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From: Archimedes Plutonium on 13 Jul 2010 16:56


Archimedes Plutonium wrote:
> 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.
>

And working on that "reasoning" to be sure. I have never had the need
or call to
readapt the Euclid Number so that it is a exponential number. So that
Euclid's
Number is different from "multiply the lot and subtract 1" What if I
adapted
Euclid's Number as that of multiply the lot and stick that lot into an
exponent for
the 2 in ( 2^p) and then subtract or add 1

Mersenne primes infinitude are a challenge in readapting the Euclid
Number for the
Indirect Proof Method. Up till now, I could get away with simply
adding and subtracting
from a multiplied lot. Here I am challenged to use an exponent.

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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