Monday, October 16, 2017



It is common these days to hear people absolutely blasting those with opposing political views, accusing them of dark motives and calling them derogatory names. Both sides are doing it. 

A word of caution: Be very careful. There’s a subtle difference between logical evaluation and judgement, and while you may think you are evaluating an issue, it’s easy to slip over the edge into judgement of other people, in which case you risk becoming subject to judgement yourself. One way to check this tendency is to ask yourself: Am I acting or reacting?

Action is a positive response to something or someone you oppose, while reaction is always negative, leading to further divisions and alienation. So, if you are offended by something someone says or does, take a deep breath and instead of calling that person names and hurling insults, present your opposing view in positive terms. You’ll be surprised how often the other person will back down, and at least partially agree with you. It is better to agree to disagree than to build walls of insults that divide you forever from someone who has a different opinion.

Your view of the world is strongly affected by your  world view, which is based on your personal belief system, which may be partially right and partially wrong. There is nothing wrong with reality. Reality functions according to the logic of the Laws of Nature. If you are religious, know that even though it may sometimes not seem so, God is always in control. So, if the world seems wrong, know that it is your attitude that needs to change, not the world. You can do only very little to change the way others think, while you do have a lot of leeway to change the way you think of others. 

Evaluate controversial issues carefully, but do not judge others if they see things differently. If necessary, take positive action to express your view, but do not react in anger and condemn others. If you do so, then you are most probably part of the problem.


Sunday, October 8, 2017


As my Face Book Friends know, yesterday I celebrated my 9x9 = 81st birthday. Thank you again for all the wonderful birthday wishes, comments and blessings. I wish to send Love and Light to each and every one of you, and please know that I am going to be around for a while longer, I still have a lot to do.

Would you believe that someone asked me a few years ago: “How long have you been retired?” My response was: “Retirement is not a meaningful word in my vocabulary.” What on Earth makes someone think I’m retired? People retire when they are tired of what they’ve been doing, and/or are getting ready to die!

Concerning death, the cerebral comedian Woody Allen said:

I’m not afraid to die, I just don’t want to be there when it happens!

I believe we are on this Earth to learn that life and death are passing dreams from which we all must one day waken, and for most of us, it takes a while. I worked as an actuarial mathematician, writing computer programs for the Univac computer for a major insurance company in downtown Los Angeles 57 years ago and saw statistics that showed that most men (I think it was about 87%) died within 6 months after retirement. I decided right then, never to retire.

How long do I plan to live? Let me answer that by quoting British biologist Thomas Huxley:

The rung of a ladder was never meant to rest upon, but only to hold a man's foot long enough to enable him to put the other somewhat higher.”

There is still so much to learn. I believe when one stops learning, one starts dying.

Try to learn something about everything and everything about something.”
- Thomas Huxley again.

I’ll go a bit further than that: Pick out something you love, it could be anything; I believe that if you really try to learn everything there is to know about something, anything real, and get even close, you’ll know a lot about everything else.

Most of you know about my efforts to get mainstream science out of the dead end of gross materialism. I’ve just finished writing a chapter for a book being published by the Academy for the Advancement of Post-Materialist Science that proves that the reconciliation of relativity and quantum physics in the new paradigm Dr. Vernon Neppe and I have developed, eliminates materialism as a valid metaphysical basis for science. Mainstream scientists who are self-acclaimed atheists hate this because it threatens their world view and overturns their life’s work. Some have resorted to calling us names.

No real scientist can possibly be an atheist. Because atheism does not meet the necessary criteria to become a scientific hypothesis. A scientific hypothesis must be “falsifiable”, i.e. it must be testable, and capable of proof or disproof. The hypothesis that God does not exist cannot be proved. On the other hand, the reality that nothing would exist without the organizing action of a higher form of consciousness is provable, - by direct experience. But mainstream scientific egos think they are authorities on the subject, declaring that because they haven’t experienced anything greater than their own egos, no one has!

 Quoting Huxley again:
Every great advance in natural knowledge has involved the absolute rejection of authority.”

Interestingly, Huxley, the grandfather of Aldous Huxley, the author of Beyond the Doors of Perception, defined himself as an agnostic. And that’s fine, every scientist should be an agnostic, especially about his own field. An agnostic is a skeptic, a doubter, a “doubting Thomas”. Concerning God, about all an agnostic can say is that God, if he exists, hasn’t appeared in front of him, or spoken to him. Of course, that is probably because most scientists don’t know how to stop thinking long enough to hear Him!

I have learned to live as if something wonderful is about to happen, - because it does, and has, repeatedly! Every day I see the sun rise is a wonder. The fact that there is something instead of nothing is an on-going miracle. Your existence as a conscious being is a miracle beyond miracles.

My plan, when the time comes to leave this body, is to exit consciously, unafraid and expecting something wonderful to happen!

Edward R. Close, October 8, 2017

Saturday, October 7, 2017


The Nine-Dimensional Doorway to Infinity
Nikola Tesla’s Magnificent Trio: The 3, 6 and 9

Solving the Mysteries of Life and the Universe with the Cube
©Edward R. Close 2017

The Rubik’s Cube® has 9 rotatable planes, 3 in each of the 3 orthogonal dimensions. This book teaches you how to use the cube to model everything from electrons and atoms to solar systems and galaxies; but more importantly, it teaches how to develop your intuition, focus and concentration to solve any problem! In the process, you can also learn how to solve the cube.  

There is more about this forth-coming book in the archives under Rubik's Cube Power point.

Saturday, September 30, 2017

Thursday, September 28, 2017

Saturday, September 23, 2017

TRUE Quantum Calculus and the Electron


By Edward R Close, PhD
Copyright September 23, 2017

Uniting Quantum Physics, Relativity and Consciousness

Thursday, September 7, 2017


©Edward R. Close September 2017

It seems that human beings are a volatile and unstable species. Our history on this planet is dotted with horrible wars and millions of war-related innocent deaths. The little valley where I was born is no exception. In the center of the Louisiana Purchase, Arcadia Valley had been affected by the French and Indian wars, the Civil War, World War I and World War II by the time I was 10 years old.

The first European settlement in the Valley, on Stout’s Creek, was burned to the ground and all the settlers massacred by Native Americans in the winter of 1780. As a child, I played in and around the earthen works of Fort Davidson, the site of the Battle of Pilot Knob, which took place in 1864, occasionally finding arrowheads, lead bullets and cannonballs, relics of the massacre of 1780 and the Civil War 84 years later. In the early 1900s the first automobiles required better roads than the horse and wagon, and when a road bed was excavated near Fort Davidson a mass grave of Civil War soldiers was discovered. The site is now a State Historical Park. Bloody war with hand-to-hand combat has not occurred in the Valley since 1864, but before another 84 years passed, residents were impacted by the effects of World I and World War II.

I have vivid memories of life in the Valley during World War II: Air-raid drills, nightly blackouts and food rationing. My father, who was Tri-City Constable in Arcadia Valley at the time, obtained full-face gas masks for himself, my mother and me. I was fascinated with the masks. I still remember the sounds of the flap valves opening and closing as I breathed in and out. We planted Victory Gardens to assure that we had food if the large cities and transportation routes were destroyed.

Even though the war in Europe seemed very far away, we were told daily by radio, newspapers and even comic books, that Adolf Hitler, Benito Mussolini and Hiro-Hito were evil men, it was even suggested that the Japanese were not really human! And that this axis of evil was bent on destroying our world. Why, I wondered, would the Japanese, Germans and Italians want to bomb our little town in the middle of the San Francois Mountains? I didn’t know at the time that we were located only about twenty miles from the largest high-grade lead mines in the world, the source of most of the bullets used by US armed forces in World War I and II. And the iron mines in our valley and the next valley north, were primary sources of the iron ore used to make everything from guns to tanks for the war effort. We were only about 70 miles from a large secret underground storage of aviation fuel, and about the same distance from a large scale Uranium 238 processing facility. We were in the middle of important targets for the enemy.

My fraternal grandfather was a son of German immigrants, and there were many families of German origin in the valley, along with a mix of people of almost every other European heritage who had been drawn to this area to work in the mines. Long before World War I, my Great-grandparents had discouraged their children and grandchildren from learning German. Their attitude was: “We are Americans now, so we must be Americans and speak English.” Still, as World War II loomed, there were German sympathizers in the Valley, and in 1936 the Nazi Bund established a local chapter in the Valley.

My dad, who was half German and half Irish, born in the major mining town of Saint Francois County Missssouri in 1908, volunteered for the U.S. Army in 1925, and served for four years in Hawaii, returning home to marry and start a family before World War II began in Europe. He knew about the NAZI Bund in the Valley, but would have nothing to do with it. In 1941, when the US entered World War II, he was still under draft age, so he volunteered for the Navy, and served as a member of the Amphibian Scouts and Raiders, known today as the US Navy Seals. He was on the Treasure Island Naval Base in San Francisco Bay with the men of Operation Olympic, set to invade the Japanese homeland, when the atomic bombs were dropped on Hiroshima and Nagasaki. President Harry S. Truman, also a Missourian, decided to drop the bombs, in part, because of the anticipation of up to 90% US casualties in the invasion.

With the fall of the NAZI’s Third Reich in May1945, and the surrender of the Japanese, signed on board the USS Missouri September 2, 1945, World War II officially ended. I remember the day well. When we received the news via Radio KMOX broadcasting from St. Louis, a wave of emotion swept across the Valley, and the jubilant residents of our town drove up and down the main street, Missouri Route 21, shouting, blowing horns and waving American flags. The WAR WAS OVER! But many waited in vain for their fathers, sons and brothers to come home. More than 400,000 US service personnel were killed in the war. I was luckier than some of my friends; my father did come home.

If President Truman hadn’t ordered the bombs to fall, Operation Olympic was scheduled to leave the West Coast of the US for Japan in early September 1945. My father, a member of a platoon of Amphibian Scouts and Raiders would have been in the first wave to land on the beaches of Japan. Their job was to sabotage key Japanese facilities in advance of fourteen Combat Divisions of Soldiers and Marines, the main invasion force, which was to land on November 1.

Why am I sharing these memories? I believe that civilization now stands at the most important threshold in all of human history: Belief in scientific materialism has produced reliance on the things of technology, and has robbed us of meaning and purpose. Without meaning or purpose, we are as likely to destroy ourselves as not, just because we can. We have a choice: We can follow the path of materialism which has resulted in war after horrible war, or we can step up to a new level of awareness. With the discovery of the existence of the third form of reality, a form that acts as the agent of Primary Consciousness in the physical world, and with empirical evidence and mathematical proof that consciousness, not matter, is primary, we can escape the dead end of materialism. 

Scroll down to the post below to read my address to the members of the Academy for the Advancement of Post-Materialist Science, August 26, 2017.

Tuesday, September 5, 2017


The following is a copy of my presentation to the founding members of the Academy for the Advancement of Post-Materialist Science, August 26, 2017

A Presentation by Edward R. Close, August 2017

First, I want to thank Dr. Gary Schwartz, Dr. Marjorie Woollacott, Dr. Charles Tart, and all who have worked so hard to make the Academy for the Advancement of Post Materialist Sciences and this meeting possible, including our anonymous benefactor. This meeting is the beginning of something I have dreamed of for many years.

I am struck by the similarities among the intellectual and psychic experiences of those gathered here today, but this should not be a surprise! It is evidence for what Erwin Schrӧdinger declared in his wonderful little book “What is Life?” published by Cambridge University Press in 1967, when he said: “There is no evidence that consciousness is plural.” Many of us know that all things are connected at a fundamental level, and, my friends, it is time for the first real scientific paradigm shift since relativity and quantum physics!

I want to start by sharing an experience I wrote about in my first book, “The Book of Atma”, published in 1977. It reveals the motivation that has propelled me throughout my life:
It was the summer of 1951. I was fourteen. I found a little book on analytical geometry written in German among some old books. Reading it, I had the distinct awareness that I already knew this mathematics. It was as if I were remembering, not learning. Also, I had just discovered the work of Albert Einstein, which had opened a whole new world for me.

One evening, in the twilight just after sunset, I walked out of the little house on my parent’s farm in the Southern Missouri Ozarks, past a line of catalpa trees, to the bank of a pond. I had been thinking about the “electrodynamics of moving objects” as described in Einstein’s special theory of relativity, and I had reached a point beyond which I could not go. Frustrated, I looked up at the sky and complained: “God, I want to know everything!”

What followed was totally unexpected, but so real that I knew it was completely natural. Suddenly, I could “hear” the silence around me. My surroundings took on a glow, as if everything were alive. My conscious mind seemed to melt, and the distinctions between my physical body and the surrounding landscape seemed to fade. I was filled with an all-pervading feeling of well-being. I knew I had received my answer! I would be a theoretical physicist!

I could spend my twenty minutes describing the series of psychic experiences and epiphanies that led Dr. Vernon Neppe and me to develop the Triadic Dimensional Distinction Vortical Paradigm (TDVP), and list the paradoxes it has resolved and the phenomena it has explained that are not explained by the current materialistic paradigm, but that would only scratch the surface. Instead, I want to address Dr. Gary Schwartz’s last item in his list of important questions: “Do we need an expanded mathematics, as Close and Neppe propose, to advance Post Materialist Sciences?”

Of course my answer is yes; but let me illustrate and emphasize this answer with a short history of the development of the new mathematics that unites number theory, geometry, relativity, quantum physics, some aspects of string theory, and the consciousness of the observer.

A paranormal experience in 1957 resulted in my discovery of the work of Pierre de Fermat. My College roommate, now Dr. David Stewart, and I were carrying out experiments in which we obtained verifiable information not available to us by normal sensory means. One of the most successful of these experiments was submitted to Dr. J.B. Rhine at Duke University. During one of our early experiments it was revealed that I had access to memories of the life of Pierre de Fermat. We obtained mathematical representations of concepts that far exceeded my training at the time, but were verified by my physics professor.

In 1637, Fermat wrote in the margin of his copy of a book on Diophantine equations, that he had found a “marvelous” proof that the equation xn + yn = zn has no integer solutions for n >2.  But his proof was never found. After receiving my degree in mathematics and physics in 1962, while teaching mathematics, I spent considerable time trying to access Fermat’s marvelous proof. Sometime during that period, I realized that Fermat’s Last Theorem, considered by most to be nothing more than a hypothesis in pure number theory, had important implications for quantum physics if x, y and z represent the radii of elementary particles that combine to form what we experience as ordinary physical reality.

This led to the realization that a quantum mathematics was urgently needed for describing the quantized reality we live in. The differential and integral calculus of Newton and Leibniz are inappropriate for describing quantum phenomena because they depend on a continuity of the variables of measurement that does not exist in a quantized world. I believe that the inappropriate application of Newtonian calculus to quantum phenomena gives rise to much of the ‘weirdness’ of quantum physics that physicists like to talk about.

I found the basis for the needed quantum mathematics in G. Spencer Brown’s calculus of indications published in his 1969 book “Laws of Form.” And it was obvious to me from the results of the Aspect Experiment resolving the Einstein/Bohr debate, that we have to have a mathematics that incorporates the consciousness of the observer. I published the basic concepts of an adaptation of Brown’s Calculus which I called the Calculus of Distinctions in my book, “Infinite Continuity,” in 1990. The Calculus of Distinctions is different from Brown’s Calculus of Indications in several ways that I do not have time to go into here. Unfortunately, that book is now long out of print, but the basic logic is published in an appendix to my 1996 book, “Transcendental Physics.”

In those references, I show that the drawing of a distinction is comprised of a triad:
1.     the object of distinction
2.     the features distinguishing the object from everything else, and
3.     the consciousness of the observer.

Thus, a distinction is inherently triadic, and the consciousness of the observer is implicit in the logic of the CoD. Therefore, application of these basic concepts inherently includes the consciousness of the observer in the equations of science. I later adapted the CoD to reflect the multi-dimensional geometry of finite distinctions and the differentiation of existing distinctions from conceptual distinctions in the Calculus of Dimensional Distinctions (CoDD).

With the help of Russian-born mathematician Vladimir Brandin in 2003, and Dr. Vernon Neppe, from 2008 to the present, application of the CoDD has allowed me to develop the definition of a true quantum equivalence unit that I call the Triadic Rotational Unit of Equivalence (TRUE), and the discovery of the third form of the substance of reality, necessary for the stability of atomic structure. This third form cannot be measured as mass or energy, but is detectable in the total angular momentum of any rotating physical system. Dr. Neppe proposed the name gimmel for the third form for a variety of interesting reasons.

We decided to call the new paradigm TDVP: Triadic because that was the nature of the underlying structure of mass, energy and consciousness. Dimensional, because to be consistent, the mathematics had to incorporate extra dimensions beyond three of space and one of time. Vortical, because of the spinning nature of elementary particles, and Paradigm to emphasize that it is a shift from the current materialistic metaphysics of modern science.

Physicists talk about a “theory of everything”. But you can’t have a theory of everything if everything is not included in it. I see the discovery of gimmel as the fulfillment of my efforts over the past 30 plus years to put consciousness into the equations of science. Gimmel has all the earmarks of consciousness, or at least of an agent of consciousness, acting through what I call the conveyance equations, to bring the logic of the multi-dimensional substrate of Primary Consciousness into the 3 Spatial dimensions, 1 Time dimension, and 1 dimension of Consciousness, i.e., the domain of physical observation.

The discovery of gimmel eliminates materialism as a viable metaphysical basis for science. It eliminates materialism because gimmel is inherently non-material, and because I have proved that it is necessary for the stability of quarks and subatomic structure. Without it there would be no physical universe. The discovery of gimmel answers Gottfried Leibniz’s unanswered first priority question: “Why is there something rather than nothing?”

I believe that gimmel is the manifestation of consciousness in physical reality. This view is justified in part because the elements and compounds supporting organic life forms prove to have the highest levels of gimmel. TRUE units and gimmel provide the necessary basis to analyze and quantify consciousness working within our physical/spiritual/conscious reality.

Through the use of TRUE unit analysis and LHC data, and applying the principles of relativity and quantum physics, several unexplained phenomena have been explained quite elegantly by TDVP. Because TDVP includes consciousness in the equations of science, and therefore is more comprehensive than materialistic theories, it can provide the mathematical basis for investigating and describing psi phenomena like those experienced by virtually everyone in this room.

My answer to Gary’s question about whether the Academy needs an expanded math is this: It is my personal belief, based on over 50 years of explorations of mathematics, physics and consciousness expansion techniques, that mathematics is not merely a tool, mathematics reflects the actual structure of reality. And if you look at the history of science, every real scientific paradigm shift of the past has been accompanied by new mathematics. The paradigm shift to the primacy of consciousness can be no exception. It is my opinion that, in this case, a new mathematics is even more crucial than ever before because of the magnitude of this shift. Post-Materialism Science cries out for a new more comprehensive mathematical paradigm, and in my opinion, that new paradigm is TDVP, and the new math is the Calculus of Dimensional Distinctions.

Monday, August 21, 2017


It was vrey dark here for a little more than two and 1/2 minutes. There was no air movement during that time, and not surprisingly the birds stopped singing. But there were no other noticeable side effects. Certainly no eartquake ---thankfully.


Posted Wednesday, June 21, 2017


The link below is to an image of the paths of the total Solar eclipses of 2017 and 2024. The X happens to mark a spot a few miles from the epicenter of the largest known earthquake ever in the US, the New Madrid quake of 1812. We're told that a large quake, 8 to 9 0n the Richter scale, has happened about every 200 years in this area according to geological evidence.  Will the combined gravitational pull of the earth and sun lined up in a total solar eclipse on August 21, 2017 be enough to trigger the next one? Incidentally, we live about the same distance from the center of the X as the epicenter of the New Madrid Missouri quake of 1812, well within the paths of both solar eclipses! We have a front row seat to view the 2 minute and 40 second maximum eclipse on August 21, and we're only a few miles from the New Madrid fault. Would you be excited, or scared?


Sunday, August 13, 2017


I think that this recent exchange with a friend who has been looking at FLT65 for some time may be of interest to  those who are following the subject of Fermat's Last theorem, which as they may know, is importent to the explanation of quantum phenomena in TDVP. This conversation may be helpful for anyone interested in understanding the logic of the 1965 proof. For this reason, I am posting my latest response here without identification of my friend or his math professor friend.


I spent some time thinking about Prop. P between meetings and events in LA last week. There really should be no confusion about when P is true and when it is false. I will attempt to clarify this while answering your latest comments, because they do reflect what I see as confusion about what FLT65 says and does. I shall also attempt to show you how distinguishing when Prop. P is true and when it is false, and when the identity sign is appropriate, leads to a better understanding of FLT65. 

From your email earlier today:

      Q: Do you believe that Prop P is ever false?

Let’s consider this. Here is Prop P as you stated it:

     P: Equating a polynomial to a constant, for the purpose of finding a  
        specific solution, automatically turns the polynomial itself into a  

This should not be a matter of “belief”, because it is easy to show that P can be either true or false, depending upon the nature of the polynomial, and the circumstances. And, it is an added benefit that the circumstances also allow us to clarify the proper use of the identity symbol ≡.

P is true, if and only if, all variables are specified as constants. For example, we know that, if Y in the FLT equation is an integer, then for any Y there is some integer factor A such that z – a = A. If we should also know that Z and Y are the specific integers Z1 and Y1, then a is determined, and we have: Z1 – a1 A, an identity.

But in FLT65, z is an unknown. Clearly, integers can be assumed for X and Y, and if so, then whether or not z can be an integer is as yet, in the proof, unknown. To assume that it is an integer at the beginning, is premature and leads to circular reasoning. For the polynomial z – a in FLT65, z is a real number, but its specific value is unknown, so when we set the polynomial z –a = A, a known integer factor of Y, P is false, because z and a can take on an infinite number of values. The only requirement imposed by setting z – a = A is that their difference is always A. The value of z can vary, and thus this equation is not an identity.

 I once sent you the judgement of a mathematician who had spent his entire working life as a mathematics professor in a university. You dismissed his judgement that FLT65 was invalid as “something you had seen before.” 

To be clear, I had seen this line of reasoning several times before, and have given it all the serious thought it deserves, so I saw it as the same knee-jerk reaction I’ve seen numerous times, and refuted every time. I felt that you should have recognized the circularity of the argument. But I apologize for my abrupt manner. I should not have been so abrupt, and I certainly should not have been condescending. I’ll copy the math professor’s comment here and try to respond more appropriately.  He said:

“Here is a possible way of putting it that might convince Mr. Close.  In his argument he sets  a = Z - A   and considers the divisor polynomial  g(Z) = Z - a, which he says is a polynomial of degree  1  in Z.  But  g(Z) = Z - a = Z - (Z - A) = A,  which is not of degree  1  but of degree 0.  When the divisor A  is a (nonzero) constant, the polynomial division algorithm over the reals just says there exists a (unique) polynomial  q(Z)  such that  f(Z) = Aq(Z) + 0, where, of course, q(Z) = f(Z)/A.  so there is no  f(a).”

This argument completely misses the point of FLT65 by assuming the definition a = Z – A which is not the case in FLT65. His statement that I set a = Z – A is false. You will not find this anywhere in FLT65. Z is the unknown, the dependent variable. Z – a is defined as a polynomial of the 1st degree, meaning that the value of Z depends on the value of A, an integer factor of Y and the value of a, which is an integer if there is an integer solution of the FLT equation, which is as yet, in the FLT65 chain of logic, unknown. By assuming that Z is an integer by definition, the conclusion that Z –a is a constant, and therefore of 0 degree, is of course, circular reasoning, as mentioned above, and stated in my more abrupt response. By the way, I meant no disrespect to your professor friend by dismissing his comments. A university professor known as a number theory expert made the same mistake, but he quickly acknowledged that it was circular reasoning when I pointed it out to him.

Thank you for your statement of what you see as a disproof of FLT65. It enables me to better understand why you kept coming up with propositions that had nothing to do with FLT65. I think other critics may have had this same misconception about how FLT65 goes about proving FLT.

Your statement has the logic of FLT65 completely backward:

 “The equation of constants, f(Z1) = Ap = (Z1 - a)p, does not imply that the variable (Z - a) is a variable factor of the variable f(Z). 

I agree! However, what FLT65 says is the converse: the fact that f(z) cannot contain z – a as polynomial factor for real number values of the variables, implies that, if there were an integer solution for the Fermat equation, then there would be an integer version of f(z)/(z – a) = q(z) + f(a) where f(a) would equal zero, and that would violate the ‘if and only if’ condition of the division algorithm for polynomials.

It appears to me that the confusion comes from considering Z to be a specific integer before it is known whether z can be an integer or not. Maybe this will be clearer if we go step-by-step:

Dividing f(z), a polynomial of degree p-1 by z - a, a 1st degree polynomial, we have unique polynomials q(z) and f(a) of degree less than p such that:

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

From this we see that the polynomial f(z) is factorable into two polynomial factors, q(z) and z-a, if and only if f(a) = 0. But f(a) = ap-1 + ap-2x + ap-3x2 +•••+ xp-1, which cannot equal zero because a and x are positive for any integer solution of the FLT equation.

Therefore, the FLT equation factor polynomial f(z) cannot be factored into two polynomials of degree less than p, one of which is z - a.

If there is an integer solution, then with the term-by-term substitution of the integer variables into the variable polynomial f(z), we obtain the variable polynomial f(Z) for any integer solution of the Fermat equation. But, for any integer solution, f(Z) = (Z-a)p, where Z-a = A, a single integer factor of Yp. So now we have:
The hypothetical integer polynomial f(Z) = (Zp-1 + Zp-2X + Zp-3X2 +•••+ Zp-1) = q(Z)(Z-a) + f(a) = (Z-a)p.

By inspection of this integer polynomial equation we see that f(a) contains Z - a as a factor, and although we don’t know what the specific values of a and X are for an integer solution, we know that they are positive constants. So f(a) = M(Z-a) where M is an integer constant, and we have f(Z) ) = (Z-a)p, = (Zp-1 + Zp-2X + Zp-3X2 +•••+ Zp-1) = q(Z)(Z-a) + M(Z-a), from which we have: f(Z) = [q(Z) + M](Z-a).

And q(Z) + M is a variable polynomial in Z of degree less than p, call it q1(Z). Thus for hypothetical integer solutions of the FLT equation, we have the variable polynomial f(Z) = q1(Z)(Z-a). But this is a violation of corollary III of the division algorithm, which tells us that the variable polynomial f(Z) cannot be divided into the factors Z - a and another polynomial of degree less than p. The only way we can avoid this contradiction is for z to be an irrational real, not an integer. 

I have demonstrated in at least three different ways, including Fermat’s favorite method of proof, infinite descent, that if we ignore this contradiction, and assume that two of the three variables x, y and z are integers, and solve the FLT equation for the third variable, then that third variable cannot be an integer.

I don’t consider adding such demonstrations to FLT65 to be necessary because the contradiction f(a) ≠ 0 versus f(a) = R = 0 is sufficient to prove FLT by itself. This contradiction is valid because it is obtained by applying the division algorithm and corollaries to variable polynomials, not constants, and a single contradiction is sufficient to prove there can be no integer solutions for zp – xp = yp.

I believe the whole confusion for most critics arises from assuming that the division algorithm and corollaries are inappropriately applied to constants. They think this because they jump to the conclusion that all three variables must be treated as integer constants from the beginning of the proof. Thanks to you, this confusion has been made clear with the analysis of Prop P! 

Once you see that P can be true or false, depending upon the nature of the polynomial, and that in FLT65, f(Z) and Z – a are still polynomials of the variable Z, even though for a hypothetical integer solution, they are equal to the constant integer factors of Yp, the logic of FLT65 becomes clear, and the search for counter examples and counter propositions becomes unnecessary and irrelevant.

Edward R. Close  August 12, 2017

Friday, August 4, 2017


in 1637, Pierre de Fermat, a judge at the French Parliament of Toulouse, wrote in the margin of a book on Diophantine equations that he had devised a marvelous proof that there are no positive whole number solutions to the equation xn + yn = zn for n greater than two. His proof, for n = 4 is known, but his general proof for all n greater than two was never found.

It is an interesting aside that Fermat was not a professional mathematician. He did not publish his findings, he simply conveyed them in letters to other mathematicians, and thus was considered an amateur. The less than modest French mathematician and philosopher Rene Descartes tried to discredit Fermat by proclaiming that he was “a troublemaker who owed his reputation to a few lucky guesses”. However, in one dispute after another, e.g. their derivations of the sine law for the refraction of light, Fermat proved to be right and Descartes wrong. While Descartes clearly considered himself to be the superior intellect of the day, a comparison of their works reveals the fact that Fermat was the better scientist and mathematician of the two.

Descartes’ arrogance shines out in the following statement: “‎I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery.” Implying that he could have explained much more. This is analogous to a classmate of mine who liked to say “I’m not conceited, I’m actually twice as smart as I say I am!”

Peter Bernstein, in his book Against the Gods, states that Fermat "was a mathematician of rare power. He was an independent inventor of analytical geometry, he contributed to the early development of calculus, he did research on the weight of the earth, and he worked on light refraction and optics. In the course of what turned out to be an extended correspondence with the Pascal, he made a significant contribution to the theory of probability. But Fermat's crowning achievement was in the theory of numbers."

Regarding Fermat's work, Sir Isaac Newton wrote that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."
Speaking of Fermat's work in number theory, Mathematician Andre Weil says that: "what we possess of his methods for dealing with curves of genus 1 is remarkably coherent; it is still the foundation for the modern theory of such curves. It naturally falls into two parts; the first one ... may conveniently be termed a method of ascent, in contrast with the method of descent which is rightly regarded as Fermat's own. … With his gift for number relations and his ability to find proofs for many of his theorems, Fermat essentially created the modern theory of numbers.

Never-the-less, popular history has treated Rene Descartes very well, while ignoring Fermat. The name Descartes is well known to every student of mathematics and science, while Fermat’s name is virtually unknown except for in relation to Fermat’s Last Theorem, -and most modern mathematicians openly doubt that he actually proved it! Why? Because for more than 300 years, professional mathematicians tried to find a proof, and failed.

The power of Fermat’s math lies in his method of infinite descent, and the principle of “efficient purpose in nature”, and in 1964-5, while teaching high school mathematics, using the same simple principles used by Pierre de Fermat, I produced a proof which I first submitted to a professional mathematician in 1966. Because my proof was completed in 1965, I call it FLT65.

To see FLT65 as it was submitted to the first reviewer on January 25 1966, copy the link below and paste it into your web browser.

This link will take you to a lengthy discussion of FLT65. To see the original submitted version, scroll down to Appendix C.

I have submitted FLT65 to more than 50 mathematicians over the years, and only a few of them who responded actually offered any mathematical arguments attempting to disprove FLT65. And while a precious few have admitted it, none of them were able to produce a valid refutation of FLT65. Because I believed it would eventually be recognized as a valid proof, I have carefully documented the submittals and responses.

You might well ask: if no one has refuted FLT65, why hasn’t it been accepted? The answer to this is an interesting story by itself. To learn the details of the history of the odyssey of FLT65, copy the link below and paste it into your web browser.

I believe that the time for Fermat to become a household name has come, because his simple methods of Diophantine analysis are totally appropriate and exactly what is needed for application to quantum physics. It is time to go beyond the calculus of Newton and Leibniz and apply the Calculus of Distinctions to TRUE quantum equivalence units to produce a better description of multi-dimensional reality. See details in the posts of TDVP and

Because of excessive academic specialization, and the fact that modern thinking has lost its metaphysical basis in Infinite Intelligence, science has gone astray and adopted atheistic materialism as its metaphysical basis. The result is a world society that is morally adrift. This lack of meaning and purpose is so dangerous that, if not corrected, it could spell the demise of the human race. It is time to revisit the simple infinite descent and principle of “efficient purpose in nature”, of Pierre de Fermat.

Edward R. Close    August 4, 2017

Wednesday, August 2, 2017



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 + bxn-1 + cxn-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:
yp = zp – xp = (z-x)(zp-1 + zp-2x + zp-3x2 +•••+ xp-1)
It is shown that z–x, and f(z), as factors of yp, can be considered to be relatively prime. Thus f(z) = Ap, 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:
(zp-1 + zp-2x + zp-3x2 +•••+ xp-1)/(z-a) = q(z) + f(a)/(z-a). Multiplying through by z-a, we have:
(zp-1 + zp-2x + zp-3x2 +•••+ xp-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 zp – xp = yp, then yp is an integer containing the integer factor f(Z) = Ap. = (Z-a)p. This gives us two equations:

1)         (zp-1 + zp-2x + zp-3x2 +•••+ xp-1) = q(z)(z-a) + f(a), an algebraic polynomial equation and
2)         (Zp-1 + Zp-2X + Zp-3X2 +•••+ Xp-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 zp – xp = yp.
For an integer solution, if one exists, equation #2 can be reduced to an equation of single integer terms:
3)         Ap = Q·A + R, which implies that the integer f(Z) = Ap 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 Yp, and f(Z) = Ap, 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 (Zp-1 + Zp-2X + Zp-3X2 +•••+ Xp-1) yields Ap, 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, Yp =ApBp =f(Z)Bpf(Z) =Ap 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