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From: Archimedes Plutonium on 13 Jul 2010 05:40


Archimedes Plutonium wrote:
> After some rumbling fits and starts, I am confident these are proofs
> of the Polignac Conjecture. Looking in Wikipedia this conjecture dates
> to the early 1800s in France
> and concerns the distribution of primes by a metric spacing of even
> numbers.
> So that Twin Primes are the N+2 primes and the Quad Primes are N+4
> primes
> and the N+6 Primes have a metric separation of 6 units.
>
> Given the list of the first few primes:
>
> 2, 3, 5, 7, 11, 13, 17, 19, . . .
>
> The first Twin Primes is 3 and 5
> The first Quad Primes is 3 and 7
> The first N+6 Primes is 5 and 11
>
> Polignac's Conjecture is that each of these sets of primes are
> infinite sets, such
> that N+2 is an infinite set and N+4 is an infinite set etc etc.
>
> Now the proof of Polignac follows from one format, the proof of the
> infinitude of Twin Primes.
> That proof is this:
>
> Infinitude of Twin Primes proof:
>
> (1) definition of prime
> (2) hypothetical assumption: suppose set of all primes is finite
> and 2,3,5, 7, 11, . ., p_n, p_n+2 is the complete list of all the
> primes with
> p_n and p_n+2 the last two primes and they are twin primes.
> (3) Form Euclid's numbers of W+1 = (2x3x5x 7x 11x . .x p_n x p_n+2) +1
> and W -1 = (2x3x5x 7x 11x . .x p_n x p_n+2) -1
> (4) Both W+1 and W -1 are necessarily prime because when divided by
> all the primes that exist into W+1 and W-1 they leave a remainder of
> 1, so
> they are necessarily prime from (1) and (2)
> (5) Contradiction to (2) that W+1 and W-1 are larger twin primes.
> (6) Twin Primes are an infinite set.
>

Now I should add a cautionary note here, or a further explanation so
as to prevent
someone from making a judgement mistake. For I can anticipate many
will read
the above and not grasp the meaning, and fail to see it as a proof.
Thinking that
I fetched only a finite set of twin primes.

They will read the above and say to themselves "hmm, I can see that
3,5 are twin
primes and that 5,7 are twin primes and 17,19 are twin primes and that
the last
two primes in the List of all primes are twin primes so how in the
world does that
prove twin primes are infinite once W+1 and W-1 are handed over as
twin primes.
The complaint will be that this is still a finite set.

They miss the obvious.

They are unhappy and feel that I have only handed over a finite set of
twin primes.

But here is how they are wrong. So they are unhappy, and now I tell
them, put the W+1
and the W-1 into the above proof and extend the List to be not just
this:

(2,3,5, 7, 11, . ., p_n , p_n+2)

but extend it to be this:

(2,3,5, 7, 11, . ., p_n , p_n+2 , W -1 , W +1)

and if not happy with that, I produce two new Euclid Numbers and add
it to the original
list, and then ad infinitum do I continue to reiterate the same proof
schemata.

So please do not complain that I only fetched a finite set of Twin
Primes, for the proof
scheme is reiterated ad infinitum.

You could in a sense, say that W-1 and W+1 are two new primes at the
"point of infinity"
meaning that I can reiterate or generate more twin primes if one is
not happy with W-1
and W+1.

Same holds true for Quad primes, N+6 primes ad infinitum

Sales Note: of course, for me, the "point of infinity" means 10^500
where
the last largest number has any physics meaning and is where the
StrongNuclear
force in physics no longer exists.

> Now I repeat the above with minor modifications for that of Quad
> Primes N+4
>
> Infinitude of Quad Primes Proof:
>
> (1) definition of prime
> (2) hypothetical assumption: suppose set of all primes is finite
> and 2,3,5, 7, 11, . ., p_n, p_n+4 is the complete list of all the
> primes with
> p_n and p_n+4 the last two primes and they are quad primes.
> (3) Form Euclid's numbers of W+2 = (3x5x 7x 11x . .x p_n x p_n+2) +2
> and W -2 = (3x5x 7x 11x . .x p_n x p_n+2) -2 with the proviso of
> deleting
> the 2 prime.
> (4) Both W+2 and W -2 are necessarily prime because when divided by
> all the primes that exist including 2 into W+2 and W-2 they leave a
> remainder, so
> they are necessarily prime from (1) and (2)
> (5) Contradiction to (2) that W+2 and W-2 are larger quad primes.
> (6) Quad Primes are an infinite set.
>
> The same format goes for N+6 primes with a deletion of the 3 prime
>
> So in turn all the primes of form N +2k are proven to be infinite sets
> by
> the Indirect Method.
>
> A few passing thoughts by the Author:
> Everyone in math knows that to understand these number theory
> conjectures is
> easily understandable to everyone, especially those not even in
> mathematics
> can digest the problem in a few minutes of time. So the wonder is why
> such
> easy problems yet never any proof. May I suggest the reason that easy
> problems of the
> Twin Prime conjecture and Polignac conjecture is that noone looked to
> see if there are
> flaws in the Symbolic Logic Structure when putting together Euclid's
> Infinitude of Primes
> Indirect Method. Not that these math problems were hard and difficult,
> to the contrary, they
> are simple and easy proofs. What made them unproveable is a lack of
> understanding that
> Euclid's Numbers are necessarily prime in the Indirect Method.
>
> 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
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