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From: Archimedes Plutonium on 13 Jul 2010 14:49


Archimedes Plutonium wrote:
(snipping)
> > 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 we have a proof of the Infinitude of Perfect Numbers and Mersenne
primes.
I leave it to the reader to look up what they mean. I am just showing
what the proof is
and expect the reader to know what the problems were. But I do make
note of the history.
This is perhaps the oldest unsolved mathematics problem, along with 1
being the only
odd perfect number. The reason that I am able to prove it, is because
of a tiny small mistake
and misunderstanding in the Indirect Proof method. In that method,
there is a step where
Euclid's Number under view is "necessarily a new prime within the
Indirect Logic structure"
This allows for the proof.

The moral theme is that a tiny toehold onto a beach assaulted by
marines in war, is enough
to in the end, secure the beach. In the long history of mathematics
from Euclid to 1990s, noone saw that there is this tiny toehold onto
the beach of infinity proofs. The toehold is
the fact that Euclid's Number is necessarily prime in the Indirect
Proof Method. So an entire
class of proofs, such as Twin Primes, Polignac, Mersenne (2^p) - 1 and
the inverse of Mersenne of (2^p) +1, Perfect Numbers are all classes
of infinitude proofs that are easily proveable once the mathematician
realizes the full nature of the Indirect proof method.

Proof of Infinitude of Perfect Numbers and Infinitude of Mersenne
Primes:
(1) definition of prime
(2) hypothetical assumption: suppose set of all primes is finite
and 2,3,5, 7, 11, . ., ((2^p) - 1) is the complete list of all the
primes with
((2^p) - 1) the last and largest prime.
(3) Form Euclid's numbers of W+1 = (2x3x5x 7x 11x . .x (((2^p) - 1))
+1
and W -1 = (2x3x5x 7x 11x . .x (((2^p) - 1)) -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
and so they are necessarily prime from (1) and (2)
(5) Contradiction to (2) that W+1 and W-1 are larger primes than
((2^p) - 1).
(6) And W+1 is a prime of form (2^p) + 1, and W -1 is a prime of form
(2^p) - 1)
Reason: you can place any form
of algebraic prime (x^p) for the last prime in the series so long as
it is -1 or +1 addition
(7) Mersenne primes are an infinite set, hence Perfect numbers are
infinite set.

In the early 1990s I looked up what the inverse Mersenne primes were
of importance,
those primes of form ((2^p) + 1). I do not recall seeing any
importance attached to them
but I know they must have some importance.

P.S. I am going to work on that Reason for why they are that form,
above in the proof.
So that I make that step alot more clear. Someone may come armed and
arsenalled like
a Marine and add more algebraic firepower to the reason.

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 13 Jul 2010 20:38
On 13 July, 19:49, Archimedes Plutonium
<plutonium.archime...(a)gmail.com> wrote:
> Archimedes Plutonium wrote:

(Usual incoherent gushings deleted)

> In the early 1990s I looked up what the inverse Mersenne primes were
> of importance,
> those primes of form ((2^p) + 1).

Presumably p stands for prime
As every p>2 is odd how can 2 to an odd power
plus 1 be prime ?

8+1 = 9
32+1 = 33
128 +1 = 129

Can AP, the dimwit supreme, see a pattern
emerging here ?
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