Fermat's Last Theorem Proof in WSEAS

[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 doesn't 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.