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malc...@gmail.com wrote:

[quote]
Just to put things in order:

I do not hate anyone. I just noted that in normal speech such kind of
mistakes are common and that it is also normal not to pay much
attention on them. Certainly one would expect any competent person in
any field to be able to grasp this kind of mistakes instantly.
However, as you have come to realize, this simply doesn>t happen.

[/quote]
When G.H. Hardy writes the Euclid Infinitude of Primes Proof in a book
"A Mathematicians Apology"
and makes the logical mistakes and when Niven, Zuckerman, Montgomery,
in a textbook
AN INTRODUCTION TO THE THEORY OF NUMBERS makes the logical mistakes.

--- quoting from my book Correctiong Euclid>s Infinitude of Primes
Proof ---
(#3) --- quoting WHAT IS MATHEMATICS? Richard Courant and Herbert
Robbins
1941 page 22 ---
The proof of the infinitude of the class of primes as given by Euclid
remains a model of mathematical reasoning. It proceeds by the
"indirect
method". We start with the tentative assumption that the theorem is
false. This means that there would be only a finite number of primes,
perhaps very many -- a billion or so -- or, expressed in a general and
non-committal way, n. Using the subscript notation we may denote these
primes by p1, p2, ...,pn. Any other number will be composite, and must
be divisible by at least one of the primes p1,p2,...,pn. We now
produce
a contradiction by constructing a number A which differs from every
one
of the primes p1, p2, ..., pn because it is larger than any of them,
and which nevertheless is not divisible by any of them. This number is
A = (p1xp2x...xpn) +1, i.e. 1 plus the product of what we supposed to
be all the primes. A is larger than any of the p>s as a divisor. Since
our initial assumption that there is only a finite number of primes
leads to this contradiction, the assumption is seen to be absurd, and
hence its contrary must be true. This proves the theorem.
--- end quoting WHAT IS MATHEMATICS? Courant and Robbins ---

One thing that Courant and Robbins do that is really good is clearly
state what they thought Euclid method was.
But then their proof pretty much dissolves away or
collapses. For they did not fetch a new prime to ever warrant them
saying
they reached a contradiction. They say that A is different and A is
absurd,
but why were they never able to say that A is necessarily a new prime.
Like the other authors listed before, if Courant and Robbins had had
to
provide
both a indirect and direct method proof, perhaps they would have
delivered
a clear and valid result instead of this incomplete attempt.
--- end quoting ---

I would say those books are serious books and if they cannot give a
waterproof proof of
Euclid IP without huge error, then Malcolm is a hatemonger that trys
to diminish and belittle
the accomplishments of Archimedes Plutonium.

In my example of the Grade Schoolers Analogy:

Add this column:
3
5
10
9
____

And if thirty Grade school children handed in their answer and twenty-
eight of them
summed to 18
while two of the children summed to 27.

Then Malcolm is going to say that all thirty had it correct and that
their mistakes were
minor language mistakes.

This is where people in academics and education no longer belong in
those fields, where
they continue to make excuses and continue to not recognize
achievement.

When we ask the question of 30 professors of mathematics about their
published "alleged"
proof of Euclid Infinitude of Primes (of course Hardy and Courant and
Polya are dead) but
of those 30 professors who are living who made the mistakes. If you
ask them this question:

If the Universe of "all the primes" were merely the set of 3 and 5,
then (3x5) + 1 = 16
then the 16 is
necessarily a new prime in that universe, and the reason for the
contradiction is because
of the starting off definition of prime. If you ask those of the
thirty who got it wrong, whether
they can agree and understand that 16 is a new prime number, then they
are admitting to
their big mistake.

Likewise, if we were to question the two Grade Schoolers why they
added 10 as 10 and not as
1 and they are able to tell you that 10 is the number after 9 while
the twenty eight kids with the
wrong answer see 10 as 1 and 0.

[quote]I think that the matter on how horrible the mistake is, is a bit
subjective. For me it is a curiosity, and I believe that it is due to
a confusion of two very similar proofs. I do not think any of those
mathematicians, Hardy included, would not have corrected their proofs
as soon as anyone would have made them note that.

[/quote]
A mistake is a mistake. If you answer that 3 + 5 + 10 + 9 = 18, then
do not
cover up your mistake. Likewise, if you claim to be giving a valid
Euclid Infinitude of Primes
Proof such as Niven, Zuckerman, Montgomery or Courant and Robbins or
Polya or Hardy,
then do not try to be making excuses for your big mistake. Come forth
and admit the
mistake is big.

[quote]I am sorry that you have felt my discussion as an attack.

[/quote]
You suffer the same disease of hatred that Jesse Hughes suffers. You
see someone you
hate, and then you make all sorts of excuses. You cannot admit that
Archimedes Plutonium corrected a
big mistake made by mathematics professors in delivering a valid proof
of Infinitude of Primes.

The moment that a person such as you, cannot admit to the
accomplishments and
achievements by others then you should depart education and science.

[quote]The question about the primeness of the strange looking ......
13121110987654321 was serious. Is there anything similar to the
fundamental theorem of arithmetic for your AP-adics?

Cheers.
[/quote]
I wrote a very long book recently called AP-adics primer, where your
question is answered. That
answer is too long to get started here. The above is a prime number
for it is Champernowne>s (spelling)
number attached to the primes 11 and 13 or any other pair of primes.
In that book I argue that
this set of numbers 1,2,3,.... the Counting Numbers are fictional.
Again, too long to start a discussion
here.

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

Tonico · Guest

On Jul 3, 7:25 pm, plutonium.archime...@gmail.com wrote:
[quote]malc...@gmail.com wrote:

Just to put things in order:

I do not hate anyone. I just noted that in normal speech such kind of
mistakes are common and that it is also normal not to pay much
attention on them. Certainly one would expect any competent person in
any field to be able to grasp this kind of mistakes instantly.
However, as you have come to realize, this simply doesn>t happen.

When G.H. Hardy writes the Euclid Infinitude of Primes Proof in a book
"A Mathematicians Apology"
and makes the logical mistakes and when Niven, Zuckerman, Montgomery,
in a textbook
AN INTRODUCTION TO THE THEORY OF NUMBERS makes the logical mistakes.

--- quoting from my book Correctiong Euclid>s Infinitude of Primes
Proof ---
(#3) --- quoting WHAT IS MATHEMATICS? Richard Courant and Herbert
Robbins
1941 page 22 ---
The proof of the infinitude of the class of primes as given by Euclid
remains a model of mathematical reasoning. It proceeds by the
"indirect
method". We start with the tentative assumption that the theorem is
false. This means that there would be only a finite number of primes,
perhaps very many -- a billion or so -- or, expressed in a general and
non-committal way, n. Using the subscript notation we may denote these
primes by p1, p2, ...,pn. Any other number will be composite, and must
be divisible by at least one of the primes p1,p2,...,pn. We now
produce
a contradiction by constructing a number A which differs from every
one
of the primes p1, p2, ..., pn because it is larger than any of them,
and which nevertheless is not divisible by any of them. This number is
A = (p1xp2x...xpn) +1, i.e. 1 plus the product of what we supposed to
be all the primes. A is larger than any of the p>s as a divisor. Since
our initial assumption that there is only a finite number of primes
leads to this contradiction, the assumption is seen to be absurd, and
hence its contrary must be true. This proves the theorem.
--- end quoting WHAT IS MATHEMATICS? Courant and Robbins ---

One thing that Courant and Robbins do that is really good is clearly
state what they thought Euclid method was.
But then their proof pretty much dissolves away or
collapses. For they did not fetch a new prime to ever warrant them
saying
they reached a contradiction. They say that A is different and A is
absurd,
but why were they never able to say that A is necessarily a new prime.
Like the other authors listed before, if Courant and Robbins had had
to
provide
both a indirect and direct method proof, perhaps they would have
delivered
a clear and valid result instead of this incomplete attempt.
--- end quoting ---

I would say those books are serious books and if they cannot give a
waterproof proof of
Euclid IP without huge error, then Malcolm is a hatemonger that trys
to diminish and belittle
the accomplishments of Archimedes Plutonium.
[/quote]
****************************************************************

Nonsense Archie.

First, you have no accomplishment other than the ones your rather wild
and pretty pitiful imagination causes you to believe.

Second, the only huge error is that you believe you can do meaningful
physics and maths instead of making an ass of yourself.

Third: aren>t you ashamed of yourself?? I honestly hope you haven>t
spawned kids: they>ll be laughed and scoffed at badly at school.

Fourth: knock it off, fruitcake!

Regards
Tonio

Guest

Hi Malcolm, I am thinking that from reading your synopsis, that you do
not see it as a big mistake because
your synopsis is wrong:

Here is your synopsis:

malc...@gmail.com wrote:

[quote]I think that what Hardy had in mind is that since natural numbers are
well ordered, to prove the infinitude of primes you just need to prove
that the subset of all prime numbers greater than any given prime is
not empty. This and the reductio ad absurdum proof are both based on
the construction of the number 'product of primes plus one', so it is
normal to somehow 'confuse' them, being each of them correct in their
own.

Mathematicians do not normally write formal proofs but instead they
sort of describe them; they tell us how we should be able to write a
formal proof. This being understood, it is normal that whenever we
talk about things we know and understand well we make mistakes in our
discourse which, from a pragmatic point of view, can perfectly be
ignored without any damage. So giving any importance to that kind of
mistakes doesn>t make anyone any sort of genius.
[/quote]
Malcolm, I do not see it as you described above. So I am asking not
for
some childish request on my part, but as an instructive quest. I am
asking
that you provide a Euclid Infinitude of Primes Proof reductio ad
absurdum
from you, purely from your mind and not something that you read and
try to
verbalize in your own words.

And if you feel good, even a Direct proof of Euclid Infinitude of
Primes; and if
not feeling good, well okay, no big deal.

So if you would care to offer up from your mind a Euclid IP reductio
ad absurdum
then I will check it out, and hopefully convince you that what I have
done is a big
accomplishment of correcting the Euclid IP proof.

Again, if you do not feel up to it, well, that is that.

Also, Jesse Hughes is probably reading and watching. If he feels up to
writing a Euclid IP
both in direct and indirect, or just one of them, I would be happy to
check it out, because I
seriously doubt that Jesse could even do a Euclid IP from his mind
without reading it and
parroting from some book.

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

Larry Hammick wrote:
(snipped)
[quote]in Archie>s bonnet. I remarked here at sci.math that there are several
differences between Euclid>s proof of IX.20
http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html
and the modern version, although the basic ideas are same and the proof is
still attributed to Euclid. In particular,

[/quote]
Not true. Apparently you arrived at the sci.math in 2002.

I had started discussing the flaw of Euclid>s IP in 1993-1994 sci.math

witness:

Sun, 20FEB1994, 21:05:13 GMT sci.math
INCONSISTENT PEANO AXIOMS AND MATH PROFESSORS
Lines: 36
Sender: k...@spdcc.com (Karl Heuer)

k...@ursa-major.spdcc.com (Karl Heuer) writes:

In article (5JChA8g2...@jojo.escape.de>

det...@jojo.escape.de (Detlef Bosau) writes:
[quote]Ludwig.Pluton...@dartmouth.edu meinte am 18.02.94
det...@jojo.escape.de (Detlef Bosau) writes:
Wrong. Your two numbers are not necessarily prime
NO, YOU ARE WRONG. Those numbers are necessarily prime, due to
UPFAT, all the primes that exist in the finite set leave a remainder
of 1.
I>ll give you a lesson of elementary arithmetics. . .
[/quote]
I really shouldn>t bother to get involved in this discussion again,
but
Ludwig is right. In logical terms, his key statement is "if P is a
finite set containing all the primes, then prod(P)+1 is prime." This
is
a true statement.

Let>s step through your alleged counterexample:

[quote]consider your set of primes to be: {2,3,5,7,11,13}, as I assert 13 to be
the largest prime. [. . .] Now, you made the assertion, that
(2x3x5x11x13) + 1 [=30031] must be prime.
[/quote]
Yes, it>s true that if 13 is the largest prime, then 30031 is prime.
Do
you disagree with that assertion?

[quote]As you stated before, there exists an unique prime decomposition of
30031. This is 59x509. It could be easily shown, that 59 and 509
both are prime.
[/quote]
If 13 is the largest prime, then 59x509 is not a factorization of
30031.

--- end quoting Karl Heuer>s post of 1994 ---

And earlier than 1993-4, I had sent a journal article to Notre Dame,
some
logic journal where I discussed the flaw of Euclid>s IP circa
1991-1993

So, Larry, how is it that you influenced me in 1991 with your posts of
2002?

If you cannot even get your dates right, you expect to get the logic
of a math proof right?

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

Larry Hammick · Guest

Tonico, I blush to admit that it was I myself who inadvertantly put this bee
in Archie>s bonnet. I remarked here at sci.math that there are several
differences between Euclid>s proof of IX.20
http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html
and the modern version, although the basic ideas are same and the proof is
still attributed to Euclid. In particular,

-- Euclid dares not speak of "infinitely many" primes (or any other actual
infinite) as we do.

-- Euclid works on what we would call an arbitrary finite non-empty family
of primes. He does not argue by contradiction and hypothesize that they are
/all/ the primes, and he does not assume that they are distinct.

Anyhow, here>s a proof of IX.20 discovered by Filip Saidak in 2005 AD!
Define inductively a sequence (x(n), y(n)) of ordered pairs by
x(1) = 2
y(1) = 3
and for n >= 2 :
x(n+1) = x(n)y(n)
y(n+1) = x(n)y(n) + 1
By induction, x and y stay positive. Also x(n) and y(n) are relatively prime
by definition. So y(n) has a prime factor which x(n) lacks, and that factor,
along with all the factors of x(n), appears in x(n+1). So by induction x(n)
has at least n distinct prime factors, for arbitrary n.

LH

Bill Dubuque · Guest

"Larry Hammick" <larryhammick@telus.net> wrote:
[quote]
Anyhow, here>s a proof of IX.20 discovered by Filip Saidak in 2005 AD!
Define inductively a sequence (x(n), y(n)) of ordered pairs by
x(1) = 2, > y(1) = 3
and for n >= 2 :
x(n+1) = x(n)y(n)
y(n+1) = x(n)y(n) + 1
By induction, x and y stay positive. Also x(n) and y(n) are relatively prime
by definition. So y(n) has a prime factor which x(n) lacks, and that factor,
along with all the factors of x(n), appears in x(n+1). So by induction x(n)
has at least n distinct prime factors, for arbitrary n.
[/quote]
SIMPLER NN + N has more prime factors than N

"Saidak>s proof" is certainly not new.

--Bill Dubuque

Guest

Bill Dubuque wrote:
[quote]"Larry Hammick" <larryhammick@telus.net> wrote:

Anyhow, here>s a proof of IX.20 discovered by Filip Saidak in 2005 AD!
Define inductively a sequence (x(n), y(n)) of ordered pairs by
x(1) = 2, > y(1) = 3
and for n >= 2 :
x(n+1) = x(n)y(n)
y(n+1) = x(n)y(n) + 1
By induction, x and y stay positive. Also x(n) and y(n) are relatively prime
by definition. So y(n) has a prime factor which x(n) lacks, and that factor,
along with all the factors of x(n), appears in x(n+1). So by induction x(n)
has at least n distinct prime factors, for arbitrary n.

SIMPLER NN + N has more prime factors than N

"Saidak>s proof" is certainly not new.

--Bill Dubuque
[/quote]
Not as neat or tidy or all-inclusive as the new Euclid Infinitude of
Primes
Proof that includes all the Natural Numbers. When Euclid or Saidak did
their proofs
of the Infinitude of Primes they were not aware of all the Natural
Numbers such as
.....999999 or ....666666777777. Where Natural Numbers = AP-adics.

So what is the easiest of all proofs that primes are infinite? And I
believe this proof is
the fastest, easiest.

We simply construct an Infinitude of Primes:
(1) 11 and 13 are prime
(2) the number ....1413121110987654321 is also prime as Champernownes
number (spelling)
(3) construct an infinite set of primes by simply attaching the "11"
and "13" on the far right
yielding ....141312111098765432111 and ....141312111098765432113 now
for the next
pair of twin primes we attach a 2 giving ....1413121110987654321211
and ....1413121110987654321213
now the next pair of twin primes we attach a 3 giving ....
14131211109876543213211 and ....14131211109876543213213
now for the next pair of twin primes we attach a 4 to the previous 3
and 2 and we keep doing this infinitely.
Now I not only proved the primes are infinite but that Twin-primes are
infinite. So this is the first
proof of Euclid Infinitude of Primes and Twin-primes in one proof.

Now I like to challenge Mr. Bill Dubuque to a case of "logical flow"
of proof coupled with brevity.
Brevity that Ian Stewart showed when he called it "multiply the lot
add 1". I
challenge Bill to give a Euclid Infinitude of Primes proof of Direct
Method and then Indirect Method.
Two proofs simultaneously, one of direct and the other indirect for
contrast. In the past,
mathematicians were never required to give both simultaneously. Maybe
if they had been required
that such a high percentage of failures 28/30 = 93% failure to render
a valid proof argument.

Are you up for a challenge Bill?

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 3 jul, 18:25, plutonium.archime...@gmail.com wrote:
[quote]malc...@gmail.com wrote:

Just to put things in order:

I do not hate anyone. I just noted that in normal speech such kind of
mistakes are common and that it is also normal not to pay much
attention on them. Certainly one would expect any competent person in
any field to be able to grasp this kind of mistakes instantly.
However, as you have come to realize, this simply doesn>t happen.

When G.H. Hardy writes the Euclid Infinitude of Primes Proof in a book
"A Mathematicians Apology"
and makes the logical mistakes and when Niven, Zuckerman, Montgomery,
in a textbook
AN INTRODUCTION TO THE THEORY OF NUMBERS makes the logical mistakes.

--- quoting from my book Correctiong Euclid>s Infinitude of Primes
Proof ---
(#3) --- quoting WHAT IS MATHEMATICS? Richard Courant and Herbert
Robbins
1941 page 22 ---
The proof of the infinitude of the class of primes as given by Euclid
remains a model of mathematical reasoning. It proceeds by the
"indirect
method". We start with the tentative assumption that the theorem is
false. This means that there would be only a finite number of primes,
perhaps very many -- a billion or so -- or, expressed in a general and
non-committal way, n. Using the subscript notation we may denote these
primes by p1, p2, ...,pn. Any other number will be composite, and must
be divisible by at least one of the primes p1,p2,...,pn. We now
produce
a contradiction by constructing a number A which differs from every
one
of the primes p1, p2, ..., pn because it is larger than any of them,
and which nevertheless is not divisible by any of them. This number is
A = (p1xp2x...xpn) +1, i.e. 1 plus the product of what we supposed to
be all the primes. A is larger than any of the p>s as a divisor. Since
our initial assumption that there is only a finite number of primes
leads to this contradiction, the assumption is seen to be absurd, and
hence its contrary must be true. This proves the theorem.
--- end quoting WHAT IS MATHEMATICS? Courant and Robbins ---

One thing that Courant and Robbins do that is really good is clearly
state what they thought Euclid method was.
But then their proof pretty much dissolves away or
collapses. For they did not fetch a new prime to ever warrant them
saying
they reached a contradiction. They say that A is different and A is
absurd,
but why were they never able to say that A is necessarily a new prime.
Like the other authors listed before, if Courant and Robbins had had
to
provide
both a indirect and direct method proof, perhaps they would have
delivered
a clear and valid result instead of this incomplete attempt.
--- end quoting ---

I would say those books are serious books and if they cannot give a
waterproof proof of
Euclid IP without huge error, then Malcolm is a hatemonger that trys
to diminish and belittle
the accomplishments of Archimedes Plutonium.

In my example of the Grade Schoolers Analogy:

Add this column:
3
5
10
9
____

And if thirty Grade school children handed in their answer and twenty-
eight of them
summed to 18
while two of the children summed to 27.

Then Malcolm is going to say that all thirty had it correct and that
their mistakes were
minor language mistakes.

This is where people in academics and education no longer belong in
those fields, where
they continue to make excuses and continue to not recognize
achievement.

When we ask the question of 30 professors of mathematics about their
published "alleged"
proof of Euclid Infinitude of Primes (of course Hardy and Courant and
Polya are dead) but
of those 30 professors who are living who made the mistakes. If you
ask them this question:

If the Universe of "all the primes" were merely the set of 3 and 5,
then (3x5) + 1 = 16
then the 16 is
necessarily a new prime in that universe, and the reason for the
contradiction is because
of the starting off definition of prime. If you ask those of the
thirty who got it wrong, whether
they can agree and understand that 16 is a new prime number, then they
are admitting to
their big mistake.

Likewise, if we were to question the two Grade Schoolers why they
added 10 as 10 and not as
1 and they are able to tell you that 10 is the number after 9 while
the twenty eight kids with the
wrong answer see 10 as 1 and 0.

I think that the matter on how horrible the mistake is, is a bit
subjective. For me it is a curiosity, and I believe that it is due to
a confusion of two very similar proofs. I do not think any of those
mathematicians, Hardy included, would not have corrected their proofs
as soon as anyone would have made them note that.

A mistake is a mistake. If you answer that 3 + 5 + 10 + 9 = 18, then
do not
cover up your mistake. Likewise, if you claim to be giving a valid
Euclid Infinitude of Primes
Proof such as Niven, Zuckerman, Montgomery or Courant and Robbins or
Polya or Hardy,
then do not try to be making excuses for your big mistake. Come forth
and admit the
mistake is big.

I am sorry that you have felt my discussion as an attack.

You suffer the same disease of hatred that Jesse Hughes suffers. You
see someone you
hate, and then you make all sorts of excuses. You cannot admit that
Archimedes Plutonium corrected a
big mistake made by mathematics professors in delivering a valid proof
of Infinitude of Primes.

The moment that a person such as you, cannot admit to the
accomplishments and
achievements by others then you should depart education and science.

The question about the primeness of the strange looking ......
13121110987654321 was serious. Is there anything similar to the
fundamental theorem of arithmetic for your AP-adics?

Cheers.

I wrote a very long book recently called AP-adics primer, where your
question is answered. That
answer is too long to get started here. The above is a prime number
for it is Champernowne>s (spelling)
number attached to the primes 11 and 13 or any other pair of primes.
In that book I argue that
this set of numbers 1,2,3,.... the Counting Numbers are fictional.
Again, too long to start a discussion
here.

Archimedes Plutoniumwww.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
[/quote]
Hi,

You say my synopsis is wrong as a whole? Please, explain.

It also seems you want me to provide a proof... Ok, here it goes.

We say a natural number is prime iff it is greater than 1 and doesn>t
have any divisor other than 1 and itself.
By the fundamental theorem of arithmetic, every number greater than 1
has a prime divisor.

Let us suppose that the subset of all prime numbers is finite.
Then the product of all primes plus one, since it doesn>t belong to
the subset of all prime numbers (it is greater than any of them) is
not a prime, but it has a prime divisor, since it is greater than 1.
But in fact, it doesn>t have any divisor from the list of all primes.
Contradiction.

I keep on believing that the key idea in this proof is the
construction of the number "the product of all primes plus one", and
not the strict application of the rules of logic, which are supposed
to work when applied correctly.

For some other proofs of the infinitude of primes I recommend:

Number Theory. An Introduction via the Distribution of Primes
Benjamin Fine
Gerhard Rosenberger
Birkhäuser

Cheers.

Guest

malc...@gmail.com wrote:

[quote]
Hi,

You say my synopsis is wrong as a whole? Please, explain.

[/quote]
Thanks for participating, because I think we can unravel much more
when
we actually do a Euclid IP rather than we commenting on the sidelines
on Euclid IP. As
an old pragmatist that I am and pragmatist saying-- The learning is in
the doing, and not the peripheral commenting.

[quote]It also seems you want me to provide a proof... Ok, here it goes.

We say a natural number is prime iff it is greater than 1 and doesn>t
have any divisor other than 1 and itself.
By the fundamental theorem of arithmetic, every number greater than 1
has a prime divisor.

Let us suppose that the subset of all prime numbers is finite.
Then the product of all primes plus one, since it doesn>t belong to
the subset of all prime numbers (it is greater than any of them) is
not a prime, but it has a prime divisor, since it is greater than 1.
But in fact, it doesn>t have any divisor from the list of all primes.
Contradiction.

[/quote]
There is a problem with this statement in your above.
"Let us suppose that the subset of all prime numbers is finite."

I doubt it is a mathematical statement but the assemblage of
contradictions in terms. Like saying "Let us suppose infinity is
finite."

Malcolm, can you write your above and parallel each statement with a
numbers example.

Here is one for my proof of Euclid IP.

(1) Definition of Prime number
(2) Suppose the set of all primes is Finite
(2*) Suppose 3 and 5 were all the primes that exist
(3) Multiply the lot and add 1
(3*) (3x5) +1 = 16
(4) This new number is necessarily a new prime since we revert to our
definition in (1)
and all the primes divided into this new number leave a remainder of 1
(4*) 3 and 5 are all the primes that exist and when divided into 16
leave a remainder
of 1 and by the definition of prime from (1), that 16 is indeed a new
prime
(5) Contradiction
(6) Set of all primes is infinite

The essence of the correction of math professors such as Conway,
Courant, Hardy, Niven,
Montgomery, Zuckerman, is that they could not understand that 16 is
prime in the Indirect
Method. No matter what the number "multiply the lot and add 1" is, no
matter what it is, whether
it is 16 or (3x5x7) +1 = 106, that 106 is a new prime when the
universe of all primes is 3,5,7.

This is my correction of the math communities horrible habit of not
giving a valid Euclid Infinitude
of Primes proof, as they seem unable to understand the logic and they
mix the direct with the indirect.

Malcolm, you see how I parallel the statements with a numbers example.

Your above is not a proof because you can never run a numbers example
parallel to your statements
for some of your statements are not even mathematical.

[quote]I keep on believing that the key idea in this proof is the
construction of the number "the product of all primes plus one", and
not the strict application of the rules of logic, which are supposed
to work when applied correctly.

[/quote]
The crux of the proof is the synchronized working arrangement between
two elements in the proof-- the definition of prime at the start and
the "multiply the
lot and add 1" Yours above Malcolm does not even use the definition of
prime.

I am hoping that Bill Dubuque offers up his version so that I can dive
into
the internal logic of the direct versus the indirect. There is much
mathematics
in the what I call "Meta-Euclid Infinitude of Primes Proof" The
analysis of the
logical underpinning of the direct and indirect.

[quote]For some other proofs of the infinitude of primes I recommend:

Number Theory. An Introduction via the Distribution of Primes
Benjamin Fine
Gerhard Rosenberger
Birkhï¿½user

Cheers.
[/quote]
Thanks, will look them up next time in a library and see if they
managed to give a valid
Euclid IP proof. The statistics for math professors is not running
good since only 2 out of
30 have managed to give a valid Euclid IP.

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

Larry Hammick wrote:
[quote]plutonium.archimedes@gmail.com> wrote in message
news:e6c0d43e-e82d-44aa-bcee-0905d9674f6c@b1g2000hsg.googlegroups.com...

Larry Hammick wrote:
(snipped)
in Archie>s bonnet. I remarked here at sci.math that there are several
differences between Euclid>s proof of IX.20
http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html
and the modern version, although the basic ideas are same and the proof
is
still attributed to Euclid. In particular,

Not true. Apparently you arrived at the sci.math in 2002.

Sounds about right.

I had started discussing the flaw of Euclid>s IP in 1993-1994 sci.math

witness:

Sun, 20FEB1994, 21:05:13 GMT sci.math
INCONSISTENT PEANO AXIOMS AND MATH PROFESSORS
Lines: 36
Sender: k...@spdcc.com (Karl Heuer)

Okay, but were you looking at Euclid himself, or at Hardy and the numerous
other writers who speak of "the set of all primes" and "Euclid>s proof"?
Anyhow, it>s a trivial quibble.
[/quote]
Larry, I doubt you can even give a valid Euclid IP both direct and
indirect without major flaws.

You want to give it a try?

Guest

The title of this post is not capricious in "all sorts of forms" There
are the two listed forms but there are many
other forms of primes such as k^2 +1 and k^2 -1 and then there are
forms such as k!+1 and k!-1.

So there are very many forms of primes, and if my memory is correct,
someone thought they proved
a certain form of primes is a finite set. I say they must review their
alleged proof because they
started with the assumption that Natural Numbers are finite integers
when in fact the true blue
Natural Numbers are the AP-adics or infinite integers.

A number such as this is a prime number 100000.......00000003 which is
an infinite integer whose
front-view is "1" and whose endview is "3". Another prime number
is ....1413121110987654321.
So in earlier times, mathematicians never realized that the set of
primes is so much larger than their
deluded picture of primes. They were playing cards with over half the
deck missing.

In a previous post I gave a proof of the infinitude of Twin Primes by
grafting 11 and 13 unto the backbone
of the above Champernownes number .....1413121110987654321. So I
constructed a infinite string of
Twin Primes.

Now, can I construct an infinite string of primes of form 2k+1? It is
a little more challenging than Twin
Primes construction.

It goes like this. I start with the prime number of .......
1111100001111000111001101. Now it is
similar to the pattern of Champernownes number only its digits are 1
and 0. It is prime because these
numbers are "Irrational Counting Numbers". Now I have to give a
precise definition of Irrational Natural
Numbers. We have definition of Composite Natural Number and Prime
Natural Number but now
we must provide definitions of Irrational Natural Numbers and Rational
Natural Numbers. I am not going
to do that here for this is already long enough.

Now this number .......1111100001111000111001101 in the form 2k+1
is .......2222200002222000222002203
which is also prime. Now the next prime I construct is to scoot down
leftwards and eliminate the first
"1" as this number .......111110000111100011100110 which is not prime
itself but when I do the
2k+1 it becomes .......222220000222200022200221 Now I infinitely scoot
down the line and produce a
new prime of form 2k+1. Thus the set of all primes of form 2k+1 is an
infinite set. Likewise for primes
of form 2k-1 and likewise for all other forms.

Even the form that some mathematicians of the past were thought to be
a finite set, because of their
alleged proof, is not questioned since they started their work with
the deluded belief that Natural Numbers
are finite integers.

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

Larry Hammick · Guest

<plutonium.archimedes@gmail.com> wrote in message
news:e6c0d43e-e82d-44aa-bcee-0905d9674f6c@b1g2000hsg.googlegroups.com...
[quote]
Larry Hammick wrote:
(snipped)
in Archie>s bonnet. I remarked here at sci.math that there are several
differences between Euclid>s proof of IX.20
http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html
and the modern version, although the basic ideas are same and the proof
is
still attributed to Euclid. In particular,

Not true. Apparently you arrived at the sci.math in 2002.
[/quote]
Sounds about right.

[quote]
I had started discussing the flaw of Euclid>s IP in 1993-1994 sci.math

witness:

Sun, 20FEB1994, 21:05:13 GMT sci.math
INCONSISTENT PEANO AXIOMS AND MATH PROFESSORS
Lines: 36
Sender: k...@spdcc.com (Karl Heuer)
[/quote]
Okay, but were you looking at Euclid himself, or at Hardy and the numerous
other writers who speak of "the set of all primes" and "Euclid>s proof"?
Anyhow, it>s a trivial quibble.

Guest

I wrote earlier today:

[quote]It also seems you want me to provide a proof... Ok, here it goes.

We say a natural number is prime iff it is greater than 1 and doesn>t
have any divisor other than 1 and itself.
By the fundamental theorem of arithmetic, every number greater than 1
has a prime divisor.

Let us suppose that the subset of all prime numbers is finite.
Then the product of all primes plus one, since it doesn>t belong to
the subset of all prime numbers (it is greater than any of them) is
not a prime, but it has a prime divisor, since it is greater than 1.
But in fact, it doesn>t have any divisor from the list of all primes.
Contradiction.

[/quote]
Malcolm, I now realize what you were trying to do above. You were
trying to
do a Direct Euclid IP. It should have looked like this, with number
examples
for steps.

Earlier today I gave the Indirect methof of Euclid IP and here it is:

[quote]
Here is one for my proof of Euclid IP.

(1) Definition of Prime number
(2) Suppose the set of all primes is Finite
(2*) Suppose 3 and 5 were all the primes that exist
(3) Multiply the lot and add 1
(3*) (3x5) +1 = 16
(4) This new number is necessarily a new prime since we revert to our
definition in (1)
and all the primes divided into this new number leave a remainder of 1
(4*) 3 and 5 are all the primes that exist and when divided into 16
leave a remainder
of 1 and by the definition of prime from (1), that 16 is indeed a new
prime
(5) Contradiction
(6) Set of all primes is infinite

[/quote]

Malcolm, here is the direct method that you were attempting to do:
Euclid IP (direct method)

(1) Definition of Prime number
(2) Set of primes is this set {2,3,5,7,11,13,....} and we are out to
prove whether it is finite or infinite
(3) Given any subset of primes, that subset has a cardinality
(3*) a subset of primes such as {3,5} had cardinality of 2 since it
has two members
(4) The proof involves the increase in set cardinality of given any
subset of primes
and since we can increase the cardinality by one more prime, means the
set of
all primes is infinite
(4*) the crux of the proof is that like numbers, given any set of
numbers, add one to
the largest and you automatically increase the set cardinality and
thus an infinite set
(5) Every particular subset of primes, we multiply the lot and add 1.
(5*) The subset {3,5} gives (3x5)+1 = 16
(6) This new number "multiply the lot and add 1" can either be prime
itself or have
a prime factor because of (1) definition of prime
(6*) (3x5)+1 = 16 so this subset of cardinality 2, we have either 16
is prime or has
a prime factor. Obviously 16 is not prime and the prime factor is 2.
(7) Thus, given any finite set of primes we can augment that set with
a new prime
of either "multiply the lot add 1" or a prime factor of "multiply the
lot add 1"
(7*) 16 is not prime but 2 is prime and not in the subset {3,5}
(8) Since any finite subset of primes has the ability of augmentation
of a new prime
not in the list, means the set of all primes is an infinite set

There Malcolm, you were striving for the Direct Method, but you made
several mistakes. You
included a Suppose reductio ad absurdum, when you should have never
done so. There is no
"suppose" and a contradiction involved in the Direct method.

What is the difference between Direct and Indirect? Both rely on the
definition as first step.
Both require the use of "multiply the lot and add 1". The major
difference is that in the Indirect,
the new number formed by "multiply the lot and add 1" is necessarily
prime, and it can not have
a prime factor. Most math professors get it wrong and deliver an
invalid proof because in the back of
their minds they remember that 30031 has factors of 59x509 for the
subset where 13 is the
largest prime, but in the indirect method the number 30031 is a new
prime since
the set of all primes is {2,3,5,7,11,13}. The definition coerces you
to conclude 30031 is a new prime
not on the list of all primes and this new prime number 30031 is the
contradiction and discharge of the
proof. This is the mistake Niven, Zuckerman, Montgomery made in their
textbook, and the mistake that
Courant and Conway and Wikipedia and 30 other math professors make.

The mistake is a lack of Logical Continuity, just as if we were to
electrically wire a new house and
if the lines are not connected correctly there will be no flow of
electricity.

So the moment you do a Indirect Euclid IP and mention a search for a
"prime factor", you lost the proof
and have fallen off the mountain.

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

Seldom has one human being (Archimedes Balonium) written so much and
said so little of value.

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.

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


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