Friday, June 30, 2017



From the first bright streaks of morning’s light,
To golden mid-day, to evening’s red, and night.

From the roaring storm with lightning strikes of fear and pain,
To the welcome pattering of soft shining silver pearls of rain,

From the depths of the Canyon Grand,
To the highest mountain’s snowy peak,

Embrace the wonder of this, our land.
We have what so many others seek.

In this land, the USA, each one is free
To speak, and sometimes rant and rave.

No one should be a victim meek,
And no should be another’s slave.

But don’t forget to have respect for the others’ points of view.
Slanderous words will reflect, and wreak their havoc upon you!

Some are easily misled by political deceit, greed and desire,
But don’t you slide into that quagmire, it’s a deep dark hole,

To rise above this awful curse, out of judgmental line of fire,
You need to listen to His Voice, right there within your soul.

Edward R. Close June 30, 2017

Tuesday, June 27, 2017



The Earth turns, the Sun rises,
What was hidden is revealed:
Awareness brooks no surprises;
Nature’s secrets are unsealed!

From quark spin to stars’ shine,
The unknown revolves into the known,
Quantum Truth is on the line,
The Lord God is on the phone!

Pick up! Pick up! While it yet rings,
Ever spreading from the Sun,
His is the Heart within all things:

Love unites, and we are One!

E. R. Close, June 27, 2017

Monday, June 26, 2017



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

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

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

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

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

Edward R. Close, June 26, 2017

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

The Basic FLT65 Proof
The following steps summarize the logic and mathematics of FLT65. For brevity, I will not present proofs of the steps here because they are so easily proved that they can be proved by a bright high school algebra student. If these steps aren’t obvious go to

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

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

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

Similarly, zp – yp = (z-y)( zp-1 + zp-2y + zp-3y2 +•••+ yp-1) = xp.

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

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

(z-x)(zp-1 + zp-2x + zp-3x2 +•••+ xp-1) = yp, we see that we can write
(z-x)= Bp, and (zp-1 + zp-2x + zp-3x2 +•••+ xp-1) = Ap, where A and B are positive integers.

STEP #5: If there is an integer solution, then x and y are specific integers X1 and Y1, and the p-1 polynomial in z, f(z)=(zp-1 + zp-2X1 + zp-3X12 +•••+ X1 p-1) = Ap, and BpAp = Y1p. That is, the two factors must be perfect p-powers, integers raised to the pth power.

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

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

Note that the case n = 4 is not addressed in this proof. It was overlooked in FLT65, but this was not a problem because there were several known proofs for n = 4, including one by Fermat himself.
So FLT65 is effectively a complete and valid proof of FLT; but approximately 90% of the mathematicians to whom the proof was submitted over the years did not respond at all. This is because, before Sir Andrew Wiles’ proof was accepted, professional mathematicians received hundreds of supposed proofs of Fermat’s last theorem per year.

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

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

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

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

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

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

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

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

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

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


Sunday, June 25, 2017



With the spinning and rotation of the Earth through the Summer Solstice 2017, and the rotation of the Milky Way Galaxy into the most energetic space domain, in more than 12,000 years, everything has changed. Many of us feel that change because of an increased awareness of the interconnectedness of all things. World events are reflecting this greater availability of positive energy. Scientists applying new math have discovered that the human brain has access to up to eleven dimensions. See and that the current mainstream scientists are using the wrong math to describe reality. See Dr. Vernon Neppe and I have been telling everyone who will listen to us this for about seven years now. See many posts on this blog and

It’s time to begin the full disclosure of the link of every conscious being to Cosmic Consciousness. Stay tuned!



There are several things that I want to bring from our contemplations of the cube in Part I, and quantum reality in part II, into Part III. Those things include the Triadic Rotational Units of Equivalence (TRUE) units of mass, energy and gimel, the universal connectedness of all things and the importance of intuition.

Without intuition, we are adrift in an infinite sea; the world is like shifting sand beneath our feet. We seem to have been thrust, without being prepared, into an on-going drama, where we must try to survive in a harsh and brutal environment. Looking back, birth may seem like awakening from a deep sleep: Upon awakening, we look around and find that we have arrived in the middle of the strange story of life on this planet, with no knowledge of how it began, and even worse, no clue about how it might end. The whole scenario is a complete mystery. Our awareness of self seems unconnected with the world we experience. The only thing we know for sure is that we exist - and that we need to know more in order to continue to exist.

In most cases, we have no conscious memory of what happened before we open our eyes as an infant, but this is an illusion. We do have a deep somatic memory hidden in the RNA/DNA structure of our forming bodies. We are not born as empty containers, little blobs of unstructured protoplasm with no knowledge of the past. Not at all. In fact, each blob of protoplasm blossoming into this world has within it a vast storehouse of memories, physically manifest in complex single and double spiral structures called RNA and DNA, containing records of the distant past and blueprints for the distant future.

Also hidden from us as long as our consciousness is focused on and attached to the physical body, is the complete record of the process of the manifestation of Primary Consciousness in the finite worlds of nine dimensions. This memory in Primary Consciousness, existing beyond time and space, is called the Akashic Record. Our sub-conscious and super-conscious connection with Primary Consciousness is the reason we have an intuitive sense of direction, of meaning and purpose. Without this intuition, we are nothing, full of sound and fury for a while, signifying nothing. Without intuition, we have no way of knowing who we are, where we came from, or where we are going. We only know that we are conscious, and truly, for us, reality begins with consciousness.

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?


Monday, June 19, 2017






COME EXPERIENCE A PRESENTATION OF THE DISCOVERIES OF Vernon M. Neppe and Edward R. Close, creators of the new science of Dimensional Biopsychophysics



Dates and locations of presentations to be announced.

Friday, June 16, 2017


 A recent email to an ISPE friend:

A good friend does not give up trying to convince someone he considers to be his friend of the truth. I believe that you are my friend, because if that were not the case, you would have given up trying to convince me that my FLT65 proof is flawed long ago. So, in return, I must not tire of trying to convince you that FLT65 is valid, as long as I see it that way.

After responding to your recent email containing the short circular argument put forth by your retired math professor friend, I had an inspiration. I believe I see a more direct way to explain FLT65. Here it is:

If one number is divisible by another, then dividing the smaller one into the larger one produces a zero remainder, while if they are not divisible, the remainder is non-zero. These simple facts are expressed by the division algorithm and its corollaries and they are true for all polynomials over the field of real numbers, whether reducible to integers or to any other real number.

Recalling FLT65, we see that when p ≥ 3, p a prime number, the FLT equation can be factored and expressed in the form (z-x)( zp-1 + zp-2x + zp-3x2 + ••• + xp-1) = yp. If there are integer solutions for zp – xp = yp, then, with specific integer values of x, y and z, g(z) = z–x and f(z)= zp-1+ zp-2x + zp-3x2 +•••+ xp-1 must be equal to relatively prime integers raised to the pth power. That is to say that, if there is an integer solution, then yp will be equal to BpAp, where Bp = z –x, and Ap = zp-1+ zp-2x + zp-3x2 +•••+ xp-1, A and B relatively prime integers. Furthermore, for an integer solution, the fact that integers are closed with respect to addition guarantees that there is always an integer a, such that z –a = A.

Of course Ap is divisible by A, so Ap = zp-1 + zp-2x + zp-3x2 +•••+ x p-1 must be divisible by A = z – a, and the division algorithm Cor. III says that f(z) is divisible by z – a, IF AND ONLY IF, the remainder, f(a), is equal to zero. Therefore, to find the value of x for any given values of z and a, we must set f(a) = ap-1 + ap-2x + ap-3x2 +•••+ x p-1 = 0 and solve for x. There are exactly p-1 solutions to this equation and for all of them, x is non-integer. This proves that for z and y equal to integers, x cannot be an integer, and FLT is proved.

I believe that the argument above is a more direct way to see FLT65, and it is completely equivalent to FLT65. I also believe that it becomes even clearer when illustrated with a numerical example, and I will use one provided by the critics.

While there are a few competent mathematicians who agree with me that FLT65 is a valid proof of FLT, more of them agree with you. For example, while reviewing my work, a Nobel Prize physicist and a very competent Israeli number-theory professor of mathematics, responded with what, in their opinions are counter examples that call FLT65 into question and, they believed, might even refute it. 

They both correctly noted that my argument in FLT65 is that when the factor of the Fermat equation f(z) = zp +xzp-1 +… + xp-1 is divided by z – a, the remainder, f(a) cannot be zero, while, for any integer solution, f(z) is definitely divisible by z – a. In fact, f(z) = Ap divided by z - a = A is Ap-1, where, if there is an integer solution to the Fermat equation, A is an integer, and this produces an inescapable contradiction. They argued that this is, or may be, incorrect because they could produce examples  for the equation when p = 3 with the remainder f(a) non-zero even though f(z) is clearly divisible by z - a when certain integers are chosen for z, x and a.

Here is one such example offered by the math professor:
Let z=7 and x=4. Thus 3 divides z2 +xz+x2, because f(z) = 49 + 28 + 16 = 93 = 3x31. So for a=4, the integer z - a = 3 divides the integer z2+xz+x2. However, in the polynomial ring R[Z], the polynomial z - a does not divide the polynomial z2+xz+x2 =z2+4z+16.  Indeed, the remainder is a2+xa+x2 > 0. Thus, he reasoned, the non-zero remainder when dividing polynomials does not prove that f(z) is not divisible by z – a = A if x, y, z, and a are integers.

There is however, a serious error in this argument. The error lies in the fact that, after choosing z = 7 and a = 4, the value for x is arbitrarily, and incorrectly chosen to make f(z) divisible by 3, allowing the production of a spurious “counter example”. The error is compounded by assuming that this supposed disparity in divisibility between the polynomial f(z) and its integer value may exist for the Fermat equation.

In fact, if z = 7 and a = 4 in the Fermat equation, then x cannot be equal to 4. This is easily and clearly demonstrated as follows:

The division algorithm expresses the essence of the fundamental operation of division for all real numbers, including integers. Corollary III of the division algorithm says that f(z) is divisible by z – a IF AND ONLY IF f(a) = 0. Therefore, in this example fabricated by the math professor, in order to see what x must be to satisfy the equation when z = 7 and a = 4, we must set f(a) = 42+4x+x2 = 0. When this equation is solved for x, we see that x cannot be equal to 4. In fact, solving this equation for x, we see that the two values of x satisfying the equation with z = 7 and a = 4 are – 2 + 2Ö3i and – 2 - 2Ö3i, which are complex numbers, and definitely not integers.

This is easily generalized for all integer values of z and a, and for all p>2 because all values of p are odd allowing the factorization into z – x and zp-1 + zp-2x + zp-3x2 + ••• + xp-1, a polynomial of p terms; and that is exactly what FLT65 does. The polynomial f(a) can never equal zero if x, z and a are integers, which they must be for an all-integer solution of the Fermat equation zp – xp = (z-x)(zp-1 + zp-2x + zp-3x2 + ••• +  xp-1) = yp. This proves FLT.

Now one must ask: Why has this simple proof, which I believe is, in essence, Fermat’s “marvelous proof”, been overlooked for more than 300 years, even by the world’s most brilliant mathematicians??? 

It appears to go back to Leonhard Euler and Carl Friedrich Gauss, arguably two of the most brilliant mathematicians of all time. Euler used complex numbers to prove FLT for p = 3, and Gauss developed modular algebra in an effort to prove or disprove the solvability of Diophantine equations including FLT. Unfortunately, like many mathematical procedures, modular algebra obscures as much about integer and non-integer polynomials as it reveals. When Gauss was unable to produce a proof using this method, he famously announced that he could produce any number of such theoretical propositions that could be neither proved nor disproved, and thus would waste no more time on it. This set the tone for many professional mathematicians in the years to follow.

Kurt Gӧdel’s incompleteness theorem proved that there are always logical propositions that cannot be proved or disproved within the mathematical system giving rise to them. This strengthened Gauss’s speculation that FLT might not be provable using basic mathematics. Add to this the increasingly extreme specialization encouraged by academia in the last 200 years, and you have a general attitude that Fermat must have been mistaken about having a proof.

Especially after Andrew Wiles and Richard Taylor produced a torturously complex proof of hundreds of pages in 1995, it was thought probable that Fermat had fooled himself into believing that he had a proof, when in fact he had not, because the complex theorems used in Taylor and Wiles’ proofs were not available to Fermat in 1637. This line of reasoning, while convincing, of course does not prove there can be no simple proof.

After many years of trying to get professional mathematicians to take my 1965 proof of FLT seriously, I had given up. When I discovered in about 1989 or 1990 that FLT had an important application in quantum physics, I revisited FLT65. In 2010, even though the quantum physics application only required FLT to be true for p £ 9, I mentioned my 1965 proof to Dr. Neppe, who was intrigued, and after studying it and proving it correct for himself, urged me to resume efforts to get it recognized and accepted.

To date, only a few competent mathematicians have agreed with me that FLT65 is a valid proof, but, importantly, no one has shown me any real proof that FLT65 is not valid. The proclivity of professional mathematicians to dismiss it because of the belief that no simple proof is possible has led even otherwise competent mathematicians to think erroneously that examples like the one presented above disprove FLT65. 

Even those who have acknowledged that such examples are not counter examples because they have no relevance to actual solutions of the Fermat equation, apparently are loathe to think that FLT65 could be valid.

I believe that the simplified FLT65 approach presented above should convince some skeptics, perhaps including you, my friend, of the truth of FLT65, if it is carefully and thoroughly considered.

With Regards,

Ed Close June 16, 2017

Saturday, June 3, 2017


I thought those of you who have followed my posts on Fermat's Last Theorem might be interested to know that another university math professor has tried to disprove my 1965 proof of Fermat's Last Theorem. I'm posting it and my reply below; but I am witholding the names because I don't want to embarass anyone.

I received this email message from a friend in ISPE who doesn't believe my 1965 proof is valid:

Dear Ed,                                                                                  June 2, 2017

I tried to send this note to you on May 31. I probably used the wrong e-mail address. I’m trying again with this address.

Because of a common interest in music for big jazz bands, I recently made friends with a retired professor who taught mathematics in a university for years. Even more recently, I showed him a copy of FLT65. He agrees with my claim that FLT65 is flawed, but this note is not just to tell you that. I want to give you his explanation.

He has shown me what purports to be a more rigorous way to describe what he and I see as the flaw. My “arm-waving” description of it never convinced you, but perhaps his approach will be more persuasive. In any case, I think that you should see it, so I’ve pasted it onto this note below.

                        Sincerely, _______

I think I may have a useful thought for you on FLT65.  Your criticism of the proof is, I believe, exactly correct.  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).

My Reply, sent  7 AM, June 3, 2017:

Dear _____,

It's nice to hear from you; I hope you are doing well. We're leaving tomorrow for a conference in Chapel Hill NC, and we're busy packing today, so I'll have to be brief. I've seen this argument a couple of times before. One time was from a university prof considered by many to be one of the best in number theory. It only took a few lines to reveal his error. FYI, he acknowledged his error, and could never produce a disproof of FLT65, but he still felt that it must be flawed! I’m sure you will agree that mathematics should not be about feelings or beliefs.

Think about it for a minute. If you argue that z - a is not a 1st degree polynomial, you destroy the division algorithm for all polynomials. Yes, for specific values of Z and a, Z - a is a constant. But that is true for any and every polynomial. And the variables of any polynomial factor of any equation will take on specific values determined by the solutions of that equation.   

Your friend also says "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 is a fine example of circular reasoning! When you set Z - a equal to a constant, which you can do for any polynomial, this statement is true, but, if it disproves FLT65, then it means that every polynomial is divisible by Z - a because every polynomial has solutions for which z - a will be constant. The point of the division algorithm is that f(Z), defined for all real numbers, is only divisible by Z - a when the remainder disappears, i.e. f(a) = 0. And the point of FLT65 is that because of the unique form of the Fermat equation’s factor, f(a) can never equal zero, and integers are reals, as stated in FLT65, a fact that no one disputes. 

All the Best,


Ed Close

NOTE (Not a part of the exchange copied in above.): The argument put forth by most who don't want to believe that FLT65 is actually a valid proof, is that the division algorithm works for polynomials of a continuous variable, but may not work for integers. This is handled in the 1965 proof by noting that the division algorithm holds over the field of real numbers, and integers are a set included in the field of real numbers. In the case of X to the nth power plus Y to the n power equals Z to the n power, when n = any prime number greater than 2, (the equation of Fermat's Last Theorem), no one has proved otherwise.