## Dr. Escultura Responds

Posted by Dan Draney on March 10, 2006

Since our last post about the “secret controversy” wherein Dr. Edgar Escultura claims to have refuted Dr. Andrew Wiles’ proof of Fermat’s Last Theorem (FLT), Dr. E has returned to the DLMSY comments section. Again, we have no conclusive proof that this is the real Dr. Escultura writing these comments, but the IP address is from the Philipines.

These comments will scroll of the HaloScan system after a few months. Also they’re scattered about and rather hard to find, so we have pasted them here in the main blog. There are three comments from Dr. E, which are somewhat duplicative. The first:

For those interested in the countably infinite counterexamples to FLT, here they are:

Let x = (0.99…)10^T,

where T is ordinary integer. y = d*, where d* = 1 – 0.99… called dark number. z = 10^T.

Then, for n > 2, x^n y^n = z^n.The numbers x, y, z, are called new integers. They consist of d* and numbers of the form N.99…, N = 0, 1, … They are isomorphic to the ordinary integers under the mapping 0 -> d*, N -> (N-1).99…,N = 1, 2, … This isomorphic embedding of the oridnary integers into the contradiction-free new real number system as integral parts of the decimals resolves the fundamental flaw of number theory, that the integers have no valid axiomatization at the present time. In other words, until this embedding they were nonsense.

The new integers x, y, z are among the countably infinite counterexamples to FLT that prove this conjecture false. The rest of the countably infinite counter examples to FLT are also countable, the triples (kx,ky,kz), where k = 1, 2, …, is an ordinary integer, since (kx)^n (ky)^n = (kz)^n.

These counterexamples and their applications are published in over two dozen scientific papers in renowned refereed internatitonal scientific journals and proceedings of international conferences. For details see my websites.

On to the second comment from Dr. E:

For those interested in the countably infinite counterexamples to FLT, here they are:

Let x = (0.99…)10^T, where T is

ordinary integer.y = d*, where d* = 1 – 0.99…

called dark number.z = 10^T.

Then, for n > 2,

x^n + y^n = z^n.

The numbers x, y, z, are called new integers. They consist of d* and numbers of the form N.99…, N = 0, 1, … They are isomorphic to the ordinary integers under the mapping

0 -> d*, N -> (N-1).99…,

N = 1, 2, … This isomorphic embedding of the oridnary integers into the contradiction-free new real number system as integral parts of the decimals resolves the fundamental flaw of number theory, that the integers have no valid axiomatization at the present time. In other words, until this embedding they were nonsense.

The new integers x, y, z are among the countably infinite counterexamples to FLT that prove this conjecture false.

The rest of the countably infinite counter examples to FLT are also countable, the triples (kx,ky,kz), where k = 1, 2, …, is an ordinary integer, since

(kx)^n + (ky)^n = (kz)^n.

These counterexamples and their applications are published in over two dozen scientific papers in renowned refereed internatitonal scientific journals and proceedings of international conferences.

For details see my websites.

Perhaps there may be some applications where this system is or will be useful. I simply don’t buy Dr. E’s assertions that: a) the Trichotomy is false or b) that redefinition of the integers is either necessary or meaningful with respect to FLT or for most of mathematics.

It can be easily shown that 0.999… is neither less than 1 nor greater than 1. The Trichotomy says the only other possiblity is that it is equal to 1. In Dr. E’s math universe there is a fourth possiblity, i.e. 0.999… is not less than 1, not more than 1, and not equal to 1, but something else entirely. Something “dark,” darker than Hilary’s disposition when she sees Bill.

This kind of thing sounds kooky, but it might have some use. The set of imaginary numbers is defined in terms of “i,” the square root of -1, which doesn’t technically exist. However, imaginary numbers are used for many real world applications, for example in describing the behavior of the NMR signals in spectroscopy (or for “MRI” as it’s called for medical imaging).

We don’t much care whether or not these alleged counter examples are “countable,” and we certainly don’t have time to count them. So let’s just consider T=1, the first example. So:

x = (0.999…) * 10

y = (0.999…) -1

z = 10

In the integer math of the rest of the world this means x=z=10 and y=0.

Indeed, if any of x, y, z is allowed to be zero, then any positive integer could be used for both of the other two variables. This is trivial, and FLT explicitly requires x, y, and z to be positive integers, and therefore not zero.

Dr. E is free to define 0.999… as different from 1, but clearly even if there were a difference it would be infinitesimally small, approximately zero. This d* would not be a non-zero integer itself under the rules of FLT. Moreover, even if 0.999… were not equal to 1, it would not be an integer either under the rules of the game everyone else is playing by.

And the third comment:

Someone missed the point again. The problem with the argument here is that the set of integers is not well-defined as a mathematical space. It falls short of basic foundational requirements. Therefore, the problem as formulated is nonsense. The only way to fix the integers is to embed them isomorphically into the contradiction-free new real number system as the integral parts of the decimals. Then FLT can be reformulated to make sense in this space.

For more discussion on FLT visit the Digital Mercenary website.

It quite a stretch to throw out Trichotomy while claiming this system is “contradiction free” and traditional math is “nonsense.”

In plane geometry two lines must intersect in a single point, all points, or not intersect at all. Change the system to spherical geometry and the lines could intersect in two points, one, all, or not at all. One cannot establish that a theorem in plane geometry is false by using the rules of spherical geometry or vice versa.

This is what Dr. E is attempting to do with FLT, as he admits in saying, “Then FLT can be reformulated to make sense in this space.” Reformulating FLT into his system, where he has an extra zero to play with, makes it a completely different (trivial) problem. So what? This has nothing to do with Dr. Wiles’ work.

How many legs does a dog have, if you call the tail a leg? It has 4, because calling a tail a leg doesn’t make it one. The dog’s legs are countable. You might say it has 3.999… legs, and we wouldn’t disagree. We’re not ready to go over to The Dark Side, though.

Dr Escultura’s web site is here, if you really can’t get enough of this stuff.

Technorati: Andrew Wiles, Fermat, Edgar Escultura, mathematics, hoax, press misconduct, Manila Times

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