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Fermat's Last Theorem Proof in WSEAS

  1. Sep 24, 2004 #1
    My paper is entitled "Fermat's Last Theorem: Proof Based on Generalized Pythagorean Diagram." This is published in WSEAS Transactions on Mathematics, July 2004, which is issue 3, vol. 3. My paper is the first in the issue, and is designated as paper 10-232 in WSEAS Transactions on Mathematics. The journal is listed in ISSN as ISSN 1109-2769. Info can be found at http://www.wseas.org All WSEAS Transactions on Mathematics papers are reviewed by 3 independent referees prior to publication.

    The journal issue was released to the public Aug. 20. Since then I have tried various outreach methods such as posting in newsgroups like this one. A summary of my paper is provided on http://www.mathforge.net Comments and critiques welcome. Also, if anyone wishes to help publicize this paper for purposes of gaining critiques and hopefully further academic review and possible endorsements supporting the method, I will appreciate it.

    Please pass this info along to anyone who may be interested, such as other physicists, math/physics experts, and mathematicians including recreational mathematicians, amateurs and school classes on math where this topic might be discussed.

    I will try to answer questions posted on MathForge.net which is another excellent place to discuss these things. I will check back here also.

    A corrected illustration page is available for the published paper. See http://www.geocities.com/jcfdillon/crx.doc

    Thank you for your consideration.
  2. jcsd
  3. Oct 22, 2004 #2
    Dont All Talk At Once

    Dont All Talk At Once
  4. Oct 22, 2004 #3
    hasn't this already been known?

    is there a pdf anywhere? :confused:
  5. Oct 22, 2004 #4


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    If you had provided a link to your paper it might have generated a response.
  6. Oct 23, 2004 #5


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    Andrew Wiles was the first to claim [credibly] a proof to Fermat's Last Theorem in 1993. This is one of the first papers regarding his proof.
    Last edited: Oct 23, 2004
  7. Oct 23, 2004 #6
    Where is the paper ?

    If you want people to read your paper, please put it in a place so that it is easy to get it. I've spent 15 minutes searching it with no success. Please provide a direct link.
  8. Nov 15, 2004 #7
    My paper is available in WSEAS Transactions on Mathematics, July 2004. Please see http://www.wseas.org and look for Journals / WSEAS Transactions on Mathematics / July 2004. This will bring you to the Table of Contents for the issue. The issue can be ordered from WSEAS.org or can be requested via Interlibrary Loan from a subscribing library. The paper is not freely distributed as a .pdf file due to copyright restrictions. A corrected illustration however is available online at my website http://www.geocities.com/jcfdillon/crx.doc Several people have requested free .pdf files. I am sorry that I cannot freely distribute this article. WSEAS was courageous enough and openminded enough to consider my article and publish it. They should be respected in regard to their copyright restrictions. I spent 24 years working on the problem starting in 1980 and was encouraged by the Wiles paper although I could not read it or understand any of it. In 1997 I made a breakthrough, going back to my original diagramming technique which I found in 1980 after ten days of work, averaging the orginal diagram as shown in the corrected illustrations, and then quickly finding a proof by my simple method. The WSEAS referees apparently agreed it represents a proof, placing the paper first in the issue. I agree, everything should be free and easy but it isnt always that way. Nobody expected a proof by anyone, few can understand the 94-95 Wiles method in depth, and no one expected a proof by any simple method but there it is. I am still awaiting the critiques promised to me by the publishers. I am not sure what the holdup is. The paper was made public Aug. 20, 2004.
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  9. Nov 16, 2004 #8

    matt grime

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    Why do you keep posting this since no one has a subscription to this journal here? I looked at the TOC, or tried to, but it was in .doc format, which immediately makes me think this journal doesn't have a clue (as if publishing amateur attempts at FLT wasn't a big enough warning anyway). There are several very reputable journals out there that are respected in mathematical circles. If you wanted recognition you should have tried publishing in one of those instead. Opening the .doc in emacs and reading between the junk it appears the journal also contains papers on:

    number theory (yours)
    non-linear analysis
    game theory
    graph theory
    applied quantum mechanics

    which also makes it less than attractive if it is so unspecialized.

    I doubt that anyone will take the time to read it or obtain it since you are tarred, perhaps unfortunately, by all the other incorrect amatuer attempts at FLT.

    Inicidentally, whilst the Wiles proof may be long and complicated at times it is understood by far more people than you appear to think.
  10. Nov 22, 2004 #9
    The paper was previously sent to more established journals starting in 1997. However, I like WSEAS because it is headquartered in Greece and so it's a good match. They requested the paper in early 2004 without my contacting them. (My paper uses only 2D Euclidean/Pythagorean geometry and related algebra.) Whether all the articles in WSEAS Transactions on Mathematics should be on number theory is not for me to judge. More established journals tend not to read or seriously consider any papers on extremely controversial topics much less papers on top problems claimed to be solved by amateurs without advanced math degrees-- much less problems of this sort solved using clear and easily understood math in only a few pages and a few lines of algebra. AMS/Denver angrily refused to look at my paper; a friend of a friend who has an advanced math degree said he never even would have looked at it if he had known the topic; Annals of Mathematics at Princeton stopped considering any papers on FLT years ago; getting any referees to look at or read papers on this topic is nearly impossible for all the obvious reasons, and of course few math amateurs or general readers would take the time although many would have the ability. It took me 17 years to find the proof method, then another 7 years after completing the paper to have it published, and during most of that time, the paper was not looked at by anyone even when it was sent to publishers. The Wiles paper is well accepted but most people cannot read it, whereas my method is understandable at grade school level. This makes it more, not less repulsive to advanced mathematicians, but some do take an interest and try to read it, including the three referees at WSEAS. If the method is correct and useable then it will be readily perfected and demonstrated by anyone who takes an interest and it will gradually be shared and acknowledged by this method. I don't expect advanced mathematicians to look at my paper and so I am very glad that I am finally published so that those who take real interest in this specialized topic and approach will consider it fairly.
  11. Nov 22, 2004 #10
    If it's only a few pages why not just write it up in tex, make a pdf, and then distribute it here? It would only take about a couple hours to do (tops). That's probably the only way you're going to get anyone here to read it.
  12. Nov 23, 2004 #11

    matt grime

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    Probably because he cannot for legal reasons redistribute now the journal has it.
    Not that anyone here of sufficiently high standing in the math community is going to bother looking at it. A perhaps sad state of affairs but as it's a proof by picture by the sound of it I doubt it is at all rigorous, and probably badly written by maths standards - we have a notoriously short attention span for such things - after all if it's so easy why is the proof so disguised sort of attitude. See sci.math responses to james harris
  13. Nov 24, 2004 #12
    Draw three square areas representing x^n, y^n, z^n, respectively. Now if these values conform to the Fermat equation, they can be placed in the form of a Pythagorean square area diagram. Do this. Note that x, y, z respectively are shorter than square roots of each. The diagram represents any true FLT-form equation in which we consider naturals x, y, z, n. This is an arbitrarily engineered diagram; I do not claim falsely that the square roots of each square area are x, y, z respectively, which would be false. In fact, the square areas have roots x^(n/2), y^(n/2), z^(n/2) respectively. Interestingly this diagram is a true representation for any true FLT-form equation including the primordial diagrams having exponents 1 or 2, such as 3 + 4 = 7 in which case the exponent is 1, and 3^2 + 4^2 = 5^2 having exponent 2. Note that this type of diagram can be drawn, in fact, for any equation of the form A + B = C as long as the three large terms are represented as square areas in this arbitrary fashion. The FLT equation is of this sort, and is to be interpreted as described above. Now, it is of little use to have this diagram as relative areas x^n and y^n may vary greatly. (cont'd)
  14. Nov 24, 2004 #13
    Because we cannot proceed to full generalization using this sort of diagram, it might be considered useless. But after some time I realized that the diagram can be helpful but only if the smaller square areas are averaged. Do this: we then have averaged areas summing to the same larger area z^n. The beauty of this type of Pythagorean diagram (or Pythagorean-style diagram) is that we now have a useful means of proof. I could leave you to do this but having been accused of deliberate obfuscation I continue. If you construct a system of nested Pythagorean-styled diagrams averaged as just suggested, you can use the same origin for all the diagrams, and keep x, y as constants shared by all diagrams. Then the diagram size will vary only when the exponent varies, and we say that the exponent n is an independent variable, with z being a dependent variable in the system. Do this. Let constants x = 3, y = 4, with exponent n varying smoothly through the positive reals. We find that the value of z is inversely related, decreasing with increasing n. In the diagram system, we can now draw z as a line segment of decreasing magnitude along the 45 degree z-axis of the system, while with all diagrams averaged, the Pythagorean-styled nested diagrams increase in area with increasing values of the exponent in the given equation of FLT. (cont'd)
  15. Nov 24, 2004 #14
    Due to the use of averaged diagrams, we can now see that even with huge disparity between the x and y values, the averaging method provides a way to generalize the proof method. Also, regardless of the magnitude of the exponent n in the system, we trap the endpoint of the z ray on the 45 degree axis of the nested diagram system. Further, when n increases, z decreases, so endpoint z as measured from shared origin 0 of the nested averaged diagrams, is located between y and z_xy2. In the case of x = 3, y = 4, for example, this means that regardless of the magnitude of the exponent, endpoint z_xyn is located between 4 and 5. In fact, y is limit of the function as n approaches infinity and z decreases from 2^(positive infinity) to 4. (I will let you figure out why i use the expression 2^(positive infinity) here. Anyway, when you start off from (3^2 + 4^2)^(1/2) = 5 and increase the exponent to 3, 4, 5, ... etc. you have shrinking z values "until" reaching limit y = 4 at n = (positive infinity). This is just one example, one doesnt prove anything by examples, but the example here is fully extrapolatable. The diagram system works and the WSEAS editors and referees seem to agree that it works. I do not know if this sort of nested system of Pythagorean-style nested diagrams has been used before. The method is readily testable however.
  16. Nov 24, 2004 #15
    Ok now once the nested diagrams have been drawn and tested, we have to generalize for all possible x, y. As stated above, the method is fully extrapolatable, so it may be redundant to do this, but anyway one can graph for one example pair x = 3, y = 4, or whatever, the curve resulting in the z value as the exponent n varies from near zero toward positive infinity. We then have a curve descending from near the vertical axis of the z values, toward the horizontal axis of the n values but leveling off at a horizontal asymptote at y = 4, which is limit of the function as n goes to infinity. Again, this sort of graph can be drawn for x = 3, y = 4 and is fully extrapolatable, but still we seek a more generalized form. (contd)
  17. Nov 24, 2004 #16
    A generalized form is suggested by the averaging process itself. The averaging process makes the diagram system scale independent. Once the averaged diagram system is set up, we have the z axis running at 45 degrees in the Pythagorean diagram system of nested diagrams. The z axis is measured in standard units from origin, and has a constraint on it due to the system of diagrams it inhabits and conforms to. This constraint is based on the standard Pythagorean diagram form: When z exists in this system, as exponent n increases, acting on the constants x, y, any value of z must be square root of z^2 and simultaneously must be nth root of z^n. Further, in the averaged system, z^2 = (z^2)/2 + (z^2)/2 and also z^n = (z^n)/2 + (z^n)/2. But remember that as the diagram was increasing with increasing n, the value of z was decreasing along the 45 degree z-axis of the system, toward lim (minimum) at y = 4, in our example with x = 3, y = 4. How can z exist as both square root of z^2 and nth root of z^n? Answer: It cannot. This is a number which cannot exist, in the engineered system we constructed. In my paper I showed that the curves resulting from two intermeshed equations require z = z_xy2 = z_xyn, which is the contradiction solving FLT for all n > 2, whereas the diagram system does allow z_xy1 and z_xy2 solutions to exist without contradiction.
  18. Nov 24, 2004 #17
    It may be hard to grasp why this system of diagrams was developed and used. Test the drawings and use the method yourself and it basically proves itself. You have to be aware of the stipulations used in the method, it is not traditional to say that a square area can be described algebraically as 13^(1,232,634) or whatever, but if x = 13, and if the exponent is the large number, in my method this would simply be represented as a square area having each side equal to 13^(1,232,634/2), and if x = 13, then in the Fermat equation x would be a constant, if you follow my stipulated method. There is no need for calculations using large numbers however, because a generalized method is shown and proven to be fully generalizable, and fully extrapolatable regardless of the magnitude of the exponent in the system. Again, via averaging of the smaller areas representing x^n and y^n, generalization of the nested diagram system really is completed. This is why I copyrighted the method when I found it in 1997.
  19. Nov 24, 2004 #18
    The two key equations in my paper in WSEAS are:

    (1) the given equation of FLT with z isolated on left:

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

    (2) the derived equation found in the nested averaged diagram system:

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

    such that a < 2 < n

    Then graphing Eqs. 1 and 2 simultaneously, as the diagram system requires simultaneous solution of these equations, we find that any true FLT-form equation with n > 2 must simultaneously have

    z = z_xy2 = z_xyn = z_xya

    while in fact we have

    y < z_xyn < z_xy2 < z_xya

    (This is the contradiction proving FLT for all n > 2, whereas this geometric contradiction does not develop with the exponents 1 or 2, as is also shown in my paper in WSEAS.)
  20. Nov 24, 2004 #19


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    I'm still trying to figure out just what you're saying, but as far as I can tell you never use the fact that x, y, z, and n are integers anywhere in your argument. This is a severe problem because, for instance, 3^3 + 4^3 = z^3 does, in fact have a (real) solution, as does 4^n + 5^n = 6^n.
  21. Nov 24, 2004 #20
    When z is considered in single-variable fashion in the diagram system as described above, there is no need for further consideration of the original tested x,y values used as constants in this proof approach. This is due to the fact that in the same averaged diagram system we can find values for half-square areas associated with x^2 and x^n respectively, and can find through Pythagorean analysis of their sides, how z itself is constrained within the system of nested diagrams. For example, in the averaged diagrams, z^2 is the sum of the two smaller half areas (z^2)/2 + (z^2)/2, each respectively having sides [(z^2)/2]^(1/2). If we now take z as constant, we cannot increase the size of the diagram; thus cannot consider higher values of the exponent in the given equation of FLT. If we take z as dependent variable, then it cannot hold the value determined previously, and must decrease with increasing exponent values in the system. So, a system of nested diagrams constructed in this fashion using the stipulations of the proof method, cannot exist solely because the conjecture of FLT is true; there are no all-whole-number solutions when n > 2.
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