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From: Archimedes Plutonium on 14 Jul 2010 00:41


Archimedes Plutonium wrote:
> 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.
>

Ah, yes yes yes yes yes, hand me any form of primes you want proved
infinitely many thereof. Show me a few cases and suppose true for case
n and if it provides n+1 then true infinitely many.

So I merely adjoin a Mathematical Induction into the Indirect Method.
But that is what I implied in the case of Quad primes and N+6 primes
anyway. I showed where I could fetch another prime of that type and
could reiterate that prime into another Euclid Number.

But primes of Mersenne form have exponents, and so the trick here is
to instill Math Induction into the proof format.

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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