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From: Archimedes Plutonium on 11 Jul 2010 22:02
Due to a conversation with a journal editor, my interest has been
rekindled in a proof of the Infinitude of Twin Primes and all even
numbered pairs of primes. The reason this attempt works, I feel, is
that I have eliminated the regular primes out of the picture by a
clever devise that was used in the Direct method of the
square root to eliminate factors.

XXXXXX
Euclid's Infinitude of Primes proof, Direct or constructive in short-
form goes like this:
  1) Definition of prime
  2) Given any finite set of primes
  3) Multiply the lot and add 1 (Euclid's number) which I call W+1
  4) Either W+1 is prime or we conduct a prime factor search
  5) this new prime increases the set cardinality by one more prime
  6) since this operation of increasing set cardinality occurs for
any
  given finite set we start with, means the primes are infinite set.

XXXXXX

Euclid Infinitude of Primes proof, Indirect in
 short-
  form goes like this:


1) Definition of prime
  2) Hypothetical assumption, suppose set of primes 2,3,5,7,.. is
  finite with P_k the last and final prime
  3) Multiply the lot and add 1 (Euclid's number) which I call W+1
  4) W+1 is necessarily prime
  5) contradiction to P_k as the last and largest prime
  6) set of primes is infinite.

XXXXXX

DIRECT Method (constructive method), long-form; Infinitude of Primes
 Proof


(1) Definition of prime as a positive integer divisible
  only by itself and 1.


(2) Statement: Given any finite collection of primes
 2,3,5,7,11, ..,p_n possessing a cardinality n Reason: given


(3) Statement: we find another prime by considering W+1 =(2x3x...xpn)
  +1 Reason: can always operate on given numbers


(4) Statement: Either W+1 itself is a prime Reason: Unique Prime
 Factorization theorem


(5) Statement: Or else it has a prime factor not equal to any of the
  2,3,...,pn
 Reason: Unique Prime Factorization theorem


(6) Statement: If W+1 is not prime, we find that prime factor Reason:
 We take the square root of W+1 and we do a prime search through all
 the primes from 2 to
 square-root of W+1 until we find that prime factor which
 evenly divides W+1


(7) Statement: Thus the cardinality of every finite set can be
 increased. Reason: from steps (3) through (6)


(8) Statement: Since all/any finite cardinality set can be increased
 by one more prime, therefore the set of primes is an infinite set.
 Reason: going from the existential logical quantifier to the
 universal
 quantification

XXXXXX

INDIRECT (contradiction) Method, Long-form; Infinitude of Primes
Proof
 and
 the numbering is different to show the reductio ad absurdum
structure
 as
 given by Thomason and Fitch in Symbolic Logic book.


(1) Definition of prime as a positive integer divisible
  only by itself and 1.


(2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
  Reason: definition of primes


(3.0) Suppose finite, then 2,3,5, ..,p_n is the complete series set
  with p_n the largest prime Reason: this is the supposition step


(3.1) Set S are the only primes that exist Reason: from step (3.0)


(3.2) Form W+1 = (2x3x5x, ..,xpn) + 1. Reason: can always operate and
  form a new number


(3.3) Divide W+1 successively by each prime of
  2,3,5,7,11,..pn and they all leave a remainder of 1.
  Reason: unique prime factorization theorem


(3.4) W+1 is necessarily prime. Reason: definition of prime, step
 (1).


(3.5) Contradiction Reason: pn was supposed the largest prime yet we
  constructed a new prime, W+1, larger than pn


(3.6) Reverse supposition step. Reason (3.5) coupled with (3.0)


(4) Set of primes are infinite Reason: steps (1) through (3.6)

XXXXXX


For years now I thought I had not delivered a proof of the Infinitude
of Twin Primes, that somehow I came up
short, but due to a email conversation, I realized that
all along I had proven the Infinitude of Twin, Quad, 6th primes
and all other even multiples Primes.

The proof is only Indirect method because only in the Indirect are you
ensured of two new primes.

Let me show you the Indirect Regular Primes Infinitude proof with a
number example:


Euclid Infinitude of Primes proof, Indirect in
 short- form with number example of 3 and 5 :


1) Definition of prime
  2) Hypothetical assumption, suppose set of primes 3,5 are all the
primes that exist with 5 the largest prime
  3) Multiply the lot and add 1 (Euclid's number) which is (3x5) +1 =
16
  4) 16 is necessarily prime due to (1) and the assumptive step
  5) contradiction to 5 as the last and largest prime
  6) set of primes is infinite.

That number example is what delivers a valid Infinitude of Twin
Primes, Quad Primes, 6th Primes, etc etc.

XXXXXX

Proof of the Infinitude of Twin Primes:

INDIRECT (contradiction) Method, Long-form; Infinitude of Twin Primes


(1) Definition of prime as a positive integer divisible
  only by itself and 1.


(2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
  Reason: definition of primes

(3) Let us instead pick the numbers of primes as
the succession of 2,3,5,7,. . , p(n), p(n+2) where
the p(n) and p(n+2) are twin primes


(4.0) Suppose twin primes are finite, then 2,3,5, ..,p_n ,
p_n+2 is the complete series set
  with p_n and p_n+2 the last and largest twin primes Reason: this is
the supposition step


(4.1) Set S are the only primes that exist Reason: from step (4.0)
This is the step in which I hesitated in calling
my proof a genuine proof because I pictured larger regular primes
beyond the p_n+2, but that was superfluous


(4.2) Form W+1 = (2x3x5x, ..,xp_n x p_n+2) + 1.
And form W-1 = (2x3x5x, ..,xp_n x p_n+2) - 1.
Reason: can always operate and
  form a new number


(4.3) Divide W+1 and W-1 successively by each prime of
  2,3,5,7,11,..p_n+2 and they all leave a remainder of 1.
  Reason: unique prime factorization theorem

Now here is where my previous proof attempts failed and here is the
patch I wish to apply to stop it from failing. If I apply a patch so
as to eliminate all the regular primes beyond p_n+2 then the proof
works.
And the way I do that is apply a square root to the
W+1 signifying that no primes above p_n+2 will be a factor of W+1 or
W-1

(4.4) W+1 and W-1 are necessarily prime. Reason: definition of prime,
step
 (1).


(4.5) Contradiction Reason: p_n+2 was supposed the largest twin prime
yet we
  constructed a new twin primes, W+1 and W-1, larger than p_n+2


(4.6) Reverse supposition step. Reason (4.5) coupled with (4.0)


(5) Set of twinprimes are infinite Reason: steps (1) through (4.6)

XXXXXX

Now a identical proof procedure works for Quad primes
of p_n and p_n+4, and for the 6th prime pairs of
p_n and p_n+6

Now, however there maybe a sticking point as to the application of the
square root so as to keep higher primes of regular primes from
interfering into the proof.

I will probably have to make piecemeal corrections in the above so do
not let the above be the final word.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
From: Archimedes Plutonium on 11 Jul 2010 22:53
I had posted these proofs many years back, without the patch of the
square root to eliminate
larger primes of regular primes from interfering in the argument.

I had felt all along from 1993, when this started in sci.math, that
the slight shade difference
of Euclid Infinitude of Primes, Indirect, where the author never
realizes that Euclid's Number
P+1 is necessarily prime in the assumptive hypothetical, versus the
author who does know
and realize P+1 is necessarily prime, has a proof of the Infinitude of
Twin Primes set to go.

I have always felt that such a slight difference-- P+1 is necessarily
prime delivers a Infinitude
of Twin Primes proof. The snags that caught me up to where I was not
confident I had delivered a proof was that the regular primes would
come into the picture and block or frustrate
me from eliminating the regular primes.

So with my new rekindled spirit, due to a conversation with a journal
editor, I mustered up again, some energy and gave it another final
shot.

I think I can safely say that with a Square Root of P+1 patch, that it
eliminates the annoying
regular primes that are infinite and larger than the twin prime
candidates. This was the sticking
point, for which in past years I never resolved.

And one other feature is that Twin Primes only has a Indirect Method
proof. So shame on those who think that whenever a proof is found that
it comes packaged with both methods.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
From: Archimedes Plutonium on 11 Jul 2010 22:56
Ha Ha Ha, look at that, there are more people rating Archimedes
Plutonium's post of
proof of infinitude of primes (7 persons within one second) than
persons reading all of
sci math posts. I hold a speed record of sorts.

Archimedes Plutonium wrote:
> Due to a conversation with a journal editor, my interest has been
> rekindled in a proof of the Infinitude of Twin Primes and all even
> numbered pairs of primes. The reason this attempt works, I feel, is
> that I have eliminated the regular primes out of the picture by a
> clever devise that was used in the Direct method of the
> square root to eliminate factors.
>
> XXXXXX
> Euclid's Infinitude of Primes proof, Direct or constructive in short-
> form goes like this:
>   1) Definition of prime
>   2) Given any finite set of primes
>   3) Multiply the lot and add 1 (Euclid's number) which I call W+1
>   4) Either W+1 is prime or we conduct a prime factor search
>   5) this new prime increases the set cardinality by one more prime
>   6) since this operation of increasing set cardinality occurs for
> any
>   given finite set we start with, means the primes are infinite set.
>
> XXXXXX
>
> Euclid Infinitude of Primes proof, Indirect in
>  short-
>   form goes like this:
>
>
> 1) Definition of prime
>   2) Hypothetical assumption, suppose set of primes 2,3,5,7,.. is
>   finite with P_k the last and final prime
>   3) Multiply the lot and add 1 (Euclid's number) which I call W+1
>   4) W+1 is necessarily prime
>   5) contradiction to P_k as the last and largest prime
>   6) set of primes is infinite.
>
> XXXXXX
>
> DIRECT Method (constructive method), long-form; Infinitude of Primes
>  Proof
>
>
> (1) Definition of prime as a positive integer divisible
>   only by itself and 1.
>
>
> (2) Statement: Given any finite collection of primes
>  2,3,5,7,11, ..,p_n possessing a cardinality n Reason: given
>
>
> (3) Statement: we find another prime by considering W+1 =(2x3x...xpn)
>   +1 Reason: can always operate on given numbers
>
>
> (4) Statement: Either W+1 itself is a prime Reason: Unique Prime
>  Factorization theorem
>
>
> (5) Statement: Or else it has a prime factor not equal to any of the
>   2,3,...,pn
>  Reason: Unique Prime Factorization theorem
>
>
> (6) Statement: If W+1 is not prime, we find that prime factor Reason:
>  We take the square root of W+1 and we do a prime search through all
>  the primes from 2 to
>  square-root of W+1 until we find that prime factor which
>  evenly divides W+1
>
>
> (7) Statement: Thus the cardinality of every finite set can be
>  increased. Reason: from steps (3) through (6)
>
>
> (8) Statement: Since all/any finite cardinality set can be increased
>  by one more prime, therefore the set of primes is an infinite set.
>  Reason: going from the existential logical quantifier to the
>  universal
>  quantification
>
> XXXXXX
>
> INDIRECT (contradiction) Method, Long-form; Infinitude of Primes
> Proof
>  and
>  the numbering is different to show the reductio ad absurdum
> structure
>  as
>  given by Thomason and Fitch in Symbolic Logic book.
>
>
> (1) Definition of prime as a positive integer divisible
>   only by itself and 1.
>
>
> (2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
>   Reason: definition of primes
>
>
> (3.0) Suppose finite, then 2,3,5, ..,p_n is the complete series set
>   with p_n the largest prime Reason: this is the supposition step
>
>
> (3.1) Set S are the only primes that exist Reason: from step (3.0)
>
>
> (3.2) Form W+1 = (2x3x5x, ..,xpn) + 1. Reason: can always operate and
>   form a new number
>
>
> (3.3) Divide W+1 successively by each prime of
>   2,3,5,7,11,..pn and they all leave a remainder of 1.
>   Reason: unique prime factorization theorem
>
>
> (3.4) W+1 is necessarily prime. Reason: definition of prime, step
>  (1).
>
>
> (3.5) Contradiction Reason: pn was supposed the largest prime yet we
>   constructed a new prime, W+1, larger than pn
>
>
> (3.6) Reverse supposition step. Reason (3.5) coupled with (3.0)
>
>
> (4) Set of primes are infinite Reason: steps (1) through (3.6)
>
> XXXXXX
>
>
> For years now I thought I had not delivered a proof of the Infinitude
> of Twin Primes, that somehow I came up
> short, but due to a email conversation, I realized that
> all along I had proven the Infinitude of Twin, Quad, 6th primes
> and all other even multiples Primes.
>
> The proof is only Indirect method because only in the Indirect are you
> ensured of two new primes.
>
> Let me show you the Indirect Regular Primes Infinitude proof with a
> number example:
>
>
> Euclid Infinitude of Primes proof, Indirect in
>  short- form with number example of 3 and 5 :
>
>
> 1) Definition of prime
>   2) Hypothetical assumption, suppose set of primes 3,5 are all the
> primes that exist with 5 the largest prime
>   3) Multiply the lot and add 1 (Euclid's number) which is (3x5) +1 =
> 16
>   4) 16 is necessarily prime due to (1) and the assumptive step
>   5) contradiction to 5 as the last and largest prime
>   6) set of primes is infinite.
>
> That number example is what delivers a valid Infinitude of Twin
> Primes, Quad Primes, 6th Primes, etc etc.
>
> XXXXXX
>
> Proof of the Infinitude of Twin Primes:
>
> INDIRECT (contradiction) Method, Long-form; Infinitude of Twin Primes
>
>
> (1) Definition of prime as a positive integer divisible
>   only by itself and 1.
>
>
> (2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
>   Reason: definition of primes
>
> (3) Let us instead pick the numbers of primes as
> the succession of 2,3,5,7,. . , p(n), p(n+2) where
> the p(n) and p(n+2) are twin primes
>
>
> (4.0) Suppose twin primes are finite, then 2,3,5, ..,p_n ,
> p_n+2 is the complete series set
>   with p_n and p_n+2 the last and largest twin primes Reason: this is
> the supposition step
>
>
> (4.1) Set S are the only primes that exist Reason: from step (4.0)
> This is the step in which I hesitated in calling
> my proof a genuine proof because I pictured larger regular primes
> beyond the p_n+2, but that was superfluous
>
>
> (4.2) Form W+1 = (2x3x5x, ..,xp_n x p_n+2) + 1.
> And form W-1 = (2x3x5x, ..,xp_n x p_n+2) - 1.
> Reason: can always operate and
>   form a new number
>
>
> (4.3) Divide W+1 and W-1 successively by each prime of
>   2,3,5,7,11,..p_n+2 and they all leave a remainder of 1.
>   Reason: unique prime factorization theorem
>
> Now here is where my previous proof attempts failed and here is the
> patch I wish to apply to stop it from failing. If I apply a patch so
> as to eliminate all the regular primes beyond p_n+2 then the proof
> works.
> And the way I do that is apply a square root to the
> W+1 signifying that no primes above p_n+2 will be a factor of W+1 or
> W-1
>
> (4.4) W+1 and W-1 are necessarily prime. Reason: definition of prime,
> step
>  (1).
>
>
> (4.5) Contradiction Reason: p_n+2 was supposed the largest twin prime
> yet we
>   constructed a new twin primes, W+1 and W-1, larger than p_n+2
>
>
> (4.6) Reverse supposition step. Reason (4.5) coupled with (4.0)
>
>
> (5) Set of twinprimes are infinite Reason: steps (1) through (4.6)
>
> XXXXXX
>
> Now a identical proof procedure works for Quad primes
> of p_n and p_n+4, and for the 6th prime pairs of
> p_n and p_n+6
>
> Now, however there maybe a sticking point as to the application of the
> square root so as to keep higher primes of regular primes from
> interfering into the proof.
>
> I will probably have to make piecemeal corrections in the above so do
> not let the above be the final word.
>
>
> Archimedes Plutonium
> http://www.iw.net/~a_plutonium/
> whole entire Universe is just one big atom
> where dots of the electron-dot-cloud are galaxies
From: sttscitrans on 12 Jul 2010 07:00
On 12 July, 03:02, Archimedes Plutonium
<plutonium.archime...(a)gmail.com> wrote:

You can't even get your definition of prime right.


> (1) Definition of prime as a positive integer divisible
>    only by itself and 1.

1 is a positive integer
1 is divisible by itself
1 is divisible by 1

Therefore by your definition 1 is prime

> (2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
>    Reason: definition of primes

Why haven't you included 1 which by your definition
is prime in this list ?

(Other muddled drivel deleted)
From: sttscitrans on 13 Jul 2010 02:53
On 13 July, 06:32, Transfer Principle <lwal...(a)lausd.net> wrote:
> On Jul 11, 7:02 pm, Archimedes Plutonium
>
> <plutonium.archime...(a)gmail.com> wrote:
> > Proof of the Infinitude of Twin Primes:
> > INDIRECT (contradiction) Method, Long-form; Infinitude of Twin Primes
> > (1) Definition of prime as a positive integer divisible
> >    only by itself and 1.

I see AP is still definining 1 to be a prime number.
Maybe he means that a natural is prime
iff it has two and only two distinct divisors.

1 = 1x1 only 1 distinct divisor => not prime
2 = 1x2= 2x1 two and only two distinct divisors => prime
3 = 1x3 = 3x1 => prime
4 = 1x4=4x1 = 2x2 three distinct divisors => not prime

> > (2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S

No 1 ?

> >    Reason: definition of primes
> > (3) Let us instead pick the numbers of primes as
> > the succession of 2,3,5,7,. . , p(n), p(n+2) where
> > the p(n) and p(n+2) are twin primes
>
> OK then, let's choose p_n = 11, so that p_n+2 = 13.

AP has bamboozled himself with his insistence on
w+1 being "necessarily" prime in the "indirect"
method.
Once a contradiction has been found, whatever
has been deduced about w+1 within the proof by
contradiction is of no further relevance.

All that AP is doing is performing the indirect
proof twice, onve with w,w+1 and once with w-1,w.

As any two consecutive naturals share no common factor other than 1,
and w is assumed to be the product of ALL
primes neither w-1 nor w+1 can be divisible by any of these primes.
w-1 and w+1 are two naturals > 1 that have NO prime divisors. This
contradicts unique factorization.

Why AP thinks this has anything to do with twin primes
only a loony doctor can ascertain.

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Prev: Dear Sir, Having consulted with my colleagues and based on the information gathered from the Nigerian Chambers Of Commerce And Industry, I have the privilege to request your assistance to transfer the sum of $47,500,000.00 (forty seven million, fi
Next: Are space and time separate issues that can be treated in a parallel way.