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.]
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11 comments:
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
Hi Jardine,
Escultura has been discredited. Check out here:
http://www.pcij.org/blog/?p=73
Mr. Escultura,
Thanks for posting your arguments. I have responded in detail to each of your points. Feel free to post your responses.
-Larry
Thanks, Larry, for the invitation to post. Since
I have responded to your comments which are
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
consists of adjacent predecessor-successor
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
Correction: expression (3) should be
(d)^(-n)a1a2…ak, n = 1, 2, ...;
the numbers are subscripts.
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
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:
http://www.reddit.com/r/math/comments/93n3i/edgar_e_escultura_and_the_inequality_of_1_and/
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
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
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
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
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
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