[NKS and Fermat's Last Theorem] - A New Kind of Science: The NKS Forum

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# NKS and Fermat's Last Theorem

Posted by: Byron Fast

When I look at Fermat's Last Theorem for cubes:
x³ + y³ = z³

I wonder why Wolfram didn't talk more about this theorem in his book since it seems to me that a simple automata makes understanding the theorem quite simple (http://www.wolframscience.com/nksonline/page-119). It is a famous, recognizable math problem that could show the value of automata in understanding such problems.

When you look at the list of cubes in binary (see page 119 image) so that they look like an automata the image shows the same randomness that any NKS fan should be familiar with. This random automata represents the set of all the possible answers for z³.

Doesn't this simple visual check prove Fermat's Theorem? As Wolfram says, it is impossible to predict what a future row in a random automata will contain. This means it should be impossible to write any statement about z³ since you would have to state exactly what would be found at some future row in a random automata. If you examine the images for other exponents (other than powers of 2, which are already disqualified) you will see that they also pass the randomness test and complete the proof of Fermat's Last Theorem.

I don't know if this can be called a "proof" but it seems like a much simpler way to view the problem. I certainly have no hope of understanding Andrew Wiles' proof in my lifetime, but in a few minutes with NKSX I was able to visually see why Fermat was right.

The inverse is if I had quickly found an image with a nested pattern I could have easily shown the theorem to be wrong.

Posted by: Charlie Stromeyer jr.

Wolfram talks more about FLT - look up "diophantine equations" in the book's index. The results by Wiles, Ribet, Taniyama and others are within the Langlands philosophy (or context), and Wiles also uses a series of inductive arguments.

Perhaps this is why Wolfram would rather see a simpler proof of FLT in terms of something like Peano arithmetic, i.e. maybe Wolfram wants a potentially more robust accounting of the axiomatic foundations used in Wiles' proof.

Btw, it was not until the 1800s that Hilbert and others revealed that the logical structure of Euclid's classical geometry was incomplete. Thus, it may be the case that the most durable and practical definition of "mathematical proof" is that a majority of the experts accept the proof as valid.

Posted by: jay dillon

WSEAS Transactions on Mathematics issue 3, vol. 3 (July 2004) has my article "Fermat's Last Theorem: Proof Based on Generalized Pythagorean Diagram." I believe there will be new methods of proof found for FLT; my own method was started in 1980. In the first 10 days of work I had developed a Pythagorean-styled square area diagram featuring as square areas the three large terms of the given equation of FLT: x^n, y^n, z^n, with the stipulation that these square areas have bases x^(n/2), y^(n/2), z^(n/2) respectively. The reason for this construction was to provide a generalized geometric model of the given equation in 2D. The fact that this diagram is legal under the stipulations should be quite obvious: any 2D Pythagorean-styled diagram can be used to model any simple equation of the form A + B = C, with the three large terms A, B, C represented as square areas. It is an automatic response to view this construction as applying only to the Pythagorean triples with exponent 2; but if the stipulations are maintained, we can test this model and it works; for example in the equation 3 + 4 = 7, the exponent is 1, and bases of the square areas are 3^(1/2), 4^(1/2), 7^(1/2) respectively. I had this diagram in 1980 but was unable to use it fruitfully until 1997 when I averaged the two smaller areas to produce a new Pythagorean styled diagram which could be nested with other averaged diagrams in a system with shared origin and shared 45 degree axis. Then the nested system could represent, with full generalization, the FLT equation with constants x, y, but allowing exponent n to vary throughout the positive reals in the positive direction toward positive infinity. Then also the new variable "a" was found, such that a < 2 < n, and by the averaged system I found z_xyn < z_xy2 < z_xya. (Subscripts indicate z as found dependent on {x,y,n}, {x,y,2}, {x,y,a} values respectively, in the FLT equation.) I would love to find any critiques of my paper since it was published in Aug. 2004, but so far no critics have read my full paper, or have not shared their critiques with me. I found two simultaneous equation curves:

z = (x^n + y^n)^(1/n) (given equation)

z = (x^a + y^a)^(1/2) (derived equation from nested averaged diagram system with x, y held constant)

Bizarrely this proof works I think for all positive reals; the requirement being that the x, y values must be held constant. (The dichotomy found develops directly from the generalized averaged diagram system, and by holding x, y constant; and this dichotomy does not require integers, so my proof goes beyond the FLT conjecture if I am interpreting this correctly.) I have been informally criticized for not discussing the integers in my paper beyond stating that x, y are to be held constant as integers; but my proof does find an insurmountable dichotomy or contradiction by my method as shown in the WSEAS paper. (The excluded maximum is found to be the only possible common solution point for the given and derived equations; excluded maximum defined as such due to the inverse relation of n and z in the system as stipulated.) I believe it can be extended further, perhaps into the negative numbers and combinations of negative and positive numbers, and perhaps imaginaries although I do not understand this aspect of the proof well yet.

Illustration help is greatly desired. I am not trained in anything, much less Mathematica; the paper will be made available in full on arXiv.org as soon as permission is granted by WSEAS. (I have requested permission to post there and requested their permission to list myself as being affiliated with WSEAS, otherwise I have no institutional affiliation as required for posting on arXiv.org.) Thank you for your consideration.

Posted by: Craig Feinstein

Jay,

I would love to see a simpler proof of FLT than Wiles', but I don't subscribe to the journal that you published yours in. Is there some other way that I can read your article?

Craig

Posted by: jay dillon

Thank you for your interest in my paper. The paper is copyrighted by WSEAS, but they have allowed me to post it online in arXiv.org

I am preparing a corrected version for posting there in arXiv. The published version in WSEAS Transactions on Mathematics (July 2004) did not show the first two color illustrations properly in black and white; these will be corrected and some minor errors in the text also need correcting. Unfortunately, I cannot directly edit the LaTeX2e file, so I cannot quickly place the paper online in arXiv.org. However if you will send me an email with your name, address, and university affiliation if any, I will try to either mail you a full paper copy, or possibly email you a .pdf file.

I am trying to follow the WSEAS copyright requirements. If I send out the paper to some few contacts for review that might be acceptable, but I cannot do it in quantity, and have to request that any review copy not be copied or distributed further. You can email me directly at jcfdillon@hotmail.com

Thanks again for your willingness to look at my paper.

I should point out that WSEAS had allowed me to post summaries online; and it is really not difficult to reconstruct my proof method from the summary given here. It would be an interesting exercise and I would be glad to try to answer any questions about this method if you have doubts.

However it should be noted that the stipulations are a bit difficult to remember; when drawing a Pythagorean-styled diagram of square areas, it must be remembered that in this proof approach the square areas are not Pythagorean in that specific meaning, rather the sides or the squares are measured as x^(n/2) etc.

If you can accept that as a stipulation, and using the fact that the Pythagorean relation holds for any three square areas A, B, C, related such that A + B = C, it is rather simple to derive the proof. In other words, the square area diagram acts as a generalized model, as such the Pythagorean relation can be used, but with the stipulations always in mind. (It has to be kept in mind that the sides are not restricted to being x, y, z respectively, in the diagram system. Also, with x, y held constant and exponent n allowed to vary in the given equation, the diagram sizes vary and, the (averaged) diagrams are nested in order to take advantage of geometric equivalence. I hope this is clear.

There are a few key steps: (1) Showing that the Pythagorean-styled square area diagram can be used to illustrate any true FLT-form equation, when each of the three large terms are interpreted as square areas; (2) Showing that the diagram can then be averaged; (3) Showing that diagram sizes can be varied even with x, y held as constants, with z as a dependent variable and n as independent variable; (4) Showing that the resulting diagrams can be overlapped or nested in a diagram system with shared origina and shared axes, with the z-ray or z-axis running at 45 degrees in the system (due to the averaging method used for the two smaller square areas).

It is via the averaging of the diagrams that the diagram system achieves full generalization. I had the first (unaveraged) diagram in 1980, but did not realize the advantage of averaging diagram areas until 1997. Without averaging the diagram areas, no proof can be found by this method because the diagrams cannot be geometrically equivalent for all true FLT form equations, without the averaging of the two smaller areas.

Also averaging puts all values of z along the z axis, making z accessible to Pythagorean analysis within the nested diagram system. This is why the derived equation (the second equation in my paper) is effective; the derived new exponent "a" in the second equation is a primordial component of the system whenever exponent "n" is allowed to range above 2. In order for "n" to exist, it must have the full continuum range from origin 0 of the system, all the way up to n. Then in that range we have exponents a and 2, and these are all related such that a < 2 < n.

Remembering that we defined the exponent as independent variable, and z as dependent variable, with x, y constants, we are forced into a contradiction because of the given and derived equations of the system. Geometrically we have a situation where a number z must be greater than, equal to, and less than itself, simultaneously, in order to fulfill the geometric requirements in the 2D system when n > 2. This contradiction does not arise when n <= 2. It should also be remembered that the the Pythagorean theorem is not usually dealt with in this manner (holding x, y as constants); this is very unusual. One might wonder why this is done. The answer is that the entire line of geometric analysis was only developed for one specific reason, which was to solve FLT. Therefore, the rather unconventional approach was found to be necessary to complete the proof. It is hard to accept from the very beginning because it is not the typical way of using the Pythagorean theorem. Sorry, that's the only way that worked.

What I am trying to say is that as bizarre as this method looks, I didnt use it to be different, but rather to engineer a working proof. It was difficult for me but I thought it could be done in a simple method. I spent 10 years (1980-1990) working usually from 2-6 hours per day, mostly on dead ends and erroneous algebraic calculations. Then again from '93-'97 I was working on it occasionally and was relieved to find that Andrew Wiles was not using any methods similar to what I thought might work. I think it was about September '96 when I realized the diagrams could be averaged.

Posted by: jay dillon

In regard to the averaged diagrams, each true FLT-form equation having n > 2 can be made to fit the generalized, averaged diagram system by averaging the two smaller terms:

z^n = (x^n + y^n)/2 + (x^n + y^n)/2

z = [(x^n + y^n)/2 + (x^n + y^n)/2]^(1/n)

and also from the derived equation:

z^2 = (x^a + y^a)/ 2 + (x^a + y^a)/2

z = [(x^a + y^a)/ 2 + (x^a + y^a)/2]^(1/2)

such that a < 2 < n

but due to inverse relation between n and z when x,y are held constant:

z_xyn < z_xy2 < z_xya

Hopefully i did not make typing errors here.

Please note that simple equations like

3 + 4 = 7

15 + 5 = 20

are seen as being true FLT-form equations having exponent n = 1. Then 3, 4, 7 can be seen as square areas, and 15, 5, 20 can be seen as square areas, fitting a Pythagorean-styled diagram in each case. Averaged diagrams would have averaged smaller areas: 3.5, 3.5, 7; and 12.5, 12.5, 25.

Posted by: jay dillon

Posted by: jay dillon

WSEAS editors will allow me to make corrections to my paper and submit the corrected manuscript for refereeing by their top mathematicians. Previously the paper was accepted by the same referee process by 3 anonymous referees. WSEAS staff advised me that if my new rewrite is perfect and is accepted as a valid proof then I will be "one of the strongest mathematicians of the 20th and 21st centuries." Needless to say, I hope that I can make all the needed corrections. No critiques were provided; so I just have to rewrite to the best of my ability and look for problem areas remaining. The paper was listed and published for about 8 months, but is not presently listed in WSEAS Transactions on Mathematics Table of Contents for July 2004. If you wish to see a copy of my uncorrected paper as it presently exists, in the form copyrighted in 1997, please write to me at jcfdillon@hotmail.com . The proof method is shown and completed in the first four pages; remaining pages, where some errors occurred, are "Supporting Arguments and Discussion," and are not vital to the rigorous proof. The paper is accessible to any math reader, not requiring any advanced math training, but will require some exercises and tests to understand the method and avoid erroneous assumptions that would violate the stipulations used.

Posted by: jay dillon

I posted here because I thought that those understanding NKS techniques might find ways to combine my geometric approach with NKS. I think that my postings here are off-topic in that they do not directly discuss NKS methods, but hope my paper will be of interest. Thank you. (I have almost zero math graphing training, and have had trouble providing clear, accurate diagrams and graphs for my previously published paper.) I am not sure how or if my "nested diagram" methods might combine with NKS graphing techniques.

Posted by: jay dillon

This year I submitted several new papers for refereeing to math journals. These include a completed proof of the Goldbach conjecture(!), a proof method for the Collatz 3x+1 conjecture, and a proof of the Beal conjecture.

Of these, I believe the first two are completed, the last is in progress and I feel hopeful that it can be completed soon.

The Collatz paper is 3 pages long, extremely simple (hopefully correct-- i think there is a good chance it is correct.. or that the general approach can be redone correctly); the Goldbach paper is 9 pages and could be cut to perhaps 4 pages or less. The Beal paper is 6 pages and can probably be cut to 3 or 4 pages, if everything works out as hoped (verification still in progress; I had misinterpreted my own method, as the referee pointed out). But the Beal proof method may still be effective if I can treat it properly.

Referees responded negatively to both the Goldbach and Beal papers, but I provided a clear and very effective reply to the referee on the Goldbach paper, and feel that the proof is now very strong, and complete (inescapable really--it's almost bizarrely powerful). I am hoping that this incredibly tantalizing problem can now be settled conclusively, and by a very accessible method. Presently starting work on the Twin Prime Conjecture but haven't made any clear progress yet.. just found a few seeming hints or clues..

The Fermat paper, previously published in WSEAS as their paper no. 10-232, was subsequently withdrawn by the publisher, an event that I only learned of when a reader contacted me and notified me that it was not available.

Not sure what prompted this reversal, other than things like a referee's recent threats to cancel library subscriptions if WSEAS considers my Fermat paper any further--; the editor said that an error was found in my published Fermat proof, but the new referees did not discuss any math errors at all, and my latest rewrite was rejected by WSEAS this summer. I did not receive any math critiques at all, over the past year that my Fermat paper was available to the public. All emails over the past year, relating to my published paper, were discarded by the editors, and were not forwarded to me. So, if math critiques exist for this paper, I have yet to see them.

So, I will attempt to cut and rewrite the Fermat paper also, and submit it to another journal within a few months; the problem being of course that the paper is already "published," so some journals may not want to consider it.

Posted by: merab

Dear All,
I think I find same solution as Phermat for Phermat’s last Theorem.
I need just 3 lines for writing it.
Whom and were I have to send my solution ? I need my right to be protected.
Regards,
Merab

Posted by: jay dillon

If you are serious, you should certainly typeset your brief proof and send it as required to Library of Congress for your copyright. Don't share it with anyone until you copyright it. You can also self-mail the manuscript in a sealed envelope (sealed officially by the post office). The manuscript needs to be dated by the post office and then when you receive the manuscript, put it in a very secure location. This is supposedly a method of self-copyright that is widely accepted as valid.

****

My Fermat manuscript is in limbo after having been published by WSEAS in July 2004. I am not sure that many readers understood my method, but I believe it is workable if carried out rigorously in accordance with the legal stipulations.

I recently submitted the Fermat (FLT) paper and my 2005 paper on Goldbach to a research journal in Sept. 2006, but have had no response yet. The papers were acknowledged as having been received by the journal editor, but I do not yet even have verification that the papers were then forwarded to any math referee(s). Because of this 6 month delay without any reply I may be forced to give up on the journal and try to submit elsewhere. However I would prefer to allow plenty of time for editors and referees to look at these papers.

As readers of NKS Forum may know, it is a bit difficult to publish a paper that has already been published and withdrawn from another journal without the slightest math critique forwarded to the author! In fact this is a gray area from hell, which I hadn't even considered as a remote possibility.

The Collatz paper is still not published and not critiqued in any mathematical sense because the referee who previously looked at it simply stated that my proof was not a proof. He or she stated that my proof of the Collatz conjecture is simply a drawing and a proof claim without verification. However, the drawing is more than just a drawing in that it is "fully generalized," i.e., applicable to any and all true cases of the conjecture.

This full generalization combined with a logical argument is in itself the proof argument; yet this argument and method was not explored or "deconstructed" as they say, in the "critique" provided by the math editor. Therefore, I can say that my Collatz paper has not yet received any math critique, whatsoever.

It is not surprising that my papers are treated contemptuously. Certainly if I were a highly trained and qualified mathematician, I would most likely give little credence and take little time with anyone who claimed a short proof of Fermat's Last Theorem, the Goldbach Conjecture, and the Collatz Conjecture. (This is "blasphemy" in mathematics, and the word is not overly dramatic to use when one experiences this difficult situation.) However, having published a paper, I would have hoped for a math critique from one or all of the papers I had submitted, especially the one which was published and then withdrawn under threats of boycott by library subscribers. (Apparently there were secret or sneaky critiques, but none of the critiques were shared with me.) I fully understand this. People have disdain for my paper on Fermat's Last Theorem, and I simply want to see the arguments against my method and see if I can improve my work. I do not doubt that I can be wrong.

Similarly, in a brief comment regarding my paper on the Goldbach Conjecture, a referee said that my paper stated the method is fully generalized, but did not prove that the method is generalized. In fact that referee said that I had provided a proof for only one number, an even more blatant insult. But the method is indeed generalized, was clearly designed as a fully generalized method, as stated in the abstract, and that is exactly what it is. I can show this full generalization, and did show it, in an addendum provided in June 2005; and via the full generalization we proceed directly to a simple proof! This is astounding to me as well as to anyone who reads the paper. I don't blame people for being puzzled by the impossible.. However the editors of that journal had by then precluded any further consideration of my Goldbach paper, and I was not allowed to have my rebuttal read by the referee. (My rebuttal was criticized for being confusingly written.) But every step of my rebuttal is, or should be, very easy to develop from the method given in the full manuscript. Here's the problem: The ease and clarity of any proof of an ancient mathematical problem is the terrifying mental block which the referees face. They must work around it or just avoid it by rejecting the manuscript without critique. So this is a common problem in math, and partly explains why more and more complex proofs in mathematics are developed even for "simple" or "simply stated" problems. The "simple" or simply developed arguments and methods to prove such problems, are then "the problem" that must be destroyed or ridiculed "by any means necessary." I don't know if this is clear to NKS readers, but "simplicity" on what is acknowledged to be a fiendishly difficult problem in number theory, is simply "blasphemous," and is aggressively treated as such.

An attempt to finalize a proof on the Beal Conjecture hit a real snag, because I was very hasty and neglected to test my method completely and use proper diagramming techniques. However, I believe my general approach was initially sound but not adequately developed through all the stages necessary for full generalization and for a full proof. So in that case, the referee(s) were completely correct, and I am presently focusing my efforts on the Beal Conjecture, because I view it as the "next" problem that is appropriate for me to attempt, and because I see my work on FLT and Goldbach to be completed unless and until a math critique on those papers arrives for my further consideration. I don't know how to respond to nonexistent math critiques though; this is very painful and difficult. (And I believe other FLT provers out there may have faced the same difficulties in the past; I only know my own method to the extent possible at this time, and only want to complete or improve it, rather than working through "simple" FLT proof attempts by others. This is because I am not really a mathematician, rather, a hobbyist and an amateur, and my FLT proof was developed for the sole purpose of proving FLT, not as a pinnacle of my achievement in the field of number theory, etc.

Interestingly, it was in the case of my Beal paper that the referees provided enough feedback to indicate where the proof broke down; I may have similar embarrassing experiences if referees will show where my other proofs may break down. Anyway, I am still willing to try to improve these papers.

I cannot send out copies of any math papers now because two are submitted to a publisher and the others are in early development.

I seriously doubt that any math experts are even looking at my work. Anyway I will be glad to correspond with amateur mathematicians on these topics, although I may not want to be very detailed at this point.

Copyright everything though, that's one thing you learn as soon as you write anything at all of import, you are surrounded by ripoff artists and scoundrels.

My FLT paper was started in 1980, was submitted to two journals and published in the second one it was submitted to. I never copyrighted or submitted any other math paper previous to this one that I published in July 2004.

Drawing a 2D square area to map or represent a number is all it takes. Try it. Use this on 5^2, 6^2, 5^100, 6^100. Represent these values as 2D square areas. Then start thinking...

A trained mathematician will say, well you cannot draw a square to represent 5^100, because a square would be 5^2. This is "rote learning." This is where the stipulations come in. We stipulate certain facts for certain reasons, in order to generalize and attack a problem.

Furthermore, a mathematician will say, you cannot hold x,y constant in the given equation of FLT, because the diagram sizes change as the exponent increases (when the large terms x^n, y^n, z^n are drawn as or mapped as square areas). Well, a mathematician who cannot allow a simple stipulation in an engineered proof argument, is not a mathematician at all. I can say this: a square area containing 5^2 square units is smaller than a square area containing 5^100 square units, or a square area containing 5^1,000,000,000,000 square units, but the root is the same. The root is 5. This 5 is what we may view as a constant in an array of nested, geometrically equivalent squares with constant vertex at one corner. These overlapping diagrams share the same root of 5 units. So in my paper I presented the stipulation that the x,y values will be held constant; they may be any positive naturals, and may differ "vastly" in value, but they must be held constant in the method of my proof. ("Vastly" is an understatement. Perhaps readers of my paper assumed that I had considered only constants x = 3, y = 4, which I used in my example. However, these values may differ by counts equalling all the grains of sand in the universe. This is the beauty of a "fully generalized approach" in number theory. But none of this was stated in my published paper in WSEAS July 2004 (later withdrawn by the publisher).

Certain aspects of my nested diagram system may be only somewhat clear or developed by myself as author. I do not know if any of the readers understood the implications of the "fully generalized" nested diagram method which I have shown in my paper. For example, with constants x = 3, y = 4, as exponent n approaches positive infinity, the value of z decreases toward min of z = 4 when n = +infinity. This may seem bizarre but it is useful under further stipulated and engineered arguments for this proof. Similarly, if x = 3, and y = 1,000,000,000,000,000,000, then as exponent n increases without limit, the value of z decreases toward min z = 1,000,000,000,000,000,000 which is what y is held constant as. In other words, the generalized proof approach gives a geometrically useful method of proof! I don't know why others don't see this as "interesting." But, this method works only if the two smaller areas of the diagram are averaged! It took me a good 17 years to go from the first diagram to the second, averaged diagram, and then when published in WSEAS, neither diagram printed correctly, because of a color/B&W printing error in typesetting (I take full responsibility.) So the puzzle is one in which the target of full generalization was developed and attacked over a period of roughly 20 years (1980-2000 approx, with a 3 year hiatus 1990-1993 because I started thinking I was getting obsessed with FLT). (I restarted a few months before I heard about Andrew Wiles's proof.)

The use of "full generalization" and holding some values as constants, was emphasized by Polya. I don't claim to know anything about his advanced work, but I did notice that emphasis in one of his books in 1980. Of course my published paper on FLT fully utilizes these two key methods; without them, there is no proof possible whatsoever.