Fermat's Last Theorem Proof in WSEAS

[jcfdillon]

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.

 

[jcfdillon]

Dont All Talk At Once

Dont All Talk At Once

 

[gazzo]

hasn't this already been known?

is there a pdf anywhere?

 

[Hurkyl]

If you had provided a link to your paper it might have generated a response.

 

[Chronos]

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.
http://www.ams.org/notices/199507/faltings.pdf

 

[T.Rex]

Where is the paper ?

The journal is listed in ISSN as ISSN 1109-2769. Info can be found at http://www.wseas.org

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.
Tony

 

[jcfdillon]

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 isn't 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.

 

[matt grime]

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
statistics
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.

 

[jcfdillon]

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.

 

[cogito²]

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.

 

[matt grime]

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

 

[jcfdillon]

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)

 

[jcfdillon]

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)

 

[jcfdillon]

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 doesn't 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.

 

[jcfdillon]

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)

 

[jcfdillon]

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.

 

[jcfdillon]

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.

 

[jcfdillon]

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.)

 

[Hurkyl]

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.

 

[jcfdillon]

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.

 

[jcfdillon]

The real number z as may be found by calculating is not stable, it is chaotic, as may be found by testing the given equation with exponent 3, in the reiterated function z(resultant) = (x^n + y^n)/[z(initial)]^2 The first calculated value of z, when plugged back into the equation, quickly makes the function chaotic, so the situation is a bit worse than simply finding that z cannot be an integer. It's not even a real number, in my opinion.

 

[jcfdillon]

Thanx for reading this-- I have not been working on this problem directly for a while because it tends to set me off. but i am glad some people are interested. i am working on another problem and may have something ready to send to a publisher shortly. The Fermat paper has illustrations which are fully generalized. I am not sure if anyone really can see how this generalization helps solve the proof. To me it is clear, but it leads to confusion when you first say a square area can be valued as x^13 or whatever, this is not standard. Then combining the other areas together, leads to more confusion. Let x^13 + y^13 = z^13. It is not readily obvious that this is provable using Pythagorean geometry. but it is obvious that any natural numbers that can be precisely jammed into the standard Pythagorean form, can be precisely analyzed using it. In fact what the Pythagorean diagram form does is link 3 values x, y, z and one exponent n which may be a positive real or a natural. I believe the same diagram can also be used to analyze negative numbers as well, for any FLT form diagram, but I have not explored this enough. There are many ways to get disgusted by this method, but I would encourage any readers to try it, even though it leads to some cognitive dissonance. For example although the line segments do not seem to work correctly, if you draw them out, measure them and test them, they do work out precisely, and can be hyperanalyzed and graphed. The method I use is not complex but I do not feel I have done it justice yet, the diagrams and graphs need improvement to show the full precision of the method.

 

[jcfdillon]

Also as to the topic of scale independence, this is rather alarming to anyone, but a simple diagram that can illustrate the FLT equation did not exist as far as I know, previous to the square area diagrams I used in my paper. These diagrams are bizarrely simple, just analogous to the Pythagorean diagrams, and they are nested. Using averaged areas, the nested diagrams can be made to force the proof. I think of it as trapping z so that it can be examined, almost like trapping a fly in a bottle, so it cannot escape. The only way it can escape is to not be there, and that's what happens. I had thought this method might have some relation to physics problems such as the Bell inequality, or the Einstein-Podolsky-Rosen paradox, Yang-Mills, etc.

 

[jcfdillon]

Envision a pythagorean diagram. the two smaller areas are then averaged. now envision a series of smaller averaged diagrams inside the first like Russian dolls, geometrically equivalent. Also envision a series of larger diagrams getting bigger, and all these share the same origin and same origin 0. Now imagine that there is some diagram in this series illustrating the simple Pythagorean triplet 3, 4, 5, with exponent 2, but with the smaller areas averaged, so that this diagram is geometrically equivalent to the smaller and larger diagrams. Imagine holding 3, 4 constant, while allowing the exponent to increase or decrease, to make larger and smaller nested diagrams, all averaged and always sharing the same origin 0 and same z axis at 45 degree angle in the system of nested diagrams. Now as exponent n increases with x and y constant at 3, 4 respectively, and with all diagrams automatically averaged for geometrical equivalence, you have a series of nested diagrams. The diagram size increases as n increases, while z decreases toward min 4 which is the larger of the values of x, y (this leads later to the need to use z in single variable analysis in a similar nested diagram system but with x, y simply removed, and with averaged areas (z^2)/2, (z^2)/2, z^2).

 

[jcfdillon]

The number "a" is a "primordial" exponent which is found when analyzing z in the nested diagrams. This value "a" is part of the independent variable n as n ranges through the positive reals. When n increases in the system, it creates larger diagrams. The smaller diagrams are still there; the constants x, y are still there; and when we look at the value of z with higher n we find that it is associated with an inner-nested diagram based on this particular number "a."

 

[jcfdillon]

In fact:

(z_xya)^a = x^a + y^a

 

[jcfdillon]

Also:

(z_xya)^a = (z_xyn)^2 = x^a + y^a

the value "a" is a primordial component of n. so as n ranges from near zero to 3, for example, with x, y held constant, the process "produces" the value "a." This is no different from saying that in the range from 0 to 100, there is a number 27 resulting from going from 0 to 100 without skipping any integers, except "a" would be termed a "real" number, not an integer.

 

[jcfdillon]

a < 2 < n

z_xyn < z_xy2 < z_xya

(with x, y held constant according to stipulations in the proof method)

then,

for example let x = 3, y = 4, and if we have n very large, we have z very close to y = 4, we have a huge number for z_xya.

These are just suggestions on how to test the method.

 

[jcfdillon]

It's hard to think of the diagram size increasing or decreasing with x = 3, y = 4 as constants, or x, y being any particular constants one may choose, however, this is the method used in the proof. The diagram size varies by changing the exponent value, but all x^n, y^n, z^n values, for all true FLT form equations, can be placed into an averaged Pythagorean-styled diagram of this sort. Then as the endpoint of z varies along the 45 degree axis of the system, we have a series of nested geometrically equivalent Pythagorean-style diagrams featuring the constants x, y, and these diagrams then create geometric conflict when n > 2. This geometric dichotomy or contradiction does not exist in the FLT form equations featuring exponents 1 or 2, because in those diagrams, the z value endpoint is not internal to any diagrams based on higher exponents.

 

[jcfdillon]

Ok suppose x, y are not integers, but are real numbers or rational numbers but not integers. But make them constants. If they are held constant, and the proof method is used as described above, the same contradiction will arise. The Fermat problem requires that we prove this for integers, but more generally, we can prove it for any number, simply noting that when x, y are held constant positive real or rational values not necessarily integers, the same dichotomy will arise; i.e., when n > 2 in the proof method, then we find a contradiction causing the dependent value z to become chaotic, and not being able to provide a solution to the given equation when the exponent is greater than 2 (even fractionally). The existence of calculated "real" solutions to FLT when n > 3 and with constant x, y positive reals, producing some real z value, does not mean that these numbers can escape the dichotomy; when these calculated values are reiterated in the given equation, they diverge and exhibit chaotic behavior. ln my opinion this is due to the fact that the geometric model is correctly analogous to the chaotic system of the given equation of Fermat's Last Theorem. By the geometric model then we can correctly show the contradiction inherent to the given equation, and this constitutes proof by contradiction. (As found by the simultaneous required solutions to the given and derived equations given above.)

 

[zefram_c]

I'm not a mathematician so feel free to ignore me, but did you just claim that for any integer n>2 and (real) numbers x,y there is no solution to x^n + y^n = z^n ?

 

[DeadWolfe]

lol

That's what Fermat's Last Theroem states. It's already accepted to be true. This person just has an aternate proof.

Although I believe x, y and z must also be integers.

 

[Hurkyl]

Yes, the requirement that the solutions be integers is rather crucial to the statement of FLT -- when the four parameters are allowed to range over real numbers, or even just one of them, it's a simple exercise to find nontrivial solutions to the equation $x^n + y^n = z^n$.

e.g.

$$1^3 + (7^{1/3})^3 = 2^3$$

and you can use the intermediate value theorem to prove

$$4^n + 5^n = 6^n$$

has a solution with n real. (Because $6^2 - 5^2 - 4^2 < 0$ and $6^3 - 5^3 - 4^3 > 0$)

So if jcfdillon's argument does indeed conclude that the equation has no solutions over the reals (and it seems it must, because he nowhere makes use of the requirement that x, y, and z be integers, and he explicitly permits n to vary over the reals), then there is either a fatal flaw with his argument, or with the mathematical field of arithmetic and geometry.

 

[CrankFan]

Although I believe x, y and z must also be integers.

Yea that's just a minor detail...

 

[jcsd]

lol

That's what Fermat's Last Theroem states. It's already accepted to be true. This person just has an aternate proof.

Although I believe x, y and z must also be integers.

An alternative proof, especially a comparitvely simple one would however be something of note, most mathemaricians believe there is no 'simple' proof of FLT.

Of course as jcfdillion has claimed that his proof proves it for the more general case of x,y,z are real numbers, staright away his proof must be incorrect as we already know that it does not hold for such a general case.