effects of a complex process that we can’t readily predict. What about the Heisenberg uncertainty principle? Is there true randomness at the quantum level? With the discovery of gimmel, the third form of reality, we can answer no, that does not introduce true randomness either.

The current dictionary definition of a random number generator, found online, is instructive: 

“A random number generator is a computational or physical device designed to generate a sequence of numbers or symbols that cannot be reasonably predicted better than by a random chance.

Notice two things: 1) First, a  definition that uses a word that is in what it is supposed to be defining is no definition. 2) Something that is “designed” to generate randomness, cannot produce a truly random sequence. The sequence is non-random by virtue of the fact that it was produced by design.

My grandfather used to say: “What is to be will be, if it never comes to pass”! When I was about 9 years old, I asked him what it meant. He said:

“Well, boy, I reckon nothing ever happens by accident. After something happens, however unlikely it might have seemed before it happened, you can always see why it happened after it happens.”

This gave me something to think about for days. But, after the experience in the Pyramid, I eventually came to understand that time is not what we think it is. When identified with a physical body, we have only the moment at hand. The past exists for us only as memories stored in the brain, and the future exists only as imagination. We only have direct access to the present. But this is because the body, with its very limited physical senses, is a super reduction valve. 

When freed of the bodily limitations, and awake in pure consciousness, we can experience past, present and future, all at the same time. 

Do we have free will? Yes, but not as much as we sometimes think we do.

Did I travel back in time to the time of the Pharaohs? No. The nature of time is such that no traveling is necessary; it is all present in the multi-dimensional eternal now.

Thursday, January 18, 2018


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:

"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 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)( 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: 

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


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!

Thursday, January 11, 2018



©Edward R. Close January 11. 2018

The year was 1685. Traveling across Germany in a Pferdekutsche (stagecoach), Gottfried Wilhelm Leibniz and a traveling companion stopped at an inn for a short rest and a drink. Leibniz was already famous. About as famous in Germany as Einstein was in the US in the 1950s. He was a polymath: a medical doctor, mathematician, linguist, philosopher, scientist; he had invented something called “the calculus”, and he had produced the world’s first mechanical calculator that could perform multiplications and divisions. When he identified himself, the people in the inn laughed. They could not believe that this mud-spattered, ordinary-looking man could be the most famous man in Europe!

“So, then, Mr. Philosopher, what is the most important question a human being can ask?” One of the scoffers queried.

Leibniz frowned, cleared his throat, and said: “The first question a man of science should ask, is ‘Why is there something rather than nothing?’”.

Everyone laughed, and returned to their bier steins, shaking their heads, not realizing that what they had just dismissed as nonsense was, in fact, the ultimate question.

Why is there something rather than nothing? Modern science’s answer is “the Big Bang”. But that is actually no answer at all. The idea of something exploding into existence from nothing, is nonsense, and you don’t have to be a scientist to know that it is nonsense.

To find the answer, we must go back another 1,435 years, to the time of Diophantus of Alexandria. Diophantus was a Greek mathematician who lived in the third century, and died at the age of 84, sometime between 285 and 299 AD. How do we know how old he was when he died? He left a mathematical puzzle on his grave stone, which, when solved, reveals his age.

Inscribed on his tombstone was the following poem (translated from the Greek):
Here lies Diophantus. Now this wonder behold,
Through art algebraic, the stone tells how old:

God gave him his boyhood for one-sixth of his life,
One twelfth more as a youth, his whiskers grew rife;

After another one-seventh, had marriage begun;
And five years later, came a bouncing new son.

Cruel Fate took the child of this master and sage
When he was only one-half his father's final age.

Consoling four years with the science of numbers,
The sage ended his life, and in this grave slumbers.

A high school algebra student today can solve this puzzle by writing and solving two equations in two unknowns. Hint: Let X = Diophantus’ age at the time of his death, Y = the son’s age at the time of his death, and use the clues given in the poem to construct simultaneous equations.

Diophantus specialized in finding whole number (integer) solutions for algebraic equations. As a result, equations for which there are integer solutions are now called Diophantine equations.

What does this have to do with Leibniz’s “ultimate” question? To answer this question within the question, we have to refer to the mathematical work of a man who was nearly 40 years old when Leibniz was born, and died when Leibniz was only 19.

In 1637, an obscure French jurist, an amateur mathematician named Pierre de Fermat, had already answered Leibniz’s ultimate question, but he probably didn’t know it! In the margin of a page in a book on Diophantine equations he wrote: “I have found a marvelous proof, but the margin of this book is to small to contain it.”

Fermat’s proof of what became known as “Fermat’s Last Theorem”, was never found, and the renowned French thinker of the day, Rene Descartes, called Fermat “an uneducated trouble-maker” and tried to discredit him as an amateur. As it turned out, Fermat was at least as good a mathematician as Descartes. You may recall Descartes’ statement: “I think; therefore, I am!” But, of course, this is no more an answer to the question of existence than the Big Bang answer, because it leads to the deeper questions of what is ‘thinking’ and what is ‘being’.

Fermat’s proof ultimately depends upon the logical method of infinite descent. If something is assumed to exist on a large scale, and can be reduced by logical steps to a version of the same thing, but on a smaller scale, then it can be reduced again and again, until you arrive at a version of the same thing at the smallest possible scale. Then it can be easily proved to be either true or false, existing, or not existing.

Notice that the Big Bang theory is a form of infinite descent. The idea that the universe is began as the result of an explosion from a dimensionless point 13.8 billion years ago is obtained by an infinite descent in space and time, starting with something, i.e., what we have now, descending to nothing. But, that makes no sense. So, maybe it was a very dense mass that existed before the big bang? But, if so, where did that come from? What existed before it exploded? Something or nothing? If it was something, then the big bang was not the beginning of everything, only the beginning of the expanding universe we have today. If it was nothing, then we have a paradox with the production of something from nothing. If your answer is “God created it”. Then God existed before the big bang, and you can’t say there was nothing before the big bang.

Is there an answer? Yes, there is an answer, and Pierre de Fermat discovered the key to it while studying Diophantus’ equations. But, as I said, Fermat himself, probably did not realize that he had the answer to the ultimate question. That realization had to wait for more than 300 years.

In 1900, Max Planck discovered that the matter and energy of the physical universe exist only in multiples of a very, very small unit. This means that in the physical universe, infinite descent to zero is impossible. Descent stops at the smallest quantum. This is, after all, what Fermat’s infinite descent is about. It stops with an integral form, not zero. The infinite part of the name infinite descent is used because it can start with any assumed object, however large, and then descends to its smallest possible integral form.

In 1986, I developed the calculus of distinctions, derived from George Spencer Brown’s calculus of indications, adapted it for use in a quantum reality, and applied it to the big-bang expanding universe theory. I published the results in a book titled Infinite Continuity in 1990.

Here are two excerpts from Infinite Continuity:

From the first paragraph of the Preface:
“The digital clock beside my bed read ‘1:11’. It was one-eleven am, January 16th, 1986, and I had just awakened from a very vivid dream. A stone had dropped into a clear, dark pool, filling it with golden ripples of spreading light, and at the center of my awareness the pieces of an intricate puzzle had suddenly fallen together. Grabbing a pencil and paper, I began writing as rapidly as I could. So vivid were the details, so clear the understanding, that the heart of the vision flowed out onto the paper that night.”

From the last paragraph of Part V, Summary and Conclusions:
“We stand again on the threshold of a new scientific frontier. As science recognizes the connectedness of all things, the need to integrate the knowledge we have gained becomes more urgent. The formalization of a new science is required. This new science will encompass the existing paradigm, and yet transcend it by including the relationship of [Primary] Consciousness to matter and energy, and the relationship of individual consciousness to existential reality.”

The calculus of distinctions applied to quantum reality became known as the calculus of dimensional distinctions and one application developed in collaboration with Vladimir Brandin, was published in Elements of Mathematical Theory of Intellect, Moscow Interphysics Laboratory, Moscow, Russia, 2003.

Finally, in 2011, in collaboration with Vernon M. Neppe, MD, PhD, internationally renowned neuroscientist, I completed the proof of the existence of a third form of reality, which Vernon named gimmel, not measurable as mass or energy, and therefore non-physical. With that, the proof was complete, and the answer to Leibniz’s question was finally clear: There is something rather than nothing because there is no such thing as nothing.

This also validates Einstein’s statement in the final appendix of his book on relativity, written only a few years before his death, where he suggested that there is no such thing as empty space.

This answer bears emphasizing:

Fermat’s method of infinite descent applied to quantum cosmology proves that there never was nothing, there is no absolute beginning or end, only changes in form. Nothingness is a fiction, a meaningless concept conjured up by confusing changes of form with beginnings and ends.