Science and Technology news... :: View topic - #545 Hardy>s and Courant>s and Polya>s Euclid IP had a big m

Guest

drmwec...@gmail.com wrote:
[quote]Seldom has one human being (Archimedes Balonium) written so much and
said so little of value.

[/quote]
I offered the Atom Totality theory and the Fusion Barrier Principle
and scores of other
new ideas. By comparison Ecker has given the world nothing as far as
new ideas of
any importance.

[quote]There is nothing wrong with the standard proof of the infinitude of
prime. It is a clear proof by contradiction. One assumes one has them
all and then constructs a new number that may or may not actually be
prime, but even if composite, its divisors are not among the assumed
finite set of all primes.
[/quote]
How silly and stupid of you, for you then imply that the direct and
indirect
are identical proofs. Even a High School student can see that you are
wrong
on that score.

[quote]
Newbies should not be taken in by this nonsense.

Dr. Michael W. Ecker
Associate Professor of Mathematics
Pennsylvania State University
Wilkes-Barre Campus
Lehman, PA 18627
[/quote]
Hatred by envy is one of the worst hatreds in the world.

When someone is in education and then deny those that have achieved
something
of Corrective value and of newness, then they no longer belong in
education. Why they
deny others of the successes they achieve is probably due to their own
careers as
being lackluster and their bitterness then lashes out at people who
accomplish
something.

Ecker does not deserve to have a degree in mathematics when he cannot
even understand
a valid Euclid Infinitude of Primes proof. Of the list of 30 or more
math professors who
I cited in my book "Correcting the Euclid Infinitude of Primes Proof,
some of them did
do a valid Euclid IP. Those that did not, are not in agreement with
Ecker. They recognize
the mistake and fault of their rendition. There was a team of authors
who I cited in my
book with a invalid Euclid IP and because of my insistence, they
corrected or at least
attempted to correct in a revised edition.

Even Wikipedia had a invalid Euclid IP and then changed it because of
my prodding.

In the above, it is apparent that Ecker is unable to do a valid Euclid
IP proof in direct and
indirect method.

In the old days before the Internet, these sour professors would
simply stew on campus
and not really be a menace to those in the sciences, but in our modern
day life, these sour
and bitter professors can magnify their bitterness in Internet
hatemongering.

When any teacher in education gets to the point where they cannot
recognize the achievement
or accomplishements of others in their respective science field, they
are more of a harm to the
science and the students and should be removed of education. If you
cannot congratulate someone
who achieved some work, then get you gone.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

Guest

Bill Dubuque wrote:
[quote]malcobe@gmail.com wrote:

I keep on believing that the key idea [in Euclid>s proof that there
are infinitely many primes] is the construction of the number
"the product of all primes plus one"

Actually the "reason" such proofs work is because the ring of integers
has relatively few units. Thus, by way of the Pigeonhole Principle,
[/quote]
Not true, because Bill is under the assumption Natural-Numbers =
Finite
Integers, which is an ill-defined set. Things are different in Natural-
Numbers
= Infinite Integers.

The reason that "multiply the lot add 1" works is that it is merely
another form
of primes such as 2k+1 or 2k-1. One form that does not work is 2k for
2 is the
only even prime number. Twin Primes are of the form k,k+2 such as 3,5
but there
are primes of form k,k+2,k+4 such as 3,5,7.

Now Bill with his old fake math of Natural-Numbers = Finite Integers
would
be unable to launch into the question of whether there is a infinitude
of primes
such as 3,5,7, but when you do mathematics with the true-blue counting
numbers
the Infinite Integers there is a simple constructive proof of the
infinitude of Triplet Primes.
An infinitude of Natural Numbers whose last digit is 9, 1, and 3 and
then those whose last digit
is 7, 9, and 1. Obviously no number whose last digit is 5 is prime. So
what Bill
thinks is the underpinning of these infinitude proofs with some
Pigeonhole Principle is far
off the mark. The underpinning is that almost any form for primes is
an infinite set, not all sets
but most.

[quote]
These are the sort of results that a curious student can easily
discover independently when pondering generalizations of Euclid>s
theorem (indeed I found them as a teenager). They make excellent
[/quote]
So you are doing Ring theory as a teenager? But when reading Hardy>s
book, unable to spot a mistake in his Euclid Infinitude of Primes
proof?

To me, a mathematics proof is like the electrical system wiring of an
entire
house. The lights and appliances work because everything is connected
and there are no shorts or grounds. But many proofs of mathematics
have so
many holes and gaps, that no electricity works for them. Many proofs
in mathematics
would be like a electrician coming to wire a house and throwing down
some wire
and then leaving.

[quote]exercises for beginning number theory students. If I can dig up
my original notes I will post further results. I hope readers
will contribute some of their favorites (there are many other
variants of proofs of Euclid>s theorem listed in Ribenboim>s
Book of Prime Number Records, but not those I presented above).

--Bill Dubuque
[/quote]
I challenge Bill to write a Euclid Infinitude of Primes proof both
direct and indirect. I challenge him
because I want to discuss Metamathematics
of the differences in the pattern of Direct versus Indirect. I cannot
discuss that if the person has no
reference to a mathematics proof that has a Direct and Indirect. Most
mathematicians believe that
if they provide a proof of a subject, that they can switch to a
indirect or direct, freely. So if they get hold
of one, they believe they can turn around and provide the other
method. I do not share that opinion, because
I know that in geometry proofs, the indirect method is virtually
nonexistent and that only one method of
proving occurrs-- some form of direct method.

The method of proofs-- direct or indirect, I feel comes more from
physics than it comes from mathematics
or logic and is meta-mathematics. This subject of an analysis of
direct versus indirect method is
seldom discussed and there is a paucity of anyone researching this
subject.

So the Euclid Infinitude of Primes is a pretty example of a proof that
has both Direct and Indirect and
allows for analysis of these two methods. I believe they are
Complimentarity methods, and that means
if a proof is of one, then it may or may not have the other.
Complimentarity is independence of one another.

So again, I challenge Bill, if not too scared, to write a Euclid IP in
direct and indirect. Then we can
analyze the Metamathematics of the two methods.

The way I see it, the direct method versus indirect method is like the
physics of electron versus
positron, where both have things in common -- same mass, but have
things opposite -- charge.
Both are independent of one another. And I hope to dispel this widely
perceived false notion
in the math community that once a proof is found, whether direct or
indirect, that they thence can
transform the proof into the other method.

Maybe Bill is too scared to do a Euclid IP both direct and indirect.
Maybe he is more of a politician and
dodging a challenge rather than simply offering his rendition. Maybe I
scare people, for fear of any
shortcoming.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

Bill Dubuque · Guest

malcobe@gmail.com wrote:
[quote]
I keep on believing that the key idea [in Euclid>s proof that there
are infinitely many primes] is the construction of the number
"the product of all primes plus one"
[/quote]
Actually the "reason" such proofs work is because the ring of integers
has relatively few units. Thus, by way of the Pigeonhole Principle,
one can avoid them and generate an infinite sequence of pairwise-coprime
nonunits. For the details see the generalizations given in the theorems
below from my prior post [0].

sttscitrans@tesco.net wrote:
[quote]
Is there any analog A to N, where the usual rules of arithmetic
apply, every non-unit is divisible by some prime, A is infinite
and yet the number of primes or irreducibles are finite ?
[/quote]
Yes, vacuously, let A = Q or any infinite field. Less trivially
force all but finitely many primes to become units, e.g. grow
Z to a subring of Q by adjoining 1/p for almost all primes p.
For example, all primes but 2 become units in the subring of Q
of all rational numbers expressible in the form m/n with n odd.
All primes but 2,3 are units in subring of form m/n, (n,6) = 1.
Such enlarged domains remain Euclidean -> PID -> UFD... since
the localization construction always preserves such structure.

Thus generalizing Euclid>s theorem to other rings will require
some hypotheses other than Euclidean, PID, UFD, etc, that are
preserved by localizations. In general 1 + pqr... can fail to
produce a new prime (or irreducible) because it may be a unit.
But we can also try 1 + d pqr... for any d in D. If all those
are units then D has a lot of units. Therefore we can eliminate
such failure by hypothesizing D has relatively few units, e.g.

THEOREM Suppose that D is an infinite integral domain where all
nonunits have an irreducible factor. If D has finitely many units
then D has infinitely many irreducibles (= primes if D is a UFD).

PROOF Via contradiction. Let m = product of all irreducibles.
Since D is infinite so is 1 + m D [ 1+md = 1+md' <=> d = d']
so it must contain a nonzero nonunit [ since only finite #units ]
with irreducible factor p | 1 + m d; but p|m => p|1 =><= QED

It is easy to generalize this theorem even further, for example:

THEOREM If a commutative ring R has a smaller cardinality subset U
containing all divisors of 0 and 1, then ~U, the complement of U,
contains an infinite number of pair-coprime nonunits (hence also an
infinite number of irreds if every r in ~U has an irred factor).

PROOF Enlarge any finite set P < ~U of pair-coprime nonunits as
follows. Let m be the product of all p in P (m = 1 if P empty).
Elements of S = 1 + m R are coprime to each p in P since p|m
Hence S is disjoint from P since coprime nonunits cannot be equal.
Now enlarge P by adjoining any s in S /\ ~U, which is nonempty
since S is too big to lie inside U by the cardinality hypothesis.
In fact #S = #R > #U because r -> 1 + m r injects R into S,
that is 1+mr = 1+mr' => mr = mr' => r = r' via m cancellable,
being a product of non-zero-divisor (so cancellable) p in P. QED

COROLLARY Elements of a finite commutative ring R divide 0 or 1.
In particular a finite integral domain is a field.

PROOF Let U = { all divisors of 0 and 1 }. Necessarily U = R
else #U < #R => ~U is infinite by Theorem, contra R finite. QED

What a surprisingly sweet little application of Euclid>s theorem!
Compare the above proof with the classic pigeonhole based proof
http://planetmath.org/encyclopedia/AFiniteIntegralDomainIsAField.html

Other ring theoretic generalizations of Euclid>s theorem can be
expressed in terms of the nilradical (intersection of all prime
ideals) and related notions such as G-domains [1]. If there are
only a finite number of prime ideals then the nilradical is
clearly nonzero: take a product of elements from each prime.
But for a wide class of domains, Krull domains (including UFDs,
Dedekind domains, etc) it is easy to show that the nilradical
is nonzero iff D has the structure in the example above, i.e.
a PID with finitely many primes. So, for example, any non-PID
in this class has infinitely many prime ideals; e.g. this
leads to Larry Washington>s proof of Euclid>s theorem via any
non-UFD number ring (if Z had a finite #primes then so would
every number ring, so every number ring would be a PID so UFD)
Closely related are various topological generalizations of
Euclid>s theorem, e.g. work by Golomb, Gotchev, Porubsky.

These are the sort of results that a curious student can easily
discover independently when pondering generalizations of Euclid>s
theorem (indeed I found them as a teenager). They make excellent
exercises for beginning number theory students. If I can dig up
my original notes I will post further results. I hope readers
will contribute some of their favorites (there are many other
variants of proofs of Euclid>s theorem listed in Ribenboim>s
Book of Prime Number Records, but not those I presented above).

--Bill Dubuque

[0] http://google.com/groups?selm=y8zbr8kv5vl.fsf_-_%40nestle.csail.mit.edu
[1] http://google.com/groups?selm=y8z7jjzdivj.fsf%40nestle.csail.mit.edu

Guest

Archimedes Plutonium wrote:
[quote]
The reason that "multiply the lot add 1" works is that it is merely
another form
of primes such as 2k+1 or 2k-1. One form that does not work is 2k for
2 is the
only even prime number. Twin Primes are of the form k,k+2 such as 3,5
but there
are primes of form k,k+2,k+4 such as 3,5,7.

[/quote]
The idea here is that most forms of primes are infinite.

Now that raises an interesting question about some alleged forms under
Natural-Numbers = finite integers are shown to be finite. I believe I
remember
this in connection with Whitehead over a discussion of the Riemann
Hypothesis
where he imagines that RH is false because of some forms of prime
slowly and
gradually end up as finite. I do not remember what form of primes it
was.

The reason I bring this issue up, is because, under Natural-Numbers =
infinite
integers, I suspect those forms that were thought to and proven as
finite, are, now,
once again truly infinite sets of primes.

I also remember trying to graph that sequence of primes to try to get
a sense of
how they stopped, never to continue. But in infinite integers I want
to reraise that issue
for I have the sneaky suspicion that such a form was really infinite
after all.

Now is the Riemann Hypothesis true or false? I cannot remember clearly
what conclusions
I had last drawn on RH, and my problem stemming from too many irons in
the fire. I believe
I ended up last time concluding RH was false. But I like to reenter
that analysis the next time
by comparing RH with e^(i x 2pi) = 1. Can we relate strictly RH with
Euler>s identity? If we can
do so, then the failure of Euler>s Identity is the fact that we have e
and pi belong to a different geometry
than does i. And realizing that, we realize that the Euler Identity is
nothing more than
n^0 = 1 and where i has the value of 0 in NonEuclidean geomety
accounting for the Euler
Identity.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

Guest

plutonium.archime...@gmail.com wrote:
[quote]Archimedes Plutonium wrote:

The reason that "multiply the lot add 1" works is that it is merely
another form
of primes such as 2k+1 or 2k-1. One form that does not work is 2k for
2 is the
only even prime number. Twin Primes are of the form k,k+2 such as 3,5
but there
are primes of form k,k+2,k+4 such as 3,5,7.

The idea here is that most forms of primes are infinite.

Now that raises an interesting question about some alleged forms under
Natural-Numbers = finite integers are shown to be finite. I believe I
remember
this in connection with Whitehead over a discussion of the Riemann
Hypothesis
where he imagines that RH is false because of some forms of prime
slowly and
gradually end up as finite. I do not remember what form of primes it
was.

The reason I bring this issue up, is because, under Natural-Numbers =
infinite
integers, I suspect those forms that were thought to and proven as
finite, are, now,
once again truly infinite sets of primes.

I also remember trying to graph that sequence of primes to try to get
a sense of
how they stopped, never to continue. But in infinite integers I want
to reraise that issue
for I have the sneaky suspicion that such a form was really infinite
after all.

Now is the Riemann Hypothesis true or false? I cannot remember clearly
what conclusions
I had last drawn on RH, and my problem stemming from too many irons in
the fire. I believe
I ended up last time concluding RH was false. But I like to reenter
that analysis the next time
by comparing RH with e^(i x 2pi) = 1. Can we relate strictly RH with
Euler>s identity? If we can
do so, then the failure of Euler>s Identity is the fact that we have e
and pi belong to a different geometry
than does i. And realizing that, we realize that the Euler Identity is
nothing more than
n^0 = 1 and where i has the value of 0 in NonEuclidean geomety
accounting for the Euler
Identity.

[/quote]
Well, people of the future can actually eyewitness the aging process
of Mr. Archimedes Plutonium.
I would never have made such a mistake of above in my 40s or younger
decades, but here as I
am now 58 years of age, I am slipping in memory. It was Littlewood, so
how could I have ever
suggested it was Whitehead? Memory is going to start to fail on all of
us, and for me, it is obvious
that at 58, that process has begun. So it is a good thing that I am in
the midst of writing all these books
before my memory gets so bad that it impinges on my organizing of all
my past thoughts.

I believe we can prove the Riemann Hypothesis is false by a
paralleling of primes of form where they
are primes and an infinite set, but which they are so spread far apart
that we only find the first such
prime of that form that we would believe they are like the Riemann
Hypothesis that all the nontrivial
zeros are on the 1/2 Real line.

Now in the Infinite Integers = Natural-Numbers, the primes are as
dense as the Natural Numbers get into
.....3333333 or say .......6767676767 as they are dense in the interval
of 0 to 10.
In fact the prime numbers ......141312111098765432109 and ......
141312111098765432111 and
......141312111098765432113 and ......141312111098765432117
and ......141312111098765432119
has five primes in a interval of equal length to 0-10 where there are
only four primes 2,3,5,7 in 0-10.

Now Littlewood said there was really no evidence to believe the
Riemann Hypothesis is true and that
there is stronger evidence to believe it is false and he used an
example of a funtion, which I was unable
to locate.

Now the Mersenne Primes are primes of form 2^k-1 where the k is prime
also. If my memory is correct
we have found only 45 such Mersenne primes.

Now, does the search for Mersenne primes remind us of the Riemann
Hypothesis? Yes indeed in that
we believe those primes are well behaved that the nontrivial zeroes
remain on the 1/2 Real line.

And since 2^k-1 has only 45 Mersenne primes, what about primes of the
form (2^k-1)^2^k-1?
In other words, what about primes of the form of Mersenne primes
raised to the power of Mersenne
primes? Does this not parallel the Riemann Hypothesis? Of course it
does, for you could travel
what seems like an eternity and never come upon such a prime, yet they
are infinite.

Likewise, the first nontrivial zero of the Riemann Hypothesis starts
at such a huge distance from
0 that we are lulled into believing RH is true when in fact it is
false.

So a parallel way of proving RH is false, is the proving that Mersenne
Primes raised to the power
of Mersenne Primes is an infinite set of primes. This is
straightforward proof in AP-adics by
using the Champernownes number as rootstock and grafting onto the
rightwards end a prime
such as .....11110001110011013.

And to prove Riemann Hypothesis is false is easily done via geometry.
The Natural Numbers lie
on a curved surface of elliptic geometry, not Euclidean geometry where
the 1/2 Real line is thought
to be a straight line. The Natural Numbers bend and curve around in
Space and come back to a zero
point. So the Riemann Hypothesis crumbles apart as nonsense as to how
the primes of the Natural
Numbers are located on a spherical surface.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

Guest

Archimedes Plutonium wrote:
[quote]
I believe we can prove the Riemann Hypothesis is false by a
paralleling of primes of form where they
are primes and an infinite set, but which they are so spread far apart
that we only find the first such
prime of that form that we would believe they are like the Riemann
Hypothesis that all the nontrivial
zeros are on the 1/2 Real line.

[/quote]
I believe the above is an alternative equivalent statement of the
Riemann Hypothesis. What it implies is
that if the Riemann Hypothesis were true, then we come to a point of a
"Prime form" wherein those
primes are no longer an infinite set and any prime forms beyond this
prime form are also finite sets.

There are only 45 known Mersenne Primes, and the proof that Mersenne
Primes are finite or infinite has
eluded mathematicians. I have recently given a method of proving
Mersenne Primes are infinite sets
in Natural-Numbers = Infinite Integers. This same method would easily
prove that Mersenne Primes
raised to the power of Mersenne Primes FORMS is also a form that is an
infinite set of primes.

So, if the Riemann Hypothesis is true, then we arrive at a FORM of
primes in Natural-Numbers = Finite
Integers for which those primes are a finite set and cannot be
infinite set and also, all forms beyond that
initial Riemann Hypothesis-Form are finite prime sets. So if the
Riemann Hypothesis is true, then
near the Mersenne Prime Form is a finite set and not infinite and all
higher forms beyond the Mersenne-Form
are also finite sets. So if the Riemann Hypothesis is true, means that
somewhere out there, perhaps
the Mersenne primes or a form slightly higher than Mersenne Primes do
all those forms have only
a finite set of primes and no more prime sets that are infinite in
cardinality.

So I think that I have linked the truth or falsity of the Riemann
Hypothesis with the FORM of primes and
their set cardinality.

If the Riemann Hypothesis is false, then, no matter how extended the
Form of prime-- say we have
Mersenne primes raised to the power of Mersenne primes or say we have
2 raised to successive powers subtract
1. No matter how attenuated or thin it is for a Prime Form, that Form
is an infinite set of primes.

So if you believe the RH is true, then there is a Form in which the
primes of that Form are finite and all higher
forms are finite. But if you believe RH is false, then no matter how
extended of a Form of primes, they are
still infinite set of primes that make up that form.

And the geometry equivalent is that the Riemann Hypothesis is lines of
latitude and those lines are formed
by an infinite set of numbers. The lines of latitude is infinite set
of latitude lines and the points on the
latitude lines are all infinite sets.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

Guest

Jesse F. Hughes wrote:

[quote]
And I challenged you to post your version of a Euclid IP both direct
and indirect. But apparently
that is too much to ask of a mere philosophy professor.

What an odd challenge!
[/quote]
To the contrary. The only thing odd is a philosophy professor who
cannot do a Euclid
Infinitude of Primes proof. When I was in University, some thirty plus
years ago, logic was
still the domain of philosophy, but maybe nowadays, logic should be
transferred out of
philosophy and placed into the subject of mathematics.

If we have philosophy professors who cannot even do a Euclid
Infinitude of Primes proof,
then we have some sort of charade in education.

Guest

Jesse F. Hughes wrote:

[quote]
I wonder why you>re so interested in my repetition of a very
well-known proof and how come you ignored my main point.

[/quote]
Because your questions stem from your lack of understanding of a valid
Euclid Infinitude of Primes proof.

You do not understand "multiply the lot and add 1" since you do not
know a valid Euclid IP proof.

So write out a Euclid Infinitude of Primes proof. Label it as to
whether you are
doing a direct method or indirect method.

I cannot fix a car if you refuse to let me under the hood.

Guest

Jesse F. Hughes wrote:
[quote]"Jesse F. Hughes" <jesse@phiwumbda.org> writes:

We could also divide both sides of that equation by 9999...99997 and
get another keen result.

Well, coherence is overrated.

[/quote]
Jesse Hughes, an adjunct professor of philosophy, at Bennett College
and Salem State
College in Arlington Mass (according to Argus Leader newspaper
29JUN08)

So I wonder, do the students at those two colleges, when they ask for
Jesse Hughes
to give a Euclid Infinitude of Primes proof, do they also get this run
around.

How many times does a student at Bennett or Salem State have to ask
Hughes to
write out a Euclid Infinitude of Primes Proof, before he does so.

Maybe he just simply cannot deliver.

Maybe Jesse just endlessly comments on math but never actually does
any math.

Guest

Jesse F. Hughes wrote:

[quote]
Could also be that Jesse has seen your silly criticisms of perfectly
good presentations of Euclid>s proof and has no desire to see you
blather and rant along the same lines again.

[/quote]
My expectations of a college professor in math, or philosophy, is that
if they
engage someone discussing Euclid Infinitude of Primes proof on the
Internet
and where Jesse makes conceptual errors, that it is very much
appropriate that
Jesse render his Euclid Infinitude of Primes proof, because it is the
root of his
misconceptions.

Yesterday I heard we had a new tennis champion other than the Swiss.
Well to be a
tennis champion, you have to sometime play the game of tennis. Noone
is going to
crown you the Wimbledon if you refuse to play tennis. For Jesse to
make erroneous comments on Euclid Infinitude of Primes proof, well,
Jesse is
going to have to post his own rendition. By refusing only means he is
incapable of
doing the proof without parroting it from some book.

And it should not matter to Jesse or anyone else what I would say of
his
rendition. I expect every math professor and every philosophy
professor to be
able to do a Euclid Infinitude of Primes proof. So it does not matter
what comments
I would add to Jesse>s rendition.
All of those who enter sci.math and enter a discussion of Euclid
Infinitude
of Primes. And all of those who are professors of math or philosophy,
are bound
by education conduct to meet that request. If not, you cease being a
professor
of math or philosophy and cease being a person in education.

What Jesse>s posts then boil down to is some hatemonger. In the Navy,
we have a saying
"we are all in the same boat". Meaning in this situation, that if you
enter a thread on
Euclid IP, you better darn well be able to do a Euclid IP for yourself
and to post it
upon request.

So Jesse is free to join in a conversation of Euclid IP, but when it
gets down to
Jesse making such clumsy blunders and errors of reason, and asked to
show his own
rendition. And refusing to do so, means one thing-- he is no longer a
professor
of anything but merely a hatemonger.

Also, I find Jesse>s signature block as rather immature for a college
professor. Where he
constantly recycles quotes of others. Most people have an idea of
standard of conduct
of a college professor, that their behaviour should be at least a cut
above the average bloke. But
Jesse>s signature block is a cut below the average bloke. I get the
impression that Jesse
is yearning for psychiatric attention with his immature signature
block.

Bill Dubuque · Guest

plutonium.archimedes@gmail.com writes:
[quote]
Where Natural Numbers = AP-adics ..
[/quote]
Archie, before I can comment you need to answer the following
questions about your ring of AP-adics, henceforth called AP.

1) Is the hypothesized ring AP a (totally) ordered ring, i.e.
does AP have a subset P of "positives", closed under + and *,
with every elt either positive (in P), negative (in -P) or 0 ?

2) Are the positive elements P well-ordered, i.e. does every
nonempty subset of P have a least element?

Lurkers please refrain from commenting until AP responds.

--Bill Dubuque

Guest

Bill Dubuque wrote:
[quote]plutonium.archimedes@gmail.com writes:

Where Natural Numbers = AP-adics ..

Archie, before I can comment you need to answer the following
questions about your ring of AP-adics, henceforth called AP.
[/quote]

Well, Bill, the reason I wanted you to give a Euclid Infinitude of
Primes proof, both
direct and indirect, two proofs of contrast, is to dive into the
question of the
direct method compared to indirect method. Euclid IP offers us a proof
where
we can talk in depth over the differences of one method versus the
other method.
In 1993 when I was discussing the Fermat>s Last Theorem and there were
three graduate
students at Princeton who were actively debating every issue I raised
as to why FLT
had counterexamples (hence FLT was false). One of those three
Princeton graduates
made a remark to me, whether Terry or Kin or Will, saying that once
you have a indirect proof
of something in mathematics, you immediately have the direct proof of
the same thing.

In other words, it is a common belief, perhaps, you Bill Dubuque,
shares this belief with
that Princeton graduate student, that whenever we have a proof in
math, either direct or
indirect method, we easily and simply can fetch the other method.

I do not buy that.

I believe these proof methods of direct and indirect are
Complimentarity, meaning that they
are fundamentally different and where most often is the case, that if
you prove something
in math with the direct method, there is no possible indirect method
for that same proof. Or
the case of where you have a indirect method proof, but no possible
direct method due to
the elements in the proof argument.

So, the reason I would like for you to give your own version of Euclid
Infinitude of Primes proof
one in the direct method, the other in the indirect method, is to
point out the Complimentarity nature
of mathematics of its proof method.

There is really an utter lack of knowledge in the literature of
mathematics concerning the underpinnings
of Direct versus Indirect, and the Euclid IP offers us a diving
platform to get at the inner workings of these
two methods.

[quote]
1) Is the hypothesized ring AP a (totally) ordered ring, i.e.
does AP have a subset P of "positives", closed under + and *,
with every elt either positive (in P), negative (in -P) or 0 ?

[/quote]
Well, Bill, as I told Dik Winter, what kind of algebraic structure
does "All Possible Digit Arrangements
of Rightward Infinite Strings" have? These are the Reals. Now the
Reals have an Algebraic structure,
ring-- field etc. But the Reals are of the very same infinity as the
AP-adics since those are All Possible
Digit Arrangements to the Leftward Infinite. Both are equinumerous
since they differ only in direction.

Here is something that you believe in Bill, but which is false. You
believe that between any two Reals
exists another Real. This is given some fancy name, but is utterly
unsound.

In All Possible Digit Arrangements there is the AP-adic of
00000....00000 and 00000....000001
commonly known as 0 and 1, yet in All Possible Digit Arrangements
there cannot exist a number between
those two numbers.

Likewise for Reals, in All Possible Digit Arrangements there is the
Real Number 0.0000....0000
and the Real Number 0.0000....000001.

Now I know that FrontView is a new concept for mathematicians to mull
over and I discovered it only
recently to be able to use it as a tool. This tool destroys the Real
Number concept that between
any two Reals is another Real.

Well, what Real Number exists between 0 and 0.00000.....000001 ??
There is none.

In fact, mathematicians of the past could have asked themselves a very
important question that would
have told them that Between Any Two Reals may not necessarily give you
a new Real.

Ask themselves this question. Are the Real Numbers larger than the set
of All Possible Digit Arrangements
of infinite rightward strings, or are the Reals the same set as All
Possible Digit Arrangements?

I think you Bill is a commonsense practical man and would say that the
Reals are the same as the
All Possible Digit Arrangement.

I forgotten what fancy name was given to this idea in mathematics
"between any two Reals
exists another Real". Well that idea was one of the worst and phonyest
ideas to come along
in all of mathematics.

So, to answer your question, Bill, about the AP-adics, you see, I have
altered much of what
you previously thought was true of the Reals and their algebraic
structure of a closed Field.

I defined multiply, add, divide and subtract on AP-adics.

So the question of Ring or Field or Algebra on AP-adics is simply the
question of
what does All Possible Digit Arrangements, whether rightwards for
Reals or leftwards for
AP-adics as infinite strings. What does All Possible Digit
Arrangements yield?

I am no expert on Galois Algebra but I would estimate that All
Possible Digit Arrangements
yields a Closed Field for both Reals and for AP-adics. Closed because,
it is commonsense
that All Possible really means All Possible. And because infinite
rightwards yields 3D Euclid
geometry, even though it has tiny holes between every number, it is
still a smooth geometry.
And AP-adics forms both Elliptic and Hyperbolic geometries. So,
because of All Possible
and because both number sets form geometries-- they are closed
algebraic fields.

Now we may have to redefine Galois Algebras before the dust settles.

[quote]2) Are the positive elements P well-ordered, i.e. does every
nonempty subset of P have a least element?

[/quote]
Well that is easy to answer. When Reals are All Possible Digit
Arrangements and ditto
for AP-adics, (only difference is that one is rightward infinite and
other is leftward infinite).
That given any subset of Reals or of AP-adics, because they are All
Possible Digit Arrangements
means that you have a least element.

[quote]Lurkers please refrain from commenting until AP responds.

--Bill Dubuque
[/quote]
Too much fanfare, Bill. All I wanted was you to give your direct and
indirect Euclid IP so that I can
discuss why Indirect-Direct is Complimentarity of Physics onto
mathematics. Like the three graduates
of Princeton with their vague and unsound notion of the difference
between Direct and Indirect.

You see, Physics is dominate over mathematics, and that Elliptic +
Hyperbolic = Euclidean
is another Complimentarity relationship of physics dominance over
mathematics, just as
Direct method versus Indirect method is another physics dominance over
mathematics.

Much of what Bill Dubuque knows of Galois Algebra needs to be revised
and parts thrown out as
pure fabrication. One part is the "between any two Reals is a new
Real" is simply a lie of modern
day mathematics.

Last call, Bill, please provide your own Euclid Infinitude of Primes
proof, direct and indirect, using
"multiply the lot and add 1"

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

Guest

plutonium.archime...@gmail.com wrote:
[quote]Bill Dubuque wrote:
plutonium.archimedes@gmail.com writes:
[/quote]
Now Bill will not like my answer that I gave earlier. Nor did Dik
Winter like any of my comments
on Galois theory as it pertains to AP-adics.

One of the problems here is that noone has really elaborated on why
Galois theory even works
in the first place and why it is a tool.

But the program of AP-adics never really uses Galois theory and
suggests that Galois theory needs
a modern day cleaning-up, revision and even some trashcanning of
trashy parts.

Most mathematicians would have started tackling AP-adics with Galois
theory, but I started them
with purely Geometry. Geometry is going to be around longer than any
Galois theory and geometry
has no chance of withering and dying away as does Galois theory. This
is one of the great avenues
of misguidance by mathematicians of the last century for they went
overboard on the importance of
Galois theory.

So I did this AP-adics program purely on the need to have these
numbers as points intrinsic to
Elliptic and Hyperbolic geometry. Only geometry as guide and careless
if anything agreed or disagreed
with Galois theory.

What makes Galois theory work? Or a better question is what is the
basic underpinning of Galois theory?
And this subject is seldom talked about by mathematicians, even those
that spend most of their time
on Galois theory.

What makes it work is that it is based on the concept of All Possible
Digit Arrangements. Galois theory
in essence is Probability theory in reverse. What I mean by that, is
that Probability theory is based on
knowing a "universe space of possibilities" and once you know that
space, they it is routine to figure out
the probability of a particular event as event/space.

So in essence Galois theory is Probability theory in reverse. And what
makes Probability theory work
is All Possible Digit Arrangements for that is a fancy title for
"universe space of possibilities".

Now what few people who worked in Galois theory failed to recognize is
that all of Galois theory is dependent
on "All Possible Digit Arrangements". So, when mathematicians were
working to find out if the Reals
are a ordered field or some sort of commutative ring or some other
algebraic structure, they thought
that their work consisted of defining operations to make it all agree
with those algebraic structures. When
in fact, all they really needed to do was to see if the collection of
numbers is a "all possible digit arrangement"

Now here is a finite set of all possible digit arrangements:
..00 up to .99. That set are two place value of Reals between zero and
1.0. But it is all possible digit
arrangements of two place value to the right of the decimal point. It
forms an algebraic structure, not because
some mathematicians can spend hours defining and redefining operations
to satisfy some algebraic structure,
but it is a algebraic structure because it is all possible digit
arrangements for two place value right of decimal
point.

And also, the Integers of two place value is 00, 01, on up to 99. So
that set is all possible digit arrangement
for two place value on integers. It also is a algebraic structure, the
same as the Real example. It is a
structure not because someone spent long hours making the operations
fit, but because from the beginning
it was "all possible digit arrangements for two place value"

So my answer to Bill, is that I developed the AP-adics purely from
geometry, and care less about whether
anything obeyed some algebraic structure.

But I know the AP-adics have a algebraic structure that is probably a
Ordered Field, simply because
the Reals are an ordered-field and both sets are "all possible digit
arrangements".

So here is a question for Bill. What Algebraic structure is the above
Reals of all possible digit arrangement
for two place value, and same question for integers of all possible
digit arrangement of two place value.

Now if I remember correctly the all possible digit arrangement of two
place value of two digits of
00, 01, 10, 11 is a Field. Now would all possible digit arrangements
of three place value of two digits
be the same structure as the two place value?

As I said so many times to Dik Winter, geometry is more important,
more vital, more guiding than
is Galois theory, and Galois theory is highly overrated.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

Guest

Bill Dubuque wrote:

Bill Dubuque wrote:

[quote]
Archie, before I can comment you need to answer the following
questions about your ring of AP-adics, henceforth called AP.

1) Is the hypothesized ring AP a (totally) ordered ring, i.e.
does AP have a subset P of "positives", closed under + and *,
with every elt either positive (in P), negative (in -P) or 0 ?

2) Are the positive elements P well-ordered, i.e. does every
nonempty subset of P have a least element?

I didn>t see an answer to my questions. Could you please plainly
answer either true or false to the above questions. It would help
if you could do so concisely, e.g. something like:

1) is ____

2) is ____

--Bill Dubuque
[/quote]
No, I gave you answers

Euclidean geometry is composed of Elliptic geom unioned with
Hyperbolic geom

Euclidean geometry are All Possible Digit Arrangements of infinite
rightward strings
containing both negative and positive signed numbers

Elliptic geom is All Possible Digit Arrangements of Positive Signed
infinite leftward strings
with the tacking on of two imaginary numbers for the North and South
Pole

Hyperbolic geom is All Possible Digit Arrangements of Negative Signed
infinite leftward
strings with one imaginary number for the zero point

Geometry is above Algebra of its Group and Ring and Field theory.

In fact, Field theory requires us to have the Reals in order to invent
Field theory with its
continuity postulate --- between any two Reals is a new Real or
derived in the Archimedean Postulate
n*e > M combined with Euclidean Completeness Postulate-- every
positive number has a positive
square root.

So Field Algebra was invented after we had Euclidean Geometry and
invented by trying to distill the
features of Euclidean Geometry. So Field Algebra is a attempt at
capturing the essence of Euclidean
geometry.

But Algebra misses the essence of Euclidean Geometry for it is merely
the All Possible Digit Arrangements.

So Bill, what is the Real Number between

0.0000....000000 and 0.0000.....000001

or, what is the Real Number between

0.00000.....0000005 and 0.00000.....0000006

In the old math where Bill comes from, they never had FrontView and
they never realized the importance
of "All Possible Digit Arrangements". Absent of FrontView and All
Possible Digit Arrangements, they
went on to build these phony house of cards that comes tumbling down.

Euclidean Geometry is All Possible Digit Arrangements of both negative
and positive signed numbers
of infinite rightward strings.

There is no continuity in Euclidean Geometry for there are holes
between every numbers, just as there
are big holes between the Counting Numbers going from 1 to 2 or 2 to
3.

What makes Euclidean geometry flat is that the strings are infinite
rightwards and contain both positive
and negative numbers.

If we separate out all the positive Reals and flip them around so the
string is infinite leftwards, what we
have done is created Elliptic geometry and with the negative Reals if
we flip them around as infinite
leftwards we have created Hyperbolic geometry. So that Elliptic and
Hyperbolic were nested inside
of Euclidean geometry.

So it is pointless of me in answering Bill>s question which is
dinosaurish math that was dead in the last
century. Pointless to ask me what AP-adics are in the fake Algebras of
the last century. Bill just may
as well ask me whether a fire breathing dragon has red eyes or green
eyes or uses butane or propane,
which is all pointless since no fire breathing dragon exists.

But this is typical of a mathematician of the last century whose mind
is submerged in falsehoods. Can
anyone show Bill that Algebras are phony baloney beyond what Galois
used them for in the quintic.
When you stretch something beyond its use, such as Galois theory or
mortgage lending or dot.com
finances, when you stretch something beyond their use, you run into
fakery and phonyness.

I built AP-adics from geometry, something true and everlasting, not
the overstretched Galois theory built
only to address the quintic problem of centuries past.

It is easy for the modern day mathematician to get sucked into
phonyness of the Reals having a Cantor
infinity or having "absolute continuity". Easily sucked in because no-
one really asked
"Well, aren>t the Reals just the same as All Possible Digit
Arrangements?"

I mean, what an utterly simple question and which virtually every
practicing mathematician used
during their lifetime of study and teaching math. I know I used that
expression "all possible digit
arrangements" thousands of times when doing mathematics.

So why did not any mathematician in the 20th century ever say "the
Reals are nothing more than
All Possible Digit Arrangements"

Now if you couple "All Possible Digit Arrangements" with FrontView,
well you really revolutionize
the subject of Mathematics.

So Bill Dubuque who loves playing games of algebra. What Real Number
is between these two
Reals:

0.00000.......00008 and 0.00000.....00009 for there is none. There is
a hole between those two
Real Numbers just as there is a hole between the largest Real in the
interval 0 to 1 as
0.99999.....99999 which is 0.0000....00001 away from 1 itself.

So how can you expect me to answer you algebra question when your
algebra is nothing but
phony fakery. You tossed away the true theory of Geometry and you
spent your life in a quagmire
of fakery of Galois theory.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

Guest

On Jul 14, 8:02 am, Bill Dubuque <w...@nestle.csail.mit.edu> wrote:
[quote]plutonium.archime...@gmail.com wrote:
Bill Dubuque <w...@nestle.csail.mit.edu> wrote:
1) Is the hypothesized ring AP a (totally) ordered ring, i.e.
does AP have a subset P of "positives", closed under + and *,
with every elt either positive (in P), negative (in -P) or 0 ?
2) Are the positive elements P well-ordered, i.e. does every
nonempty subset of P have a least element?
Lurkers please refrain from commenting until AP responds.
Well, Bill, the reason I wanted you to give a Euclid Infinitude of
Primes proof, both Integers, which is an ill-defined set [...]
I didn>t see an answer to my questions. Could you please plainly
answer either true or false to the above questions. It would help
if you could do so concisely, e.g. something like:
1) is ____
2) is ____
[/quote]
Archimedes Plutonium gives a partial answer to this
question in another thread, post #581:

[quote]The way I have set up the AP-adics is that they form the intrinsic
numbers that lie one the surface of
a ellipsoid (set of all positive AP-adics) and lie on the surface of
a pseudosphere (set of all negative
AP-adics).
[/quote]
So there definitely exist positive and negative AP-adics.


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