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

[gazzo]

hasn't this already been known?

is there a pdf anywhere? :confused:

[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]

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 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.

[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 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.

[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 genral case.

[Hurkyl]

To be fair, that doesn't necessarily imply his proof is incorrect -- the other possibility is that arithmetic/geometry are logically inconsistent.

Either way, it cannot be counted as a proof of FLT.

[cogito²]

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 harrisI figured this much. I only said it in an attempt to get him to detail the proof a little more (and apparently it worked).

[jcfdillon]

I'm not sure I interpret everything correctly in my own method. But yes if x, y are held constant, the method should work because there is divergence found immediately in the "z" value when exponent n is allowed to vary over the positive reals. The use of positive real range for the exponent, in my approach, was just a way of exploring how z varies as a dependent variable, with x, y both constants as any given positive integers, or perhaps as any given positive real values, and with exponent n allowed to vary as an independent variable. I found that the z value varies inversely to n, but also found a simple method of illustrating these relationships geometrically in 2D, using a Pythagorean-styled diagram. Therefore, if the diagram is drawn and interpreted as stipulated, then it may represent a simple proof method in 2D geometry. This is not a small claim of course... (cont'd)

[jcfdillon]

So I understand that most mathematicians will obey statistical probability and ignore my work, and also will have extreme hostility toward it. In the above critique one person wrote an expression using the number 1 as the value for x, which I had thought was usually thought to be a "trivial" solution, whereas most people working on FLT concern themselves with "nontrivial" solutions and would not even consider the use of x = 1 in the given equation of FLT.

I had thought that most likely, the testing of real values as constants for (nontrivial) x, y will lead to the same contradiction found in my paper. However, I may be overreaching.. My paper did set x, y as constant integer values, as the first step in the stipulations of my proof method. I am not sure I can properly make all the claims I have mentioned on this website and my interpretations may be flawed. So if that is the case I apologize.

I was hoping some people might want to take my approach and run with it, test it out, and see if they can validate it. However, if people pop in, say here's the fallacy and use "trivial" solution values in combination with attacking my own overstatements, there's no way to make any progress.

The published paper is available but I am sorry it is not readily accessible to everyone. I think it might draw some critiques from actual readers of the paper but so far no one has presented me with any actual written critiques in any organized fashion.

If people here want to test it out on a blackboard and see where the method leads please feel free. I think if anyone draws these diagrams as I have described and then tests the behavior of the given and derived equations, the same divergence and dichotomy (contradiction) can be readily seen.

The general approach can be seen if you look at http://www.geocities.com/jcfdillon/crx.doc Then imagine that as diagrams increase in size with constants x,y and increasing real exponent n, for all resulting true FLT-form equations, ...(cont'd)

[matt grime]

There are seemingly non-trivial solutions in real x,y,z that your paper says can't exist, and in fact x=1 is not a trivial solution in any sense.

Your method has been shown to be wrong since it apparently implies, for instance that there is no cube root of 16 in the real numbers, which is obviously nonsense.

So not only is there a stastical knee-jerk reaction to your work, but said reaction seemingly turns out to be justified.

You may claim that x and y are integers (and z) but if you don't actually use any property of them being integer then that supposition is unnecessary.

[jcfdillon]

.. with averaged diagrams as shown, and with these diagrams overlapped with geometric equivalence (as with transparencies), using common origin 0, you can show that there will be two equation/curves requiring simultaneous solution:

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

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

such that a < 2 < n.

These curves can then be graphed for any specific given x, y constant values, and with n and z varying inversely to each other.

In the second equation, we find the "constant" 2 appearing in the exponent 1/2; also we have the exponent a, which is some specific positive real value representing the exponent in the system when we consider z^2 for higher exponent n. In other words when the exponent is allowed to increase beyond 2 in FLT, we find by geometric construction and interpretation that the lesser exponent a appears in the system as a result of the exponent ranging smoothly from 0 to 2 to higher exponent n. In the geometric construction with constants x, y we then have a situation in which the dependent variable z is forced to (but cannot) provide simultaneous solution to the given and derived equations.

[jcfdillon]

One interesting fact that I discovered, and which led me to think along the lines of using averages to complete the proof, is that there cannot be any solutions for the Fermat equation when the exponent is an integer 2 or greater, and with x = y.

If x = y in the given equation of FLT, then

2(x^n) = z^n

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

2^(1/n) = z/x

But for any solution to FLT with all integers, z/x must be rational, whereas, is it not true that for any n > 1, the value

2^(1/n)

is irrational?

[jcfdillon]

I certainly didnt discover this fact first, but discovered it for myself, i should say..!

[shmoe]

So I understand that most mathematicians will obey statistical probability and ignore my work, and also will have extreme hostility toward it.

I've read parts of your multiple posts. When I see things like "How can z exist as both square root of z^2 and nth root of z^n? Answer: It cannot." I tend to stop reading, and I suspect you won't find many mathematicians who will take you seriously with statements like this.

[matt grime]

Your discorver two posts back that you don't claim is original? Well, some of us might have pointed out that it was flippin' obvious otherwise sqrt(2) is rational (integer even). If you didn't sport even that, then as shmoe says, we ain't going to bother taking you seriously.

[jcfdillon]

There are seemingly non-trivial solutions in real x,y,z that your paper says can't exist, and in fact x=1 is not a trivial solution in any sense.

I think solutions using 1 in certain cases of any problem considering "infinite" or open-ended ranges, will be considered trivial solutions. The following definition of "trivial" is quoted from MathWorld:

Trivial


Related to or being the mathematically most simple case. More generally, the word "trivial" is used to describe any result which requires little or no effort to derive or prove. The word originates from the Latin trivium, which was the lower division of the seven liberal arts in medieval universities (cf. quadrivium).

According to the Nobel Prize-winning physicist Richard Feynman (Feynman 1997), mathematicians designate any theorem as "trivial" once a proof has been obtained--no matter how difficult the theorem was to prove in the first place. There are therefore exactly two types of true mathematical propositions: trivial ones, and those which have not yet been proven.

**********
Your method has been shown to be wrong since it apparently implies, for instance that there is no cube root of 16 in the real numbers, which is obviously nonsense.

I did not say there is no cube root of 16 in the real numbers. But why not test specific real numbers using my method, such as x = SQRT(2) and y = pi, (both held as constants, estimated to some specific degree of accuracy) and allow the exponent n to range through the positive reals in the given equation of FLT. Then z must vary inversely, decreasing as the exponent n increases, and the same dichotomy arises as detailed above.

******************

So not only is there a stastical knee-jerk reaction to your work, but said reaction seemingly turns out to be justified.

***********

I am just saying I dont blame people for not wanting to learn about my simple paper, when they know that so many FLT proof attempts have failed, and the one accepted proof is fiendishly complex. So my paper is akin to blasphemy.

***********


You may claim that x and y are integers (and z) but if you don't actually use any property of them being integer then that supposition is unnecessary.


If a dichotomous or divergent phenomenon is proven and shown for the Fermat equation, using constants x, y, then indeed it does not matter whether x, y are integers or positive real numbers of any kind, as long as x, y are held constant.

The proof's requirement for integer values to be considered and used, is simply the way the problem is stated and how it was historically defined by Fermat. However if the same dichotomy causing failure of solutions in integers for higher exponents, can be shown to exist using x, y as positive reals, then there is no need to specify or require that x, y, n, z must be integers simultaneously, in the completed proof. (Prove first for integers, then test the same method for x, y as some constant reals). This makes my proof unusual because via complete 2D geometric generalization we find the same dichotomous or chaotic behavior in the system using x, y as constants whether integers or not (but they must be held constant, by my proof method).

If some specific z value is found, solving FLT for some given positive reals x, y, then we still have the problem associated with the derived exponent a which is associated with an inner-nested diagram. Then one might say that the exponent a is both independent and dependent in the system; because exponent "a" is a primordial component of the exponent range from 0 to 2 to n, and a < 2 < n, and in the geometric construction, this primordial exponent a causes chaotic behavior in the system because z cannot simultaneously satisfy the given and derived equations. (The derived equation is from the geometric construction, which is generalized -- fully generalized -- and thus accurately represents all true FLT-form equations.)

[jcfdillon]

I've read parts of your multiple posts. When I see things like "How can z exist as both square root of z^2 and nth root of z^n? Answer: It cannot." I tend to stop reading, and I suspect you won't find many mathematicians who will take you seriously with statements like this.

This is hilarious. You act as though I dont know that a number can be square root of its square and also nth root of its nth power. The problem is not with individual terms like z^n, but with its existence in the given equation of FLT, which is more complex.

The given equation of FLT is:

x^n + y^n = z^n

so

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

Now z is the nth root of z^n.

z is also square root of z^2.

The problem arises in the fact that, in my proof method, z and n are inversely related, due to my stipulation of holding x, y constants.

This is the background of what I said. If you want to ignore stipulations in a proof method, you are not reading fairly at all.

[matt grime]

But there are infinitely (uncountably) many positive real solutions to the equation

x^n+y^n=z^n

for all n in N.

So your proof which allegedly shows these don't exist, is wrong. That is the reason why no one appears to be taking it seriously your proof can be used to show that something is false when it is actually true.

I don't understand what on earth your getting at with claiming a counter example with x=1 is trivial, and therefore not something to be considered. It is a counter example to a claim you made. It is *trivial* in the sense that it easy to show, however x=1 was not picked for any special reason other than it was a simple small example. (n=1 is trivial in the sense of FLT.)

[jcfdillon]

Your discorver two posts back that you don't claim is original? Well, some of us might have pointed out that it was flippin' obvious otherwise sqrt(2) is rational (integer even). If you didn't sport even that, then as shmoe says, we ain't going to bother taking you seriously.

I dont have a math degree, it was interesting to me that this relationship was there when x = y--, and it seemed relevant to my proof method when I noticed it in '96-- that was prior to going back to a diagram found in 1980 and simply averaging the thing! A simple step, but it had apparently never been done before, and best of all it led to a fully geometrically generalized approach for illustrating, diagramming and analyzing FLT.

How the heck did my paper get listed #1 in a journal headquartered in Greece. I still find it hard to believe, but the strength of the method is such that it can do all this even with a bunch of typos and inexpert preparation of diagrams and graphs on my part.

[matt grime]

I dont have a math degree,


No, I don't think that comes as a surprise.

Do you agree that your proof states that there are no real solutions to

z^3=2^3+2^3?

I just want to get this straight, because that's the conclusion some people have drawn.

[jcfdillon]

But there are infinitely (uncountably) many positive real solutions to the equation

x^n+y^n=z^n

for all n in N.

So your proof which allegedly shows these don't exist, is wrong. That is the reason why no one appears to be taking it seriously your proof can be used to show that something is false when it is actually true.

I don't understand what on earth your getting at with claiming a counter example with x=1 is trivial, and therefore not something to be considered. It is a counter example to a claim you made. It is *trivial* in the sense that it easy to show, however x=1 was not picked for any special reason other than it was a simple small example. (n=1 is trivial in the sense of FLT.)

If there are so many of these equations out there with solutions why not provide one, using some small numbers. I suggested sqrt(2) = x, and pi = 7. then, if you find a value for z with n > 2,

z = [sqrt(2)^n + pi^n]^(1/n)


For a simple solution we can find this value with n = 7:

z = (11.3137085 + 3020.293228)^(1/7)

z = 3.143271116

this is a little bit more than the original constant value of y = pi


(cont'd)

[shmoe]

This is hilarious. You act as though I dont know that a number can be square root of its square and also nth root of its nth power.

I'm only going by what you wrote.


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

Now z is the nth root of z^n.

z is also square root of z^2.

The problem arises in the fact that, in my proof method, z and n are inversely related, due to my stipulation of holding x, y constants.


I did read this and there is no condraction here. x and y are fixed, z is now considered a function of n (or vice versa), big deal.

[jcfdillon]

oops.. hasty typing.

[jcfdillon]

If there are so many of these equations out there with solutions why not provide one, using some small numbers. I suggested sqrt(2) = x, and pi = y. then, if you find a value for z with n > 2,

z = [sqrt(2)^n + pi^n]^(1/n)


For a simple solution we can find this value with n = 7:

z = (11.3137085 + 3020.293228)^(1/7)

z = 3.143271116

this is a little bit more than the original constant value of y = pi


(cont'd)

[mathwonk]

helloooo? where is this going? I suspect what we have here is another casualty of "calculator math", (and a failure to communicate).

by the way i also have an elementary proof of riemann's hypothesis, using only matchsticks, but there was not room in this "quick reply" box to hold it.

[jcfdillon]

anyway... lets see here...

suppose we have z = 3.143271116, as calculated.

Then in my proof method


z = (x^7 + y^7)^(1/7) (a real number, and this seems acceptable, at first..)

however also

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

with

a < 2 < n

And z cannot simultaneously provide solution for these two equations:

z = (x^7 + y^7)^(1/7)

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

because exponent "a" is a primordial component of exponent "n." (This value "a" can be calculated, is found to be a positive real, but the graphing of these equations and curves simultaneously shows the dichotomy:

the only common solution point of these curve/equations is at exponent 2 in the system, which is the excluded maximum of the system due to the fact that z and n are inversely related in the system.

[matt grime]

Erm. What the heck are you getting at? Please answer my question.

[jcfdillon]

In other words, finding real "z" seems innocuous at first, but this derived value z, when n > 2, must also fit the equation found via geometric analysis in the averaged diagram system, which is the second equation of my paper:

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

Via geometric equivalence, which is how we can derive and estimate "a" in the system, z is forced to fit these two equations simultaneously due to the stipulation of the proof method, that we hold x, y as constants.

Therefore, real or integer values show the same dichotomy and contradiction in the method, which again is extrapolatable and fully generalizable (via the averaged Pythagorean-styled diagram system) to all positive reals x, y, n, z.

One may stipulate that we consider only the natural numbers, but this proof actually goes beyond that. I have a section of my proof where via graphing it is shown that this method can be extended (perhaps) throughout the negative reals as well, though this needs work. hey dont we all

[mathwonk]

is there a "mercy rule" here like in softball?

[jcfdillon]

ok anyway the inclusion of positive reals seems problematic as people wrote above, and i agree it is weird, why cant we just have that real number there, and there's your solution, as in the very simplest case

3^3 + 4^3 = z^3

then

z = 4.497941445 (approx)

and there's your solution!

However, if my method works for integers (as it does seem to) there is still no reason why a real number should be an acceptable solution here, because the proof method, as mentioned, doesnt even really focus on what an integer is, or what the requirements are, specifically for an integer solution as compared to this real solution, which looks acceptable.

The problem again is that the real number z estimated above, must again exist as simultaneous solution within the diagram system I have developed, with the stipulation that x, y remain as constants (here they are constants 3, 4 respectively).

(The diagram sizes increase directly with increasing n, which is legal because we stipulate that the geometrically equivalent square area diagrams, nested on shared axes and origin 0 shared also, increase not by varying x, y but only by varying exponent n)

(cont'd)

[jcfdillon]

So, the method used, which seems to ignore the integer requirement of FLT, does so safely because the same method can work with any constant positive reals x, y, using this method..

So by ignoring the integer requirement via full generalization, we prove not only for all integer cases but also for all positive reals, which of course contain the positive integers as a subset.

Anyway to sum up, if we test some specific integers x, y, we find a particular "real" solution, which doesnt satisfy FLT for all-integers. If we test constant positive reals x, y, we may find an integer or a non integer, but this will not satisfy FLT for all integers; but further if we test x, y positive reals (nonintegers) we find some specific positive value (let us suppose it is integer or noninteger) but it still cannot satisfy the given and derived equations simultaneously. This is why I think that the FLT solution z values cannot even be considered "real" in the sense that the value of pi or sqrt(2) is found to an infinite degree of precision of extended decimal estimates. Because the z value in FLT is not stable as shown by iterating the given equation with

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

We find z diverging more and more with each iteration, whenever exponent n is even fractionally greater than 2.

[matt grime]

Look, here's the very simple explanation for you:
You are claiming that if there were *any real solutions to

x^n+y^n=z^n

then they'd have to satisfy something that clearly nothing satisfies (your averaging, thing).

Now, clearly there are infinitely many real valued triples satisfying x^n+y^n=z^n

( r2^{4/3},2r,2r)=(z,x,y) for the case n=3 and for any r in R, for instance.

so your "proof" is completely wrong.

The claim is that at no point do you actually require that x,y, or z are integers in your proof, and merely stating that you assume they are does not mean your "proof" doesn't apply to other reals as well. Do you dispute that?

[jcfdillon]

In my proof as published I state that x, y are to be held constant as any positive integer values. (This is a clear stipulation of my proof.)

The idea that the positive reals can be tested and can show the same dichotomy came later, but I am not saying that real solutions (as estimated real numbers or estimated rational numbers), cannot be "found." These values however, will not satisfy the system of diagrams and simultaneous equations inherent to FLT and which are illustrated in my paper. -- Not because I ignore the obvious (that a calculation of this value can be made-- but because of a requirement of a real number, that it must at least be equal to itself-- and with higher n in FLT, this is not dependably the case, via chaotic interactions.

The sense that these found values will not work, is that they cannot, in geometric construction and interpretation, simultaneously satisfy the two curve/equations, except at the excluded maximum of the system where z = z_xy2, whereas, by the stipulations we have:

z_xyn < z_xy2 < z_xya

such that

a < 2 < n.

Because this works for all positive reals, we might exclude the requirement of integers in the proof approach to FLT, but this is not traditional and in writing this in '97 I never thought it could cover the reals.

The requirement of my method is that x, y be held constant; then the dichotomy can be shown readily by this method.

[matt grime]

"These values however, will not satisfy the system of diagrams and simultaneous equations inherent to FLT and which are illustrated in my paper. -- Not because I ignore the obvious (that a calculation of this value can be made-- but because of a requirement of a real number, that it must at least be equal to itself-- and with higher n in FLT, this is not dependably the case, via chaotic interactions."

Can we put this somewhere special please? I think it deserves to go down in posterity.

[jcfdillon]

Gee thanx, .. Smithsonian perhaps.

Anyway, .. that's the way I explain it, I talked with a friend who had studied chaos theory and he thought it was an indication of "chaos" that the iterations led to diverging values, values of z bouncing in the positive and negative directions, with each iteration.

Simple iterated equations as in fractals exhibit this behavior, so it seems that the Fermat equation also does this, as is already well known I suppose.

The structure of the diagrams and system of diagrams in 2D is just a simple way of illustrating and analyzing this problem. It's no big deal but was just missed for a long time. I started in 1980, had the first diagram in 10 days, but didnt see any further using that method until '97 unfortunately, which is when I tried the averaged diagrams.

[Hurkyl]

This is going noplace -- thread locked.