## Thursday, May 05, 2005

### Coprime Numbers

In the last blog, I spoke about Diophantus's problem: to divide a square into the sum of two smaller squares.

In other words, to find solutions for x,y,z where:
x2 + y2 = z2.

The first step in solving this problem is to realize that we can assume that x,y,z are coprime (or another way to say it, relatively prime). That is, no two of these values are divisible by the same prime. So, if p is a prime that is a factor of x, then we know that it is not a factor of y and not a factor of z.

When we have a situation where the three numbers are not coprime (for example, 6,8,10), we will be able to divide out common factors and end up with three numbers that are.

In the case of 6,8,10, the three numbers share the prime 2. If we divide out 2, then we are left with 3,4,5 which are coprime.

This assumption is important because it greatly simplifies the task of analyzing the conditions for when a solution exists. In my next blog, I will show how this assumption gives us the solution to Diophantus's problem.

Interestingly, we can apply this same assumption to Fermat's Last Theorem. From this point on, we will only need to consider the case where x,y,z are relatively prime.

One of my goals in this project is to provide complete proofs each of the conclusions presented. This blog relies on one lemma. A lemma is an intermediate statement that requires proof and is used in a larger theorem.

Lemma: All solutions to xn + yn = zn can be reduced to a form where x,y,z are coprime. [Here is the proof.]

Jdavies said...

Do check this out re: an article on the refutation on Andrew Wiles proof by E.E. Escultura...http://www.manilatimes.net/national/2005/may/05/yehey/top_stories/20050505top4.html

I'm sure it is worth reading

Larry Freeman said...

Hi Jardine,

Escultura has been discredited. Check out here:
http://www.pcij.org/blog/?p=73

E. E. Escultura said...

MY STRATEGY FOR CAPTURING FLT.

I wondered why FLT remained unresolved for centuries and concluded that its underlying fields –foundations, number theory and the real number system – are defective. Therefore, I embarked on their critique-rectification that yielded the following:

1) There are sources of contradiction in mathematics including ambiguous and vacuous concepts, large and small numbers (depending on context), unbounded or infinite set and self-reference. Here is an example of vacuous concept: A triquadrilateral is a plane figure with three vertices and four edges. The Richard paradox is an example of self-reference: The barber of Seville shaves those and only those who do not shave themselves; who shaves the barber? Incidentally, the indirect proof is flawed, being self-referent.

2) Among the requirements for a contradiction-free mathematical space are the following:
a) It must be well-defined by consistent axioms and every concept must be well-defined by them. A concept is well-defined if its existence, properties and relationship with other concepts are specified by the axioms. A false proposition cannot be an axiom as it introduces inconsistency. For example, this proposition cannot be used as an axiom of any mathematical space: There exists a triangle with four edges.
b) The rules of inference (mathematical reasoning) must be specific to and well-defined by its axioms.
c) Any proposition involving the universal or existential quantifiers on infinite set is not verifiable and, therefore, cannot be used as an axiom for it would not endow certainty to the conclusion of a theorem.

3) The real number system does not satisfy the requirements for a contradiction-free mathematical space. In particular, the trichotomy axiom is false since it is equivalent to natural ordering which the real number system has none because most of its concepts are ill-defined. Therefore, the real number system is ill-defined or nonsense and FLT being fomulated in it is also nonsense. Consequently, to resolve FLT the real number system must be fixed first and FLT must be reformulated in it. Andrew Wiles failed to do this and his work collapses altogether.

4) It is alright to introduce ambiguity provided it is 'approximable" by certainty. For example, a nonterminating decimal is ambiguous since not all its digits are known but it can be approximated by a segment at the nth decimal digit at margin of error 10^-n.

5) The rectification is to build a new real number system R* with three simple axioms and two operations + and x: 1) R* contains the basic integers 0, 1, ..., 9, and the operations + and x are well-defined by 2) the addition and 3) multiplication tables of arithmetic that we learned in primary school. The rest of the elements of R* are the terminating decimals first which are later extended to the nonterminating decimals.
A new real number is well-defined if every digit is known or computable, i.e., there is some rule or algorithm for determining it uniquely. Note that the periodic decimals including the terminating decimals are well-defined new real numbers and the real numbers, the terminating decimals, are embedded in the new real number system. The integers are embedded isomorphically into the integral parts of the decimals and are, therefore, well-defined by the axioms of R*. This remedy’s the major flaw of number theory, namely, the fact that the integers have no adequate axiomatization.

6) The new elements of the new real number system are the dark number d* = 1 – 0.99… - N – (N–1), N = 0, 1, … (the ordinary integers), and u* the equivalence class of divergent sequences. The mapping 0 – > d*, N – > (N–1).99…, where N = 1, 2, …, maps the integers isomorphically into the new integers which means that they have almost identical behavior, the only difference being that d* > 0.

7) Then the counterexamples to FLT are as follows: Let x = (0.99...)10^T, y = d*, z = 10^T, where T is an ordinary integer, T = 1, 2, ... Then x, y, z satisfy Fermat's equation, for n > 2,

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

Moreover, if k = 1, 2, ..., is ordinary integer, kx, ky, kz also satisfy Fermat's equation. They are the counterexamples to FLT. They prove that FLT is false and Wiles is wrong.

The critique-rectification of the underlying fields of FLT, the construction of the counterexamples and applications of this new methodology, especially in physics, are developed in the following articles in renowned refereed international journals and conference proceedings:

 Escultura, E. E. (1996) Probabilistic mathematics and applications to dynamic systems including Fermat's last theorem, Proc. 2nd International Conference on Dynamic Systems and Applications, Dynamic Publishers, Inc., Atlanta, 147 – 152.
 Escultura, E. E. (1997) The flux theory of gravitation (FTG) I. The solution of the gravitational n-body problem, Nonlinear Analysis, 30(8), 5021 – 5032.
 Escultura, E. E. (1998) FTG VII. Exact solutions of Fermat's equation (Definitive resolution of
Fermat's last theorem, J. Nonlinear Studies, 5(2), 227 – 254.
 Escultura, E. E. (1999) VIII. Superstring loop dynamics and applications to astronomy and biology, J. Nonlinear Analysis, 35(8), 259 – 285.
 Escultura, E. E. (1999) FTG II. Recent verification and applications, Proc. 2rd International Conf.: Tools for Mathematical Modeling, St. Petersburg, vol. 4, 74 – 89.
 Escultura, E. E. (2000) FTG IX. Set-valued differential equations and applications to quantum gravity, Mathematical Research, Vol. 6, 2000, St. Petersburg, 58 – 69.
 Escultura, E. E. (2001) FTG X. From macro to quantum gravity, J. Problems of Nonlinear Analysis in Engineering Systems, 7(1), 56 – 78.
 Escultura, E. E. (2001) FTG XI. Quantum gravity, Proc. 3rd International Conference on Dynamic Systems and Applications, Atlanta, 201 – 208.
 Escultura, E. E. (2001) FTG. XII. Turbulence: theory, verification and applications, J. Nonlinear Analysis, 47(2001), 5955 – 5966.
 Escultura, E. E. (2001) FTG III: Vortex Interactions, J. Problems of Nonlinear Analysis in Engineering Systems, Vol. 7(2), 30 – 44.
 Escultura, E. E. (2001) FTG IV. Chaos, turbulence and fractal, Indian J. Pure and Applied Mathematics, 32(10), 1539 – 1551.
 Escultura, E. E. (2002) FTG V. The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
 Escultura, E. E. (2003) FTG VI. The theory of intelligence and evolution, Indian J. Pure and Applied Mathematics, 33(1), 111 – 129.
 Escultura, E. E. (2003) FTG XVII: The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
 Escultura, E. E. (2003) FTG XVIII. Macro and quantum gravity and the dynamics of cosmic waves, J. Applied Mathematics and Computation, 139(1), 23 – 36.
 Escultura, E. E. (2001) FTG. XIV. The mathematics of chaos, turbulence, fractal and tornado breaker, deflector and aborter, Proc. Symposium on Development through Basic Research, National Research Council of the Philippines, University of the Philippines, 1 – 13.
 Escultura, E. E. FTG XIX. Recent results, new inventions and the new cosmology, accepted, J. Problems of Nonlinear Analysis in Engineering System.
 Escultura, E. E. FTG XV. The new nonstandard analysis and the intuitive calculus, submitted.
 Escultura, E. E. (2002) FTG VI. The philosophical and mathematical foundations of FLT’s resolution, rectification and extension of underlying fields and applications, accepted, J. Nonlinear Differential Equations.
 Escultura, E. E. (2002) FTG XXII. Extending the reach of computation, submitted.
 Escultura, E. E. (2003) The theory of learning and implications for Math-Science Education, submitted.
 Escultura, E. E. (2003) FTG XXIII. The complex plane revisited, accepted, Journal of Nonlinear Differential Equations.
 Escultura, E. E. (2002) FTG XXIV. Columbia: the crossroads for science, accepted, J. Nonlinear Differential Equations.
 Escultura, E. E. (2003) FTG XXV. Dynamic Modeling and Applications, Proc. 3rd International Conference on Tools for Mathematical Modeling, State Technical University of St. Petersburg, St. Petersburg.
 Escultura, E. E. (2004) FRG XXVII – XXVIII. Part I. The new frontiers of mathematics and physics. Part I. Theoretical Construction and Resolution of Issues, Problems and Unanswered Questions.
 Escultura, E. E. (2005) FRG XXVII – XXVIII. The new frontiers of mathematics and physics. Part II. The new real number system: Introduction to the new nonstandard analysis, Nonlinear Analysis and Phenomena, II(1), January, 15 – 30.
 Escultura, E. E. (2005) FTG XXVI. Dynamic Modeling of Chaos and Turbulence, Proc. 4th World Congress of Nonlinear Analysts, Orlando, June 30 – July 7, 2004.
 Escultura, E. E. FTG. XXVII (2005). The theory of everything, Nonlinear Analysis and Phenomena, II(2), 1 – 45.
 Escultura, E. E. Escultura (2006) FTG XXXIV. Foundations of Analysis and the New Arithmetic,
Nonlinear Analysis and Phenomena, January 2006.
[47} Escultura, E. E. FTG XXXV (2006) The Pillars of FTG and some updates, Nonlinear Analysis and Phenomena, III(2), 1 – 22.
 Escultura, E. E. FTG XXXVI (2006) The New Nonstandard Calculus, accepted, Nonlinear Analysis.

TAKE IDEAS FROM THE TOP, NOT FROM THE FLAT OF ONE’S FOOT.

The arbiter of scientific knowledge is the network of refereed scientific journals. Internet messages do not count, especially, when authored by those who hide behind anonimity and fake names because they are embarrassed by their own feeble ideas and empty publication list. Thank you for the compliment that I have influence over editors of renowned refereed international journals. Frankly, I don't. Anyway, here is an aspirin to calm down those who are jittery over the achievements of others:

The ScienceDirect TOP25 Hottest Articles is a free quarterly service from ScienceDirect. When you subscribe to the ScienceDirect TOP25, you'll receive an e-mail every three months listing the ScienceDirect users' 25 most frequently downloaded journal articles, from any selected journal among more than 2,000 titles in the ScienceDirect database, or from any of 24 subject areas.

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TOP25 articles within the journal:
Nonlinear Analysis

OCT - DEC 2005

1. Initial value problems for higher-order fuzzy differential equations • Article
Nonlinear Analysis, Volume 63, Issue 4, 1 November 2005, Pages 587-600
Georgiou, D.N.; Nieto, J.J.; Rodriguez-Lopez, R.
2. Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces • Article
Nonlinear Analysis, Volume 64, Issue 3, 1 February 2006, Pages 558-567
3. Variational approach to nonlinear problems and a review on mathematical model of electrospinning • Article
Nonlinear Analysis, Volume 63, Issue 5-7, 1 November 2005, Pages e919-e929
He, J.-H.; Liu, H.-M.
4. Approximation of common fixed points for a family of finite nonexpansive mappings in Banach space • Article
Nonlinear Analysis, Volume 63, Issue 5-7, 1 November 2005, Pages 987-999
Wu, D.; Chang, S.-S.; Yuan, G.X.
5. Nonlinear differential equations with nonlocal conditions in Banach spaces • Article
Nonlinear Analysis, Volume 63, Issue 4, 1 November 2005, Pages 575-586
Xue, X.
6. A survey on piecewise-linear models of regulatory dynamical systems • Article
Nonlinear Analysis, Volume 63, Issue 3, 1 November 2005, Pages 336-349
Oktem, H.
7. Fixed point theorems in metric spaces • Article
Nonlinear Analysis, Volume 64, Issue 3, 1 February 2006, Pages 546-557
Proinov, P.D.

8. Dynamic modeling of chaos and turbulence • Article
Nonlinear Analysis, Volume 63, Issue 5-7, 1 November 2005, Pages e519-e532
Escultura, E.E.

9. Existence result for periodic solutions of a class of Hamiltonian systems with super quadratic potential • Article
Nonlinear Analysis, Volume 63, Issue 4, 1 November 2005, Pages 565-574
Karshima Shilgba, L.K.
10. Multiple positive solutions of a boundary value problem for nth-order impulsive integro-differential equations in Banach spaces • Article
Nonlinear Analysis, Volume 63, Issue 4, 1 November 2005, Pages 618-641
Guo, D.
11. Fixed points of uniformly lipschitzian mappings • Article
Nonlinear Analysis
Dhompongsa, S.; Kirk, W.A.; Sims, B.
12. High regularity of the solutions of the telegraph system subjected to nonlinear boundary conditions • Article
Nonlinear Analysis, Volume 63, Issue 4, 1 November 2005, Pages 491-512
Apreutesei, N.
13. Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces • Article
Nonlinear Analysis
Jung, J.S.
14. New modelling approach concerning integrated disease control and cost-effectivity • Article
Nonlinear Analysis, Volume 63, Issue 3, 1 November 2005, Pages 439-471
Tang, S.; Xiao, Y.; Clancy, D.
15. Existence and approximation of solutions of second-order nonlinear four point boundary value problems • Article
Nonlinear Analysis, Volume 63, Issue 8, 1 December 2005, Pages 1094-1115
Khan, R.A.; Lopez, R.R.
16. Autonomous steering control for electric vehicles using nonlinear state feedback H"~ control • Article
Nonlinear Analysis, Volume 63, Issue 5-7, 1 November 2005, Pages e2257-e2268
Moriwaki, K.
17. A Petri net-based object-oriented approach for the modelling of hybrid productive systems • Article
Nonlinear Analysis, Volume 62, Issue 8, 1 September 2005, Pages 1394-1418
Villani, E.; Pascal, J.C.; Miyagi, P.E.; Valette, R.
18. Existence and uniqueness of a wavefront in a delayed hyperbolic-parabolic model • Article
Nonlinear Analysis, Volume 63, Issue 3, 1 November 2005, Pages 364-387
Ou, C.; Wu, J.
19. Positive solutions of second-order two-delay differential systems with twin-parameter • Article
Nonlinear Analysis, Volume 63, Issue 4, 1 November 2005, Pages 601-617
Bai, D.; Xu, Y.
20. Multiple positive and sign-changing solutions for a singular Schrodinger equation with critical growth • Article
Nonlinear Analysis, Volume 64, Issue 3, 1 February 2006, Pages 381-400
Chen, J.
21. Strong convergence of the CQ method for fixed point iteration processes • Article
Nonlinear Analysis
Martinez-Yanes, C.; Xu, H.-K.
22. Nonlinear analysis of arterial blood flow-steady streaming effect • Article
Nonlinear Analysis, Volume 63, Issue 5-7, 1 November 2005, Pages 880-890
Jayaraman, G.; Sarkar, A.
23. Existence and exponential stability of periodic solution for impulsive delay differential equations and applications • Article
Nonlinear Analysis, Volume 64, Issue 1, 1 January 2006, Pages 130-145
Yang, Z.; Xu, D.
24. Uniqueness results for nonlinear elliptic equations with a lower order term • Article
Nonlinear Analysis, Volume 63, Issue 2, 1 October 2005, Pages 153-170
Betta, M.F.; Mercaldo, A.; Murat, F.; Porzio, M.M.
25. Positive solutions of a second-order singular ordinary differential equation • Article
Nonlinear Analysis, Volume 61, Issue 8, 1 June 2005, Pages 1383-1399
Bonheure, D.; Gomes, J.M.; Sanchez, L.

View this website at: http://top25.sciencedirect.com/index.php?subject_area_id=16&journal_id=0362546X

Paper No. 8, Dynamic Modelling of Chaos and Turbulence, presented by the author at a plenary session of the 4th World Congress of Nonlinear Analysts, Orlando, FL, July 5, 2004, provides, for the first time, the most comprehensive explanation of gravity as part of the dynamics of turbulence, specifically, vortex flux of superstrings (the superstring is the basic constituent of matter), such as galaxy, star, planet and moon. Dynamic modelling (an off-shoot of the resolution of FLT), the new methodology introduced by the author as an alternative to the present methodology of physics called mathematical modelling (that describes nature in terms of numbers, equations, functions, etc.) explains nature in terms of its laws. To-date 43 laws of nature have been discovered using this new methodology.

E. E. Escultura

Larry Freeman said...

Mr. Escultura,

Thanks for posting your arguments. I have responded in detail to each of your points. Feel free to post your responses.

-Larry

E. E. Escultura said...

Thanks, Larry, for the invitation to post. Since
quite thorough, and I commend you for it, I’ll
just give an update of the new real number
system. In case there were points you raised
that I missed to respond to please remind me.
But first let me check if my fonts and symbols
convert right.

Here are some important points about the
new real number system.

1) In both the real and new real number
systems the only well-defined decimals are
the terminating ones; the nonterminating
decimals are simply arrays of digits
most of which are unknown.

2) In the new real number system the
nonterminating decimals are defined, for the
the first time, in terms of the terminating
decimals R as follows:

a) Consider the sequence of terminating
decimals of the form,

N.a1, N.a1a2, …, N.a1a2…an, …, (1)

the sequence (1) is called standard
generating or g-sequence. Its nth g-term,
N.a1a2…an, which is a terminating decimal,
defines and approximates the g-limit, the
nonterminating decimal,

N.a1a2…an…, (2)

at margin of error (maximum error) 10–n.

b) If the nth digit of the g-limit (2) is not 0
for all n beyond a certain integer k then (2)
defines a nonterminating decimmal.
Note that the nth g-term repeats all the
previous digits of the decimal in the same
order so that if finite terms of the g-sequence
are deleted, the nonterminating decimal it
defines, i.e., its g-limit, remains unaltered.

c) In analysis we define limit in terms of
some norm. We define the g-norm of a
nonterminating decimal as the decimal
itself so that the g-limt is also defined in
terms of the g-norm. Computation with the
g-norm has advantages one of which being
that the result is obtained directly as a
decimal digit by digit so that the
intermediate steps of approximatio is
avoided.

3) Consider the sequence of decimals,

(d)na1a2…ak, n = 1, 2, …, (3)

where d is any of the decimals,
0.1, 0.2, 0.3, …, 0.9, and a1, …, ak
finite basic integers (not all 0 simultaneously).
For each combination of d and the ajs,
j = 1, …, k, in (3) the nth term, which
we now refer to as the nth d-term of
this nonstandard d-sequence, is not a
decimal since the digits are not fixed.
As n increases indefinitely it traces the
tail digits of some nonterminating
decimal (note that the nth g-term recedes
to the right with increasing n), becomes
smaller and smaller until it becomes
indistinguishable from the tail digits of the
other decimals. We call the sequence (3)
nonstandard d-sequence since the nth term
is not a standard g-term but has a standard
limit, i.e., limit in the standard norm, which
is 0. Like the g-limit, the d-limit exists since
it is defined by its nonstandard d-sequence
of terminating decimals; we call it a dark
number d’, the d-limit of the nonstandard d-
sequence (3). Moreover, while the nth term
becomes smaller and smaller with increasing
n it is greater than 0 no matter how large n is
so that if x is any decimal, 0 < d < x. The set
of d limits of all nonstandard d-sequence is
the dark number d*

4) We state some important results:

Theorem. The d-limits of the tail digits of
all the nonterminating decimals traced by
the nth d-terms of the d-sequence (3) form
the continuum d*.

Theorem. The d-limits of the tail digits of
all the nonterminating decimals traced by
the nth d-terms of the d-sequence (3) form
the continuum d*.

Theorem. In the lexicographic ordering R
pairs of decimals (each joined by d*) so
that the closure R* in the g-norm is a
continuum.

Note that the trichotomy axiom follows
from the lexicographic ordering of R*
which is not defined on the real numbers
since noterminating decimals are not
well-defined there.

Corollary. R* is non-Archimedean and
non-Hausdorff but the decimals are
Archimedean and Hausdorff in the standard
norm.

Theorem. The rationals and irrationals are
separate, i.e., they are not dense in their union
(this is the first indication of discreteness
of the decimals).

Theorem. The largest and smallest elements
of R* in the open interval (0,1) are 0.99… and
1 – 0.99…, respectively; note that d* = 1 – 0.99…

(8) Theorem. An even number greater than 2
is the sum of two prime numbers.

(This post is excerpted from my keynote
address at the 5th World Congress of
Nonlinear Analysts, The Mathematics of
the Grand Unified Theory, July 5, 2008,
Orlando, Florida, to appear in Nonlinear
Analysis, Series A, Theory, Methods and
Applications)

E. E. Escultura

E. E. Escultura said...

Correction: expression (3) should be

(d)^(-n)a1a2…ak, n = 1, 2, ...;

the numbers are subscripts.

E. E. Escultura said...

Summation of the Debate on the New Real Number System and the Resolution of Fermat’s last theorem – by E. E. Escultura

The debate started in 1997 with my post on the math forum SciMath that says 1 and 0.99… are distinct. This simple post unleashed an avalanche of opposition complete with expletives and name-calls that generated hundreds of threads of discussion and debate on the issue. The debate moved focus when I pointed out the two main defects of Andrew Wiles’ proof of FLT and, further on, the discussion shifted to the new real number system and the rationale for it. Naturally, the debate spilled over to many blogs and websites across the internet except narrow minded ones that accommodate only unanimous opinions, e.g., Widipedia and its family of websites as well as websites that cannot stand contrary opinion like HaloScan and its sister website, Don’t Let Me Stop You. SciMath stands out as the best forum for discussion of various mathematical issues from different perspectives. There was one regular at SciMath who did not debate me online but through e-mail. We debated for about a year and I learned much from him. The few who only had expletives and name-calls to throw at me are nowhere to be heard from.

E. E. Escultura

E. E. Escultura said...

There was one unsigned feeble attempt from the UP Mathematics Department to counter my arguments online. But it wilted without a response from the science community because it lacked grasp of what mathematics is all about.

The most recent credible challenge to my positions on these issues was registered by Bart van Donselaar in the online article, Edgar E. Escultura and the Inequality of 1 and 0.99…, to which I responded with the article, Reply to Bart van Donselaar’s article, Edgar E. Escultura and the inequality of 1 and 0.99…; a website on the Donselaar’s paper has been set up:

and the discussion is coming to a close as no new issues are being raised. Needless to say, none of my criticisms of Wiles’ proof of FLT or my critique of the real and complex number systems have been challenged successfully on this website or across the internet. In peer reviewed publications there is not even a single attempt to refute my positions on these issues.

Ed. E. Escultura

E. E. Escultura said...

We highlight some of the most contentious issues of the debate.
1) Consider the equation 1 = 0.99… that almost everyone accepts. There are a number of defects here. Among the decimals only terminating decimals are well-defined. The rest are ill-defined or ambiguous. In this equation the left side is well-defined as the multiplicative identity element while the right side is ill-defined. The equation, therefore, is nonsense.
2) The second point is: David Hilbert already knew almost a century ago that the concepts of individual thought cannot be the subject matter of mathematics since they are unknown to others and, therefore, cannot be studied collectively, analyzed or axiomatized. Therefore, the subject matter of mathematics must be objects in the real world including symbols that everyone can look at, analyze and study collectively provided they are subject to consistent premises or axioms. Consistency of a mathematical system is important, otherwise, every conclusion drawn from it is contradicted by another. In order words, inconsistency collapses a mathematical system. Consider 1 and 0.99…; they are certainly distinct objects like apple and orange and to write apple = orange is simply nonsense.

E. E. Escultura

E. E. Escultura said...

3) The field axioms of the real number system is inconsistent. Felix Brouwer and myself constructed counterexamples to the trichotomy axiom which means that it is false. Banach-Tarski constructed a contradiction to the axiom of choice, one of the field axioms. One version says that if a soft ball is sliced into suitably little pieces and rearranged without distortion they can be reconstituted into a ball the size of Earth. This is a topological contradiction in R^3.
4) Vacuous concept generally yields a contradiction. For example, consider this vacuous concept: the root of the equation x^2 + 1 = 0. That root has been denoted by i = sqrt(-1). The notation itself is a problem since sqrt is a well-defined operation in the real number system that applies only to perfect square. Certainly, -1 is not a perfect square. Mathematicians extended the operation to non-negative numbers. However, the counterexamples to the trichotomy axiom show at the same time that an irrational number cannot be represented by a sequence of rationals. In fact, a theorem in the paper, The new mathematics and physics, Applied Mathematics and Computation, 138(1), 127 – 149, says that the rationals and irrationals are separated, i.e., the union of disjoint open sets.
At any rate, if one is not convinced of the mischief that vacuous concept can play, consider this:
i .= sqrt(-1) = sqrt1/sqrt(-1) = 1/i = -i or i = 0. 1 = 0, and both the real and complex number systems collapse.

E. E. Escultura

E. E. Escultura said...

5) With respect to Andrew Wiles’ proof of FLT it has two main defects: a) Since FLT is formulated in the inconsistent real number system it is nonsense and, naturally, the proof is also nonsense. The remedy is to first remove the inconsistency of the real number system which I did and reformulate FLT in the consistent number system, the new real number system. b) The use of complex analysis deals another fatal blow to Wiles’ proof. The remedy for complex analysis is in the appendix to the paper, The generalized integral as dual to Schwarz Distribution, in press, Nonlinear Studies.

6) By reconstructing the defective real number system into the contradiction-free new real number system and reformulating FLT in the latter, countably infinite counterexamples to it have been constructed showing the theorem false and Wiles wrong.

E. E. Escultura

E. E. Escultura said...

7) In the course of making a critique of the real number system some new results have been found: a) Gauss diagonal method of proving the existence of nondenumerable set only generates a countably infinite set; b) as of this time there does not exist a nondenumerable set; c) only discrete set has cardinality, a continuum has none..

8) The new real number system is a continuum, countably infinite, non-Hausdorff and Non-Archimedean and the subset of decimals is also countably infinite but discrete, Hausdorff and Archimedean. The g-norm simplifies computation considerably.

E. E. Escultura

E. E. Escultura said...

References

 Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 - 61.
 Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
 Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
 Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
 Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
 Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
 Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
 Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
Theory, Methods and Applications; online at Science Direct website
 Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
 Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May 2009, 101-107.
 Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris.
 Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
 Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.

E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University
http://users.tpg.com.au/pidro/

E. E. Escultura said...

CLARIFICATION ON THE COUNTEREXAMPLES TO FERMAT’S LAST THEOREM
By E. E. Escultura

Although all issues related to the resolution of Fermat’s last theorem have been fully debated worldwide since 1997 and NOTHING had been conceded from my side I have seen at least one post expressing some misunderstanding. Let me, therefore, make the following clarification:

1) The decimal integers N.99… , N = 0, 1, …, are well-defined nonterminating decimals among the new real numbers  and are isomorphic to the ordinary integers, i.e., integral parts of the decimals, under the mapping, d* -> 0, N+1 -> N.99… Therefore, the decimal integers are integers . The kernel of this isomorphism is (d*,1) and its image is (0,0.99…). Therefore, (d*)^n = d* since 0^n = 0 and (0.99…)^n = 0.99… since 1^n = 1 for any integer n > 2.

2) From the definition of d* , N+1 – d* = N.99… so that N.99… + d* = N+1. Moreover, If N is an integer, then (0.99…)^n = 0.99… and it follows that ((0.99,..)10)^N = (9.99…)10^N, ((0.99,..)10)^N + d* = 10^N, N = 1, 2, … .

3) Then the exact solutions of Fermat’s equation are given by the triple (x,y,z) = ((0.99…)10^T,d*,10^T), T = 1, 2, …, that clearly satisfies Fermat’s equation,
x^n + y^n = z^n, (F)

for n = NT > 2. The counterexamples are exact because the decimal integers and the dark number d* involved in the solution are well-defined and are not approximations.

4) Moreover, for k = 1, 2, …, the triple (kx,ky,kz) also satisfies Fermat’s equation. They are the countably infinite counterexamples to FLT that prove the conjecture false . They are exact solutions, not approximation. One counterexample is, of course, sufficient to disprove a conjecture.

The following references include references used in the consolidated paper  plus  which applies 

References

 Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 - 61.
 Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
 Corporate Mathematical Society of Japan , Kiyosi Itô, Encyclopedic dictionary of mathematics (2nd ed.), MIT Press, Cambridge, MA, 1993
 Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
 Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
 Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
 Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
 Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
 Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
Theory, Methods and Applications; online at Science Direct website
 Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
 Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May 2009, 101-107.
 Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris.
 Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
 Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.

E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University