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From: Archimedes Plutonium on 14 Jul 2010 21:14


Archimedes Plutonium wrote:
> Archimedes Plutonium wrote:
> > Alright, I am in no sort of rush. The proof of the Infinitude of
> > Perfect Numbers has
> > been waiting since Pythagoras of 3,000 years old, so what is 3 weeks
> > more.
> >
> > When a new technique is found in mathematics that clears out a
> > problem-- infinitude
> > of Twin Primes and Polignac Conjecture, you can usually bet that the
> > new technique
> > does alot more, that it can clear out a whole class of unsolved
> > problems. The new
> > technique I speak of is the fact that in Euclid's Infinitude of Primes
> > proof when in Indirect
> > delivers two new necessarily primes as Euclid's Numbers. It was a
> > mistake found
> > in the Logical setup of Indirect Method that ekes out two new
> > necessarily primes and
> > with this found mistake one easily proves infinitude of Twin Primes
> > and Polignac conjecture.
> > But can this new technique be marshalled to conquer Infinitude of
> > Perfect Numbers and a
> > entire gigantic list of primes of specific form? That is what I am
> > trying to resolve.
> >
> > Whether I can extend the new technique to conquer most conjectures of
> > prime form, begging
> > for a proof of infinity.
> >
> > With Twin Primes and Polignac conjecture they are easily fit into the W
> > +1, W-1, then into
> > W+2, W-2 then into W+3, W-3, etc etc.
> >
> > But what about when Mersenne primes of form (2^p)-1 pop up on the
> > radar? I believe the new
> > technique is powerful enough for it delivers two new necessarily
> > primes. The problem is to
> > finagle the form (2^p)-1 into Euclid Numbers. If I can finagle them
> > into being Euclid Numbers
> > then the proof of Mersenne primes and Infinitude of Perfect Numbers
> > falls out.
> >
> > So here I have a new technique-- Indirect Method yields two new
> > primes. Now I attach
> > another tool, that of Mathematical Induction. I do this so that I can
> > finagle the Euclid Numbers
> > to be of form (2^p)-1.
> >
> > Now I am rusty on Mathematical Induction but let me just spearhead a
> > attack.
> >
> > Indirect Method
> > (1) definition of prime
> > (2) hypothetical assumption step; suppose .. where last number in list
> > is largest prime
> > (3) form Euclid's Number/s
> > (4) Euclid's Number/s are necessarily prime
> > (5) contradiction to largest prime of list
> > (6) set infinite
> >
> > So the above worked splendidly for Twin Primes and the Polignac
> > Conjecture of all prime
> > pairs of form P, P+2k.
> >
> > But now we tackle primes of more complicated form such as Mersenne
> > primes (2^p)-1
> >
> > The first few Mersenne primes are 3,7,31, 127
> >
> > So the initial case of a Math Induction works for Euclid's Number as W
> > +1
> >
> > {2,3} are all the primes that exist yields Euclid Number (2x3)+1 = 7
> > {2,3,5} are all the primes that exist yields Euclid Number (2x3x5)+1 =
> > 31
> >
> > So I have the initial case of a Mathematical Induction on Mersenne
> > Primes
> >
> > Now I suppose true for case N on Mersenne Primes:
> >
> > {2,3,5,7,11, . . . , p_N} yields (2x3x5x7x11x . . . x p_N) + 1 is a
> > prime of form (2^p)-1
> >
> > Now I have to show that {2,3,5,7,11, . . . , p_N, p_N+1} is also a
> > prime of form (2^p)-1
> >
> > Now pardon me, and hope you do not mind this next trick up my sleeve,
> > but it looks
> > to me as though there is a repeat of the Twin Primes proof here where
> > I look upon
> > p_N as that of W-1 and look upon p_N+1 as W+1
> >
> > And thus achieve the infinitude of Mersenne Primes which thus achieves
> > Infinitude
> > of Perfect Numbers.
> >
>
>
> So I have no worries, none whatsoever that the Euclid Number is prime
> for
> the method guarantees it prime. My only worry and concern is whether
> I
> retrieve another Mersenne prime of form (2^p) -1
>
> By Math Induction supposed p_N Euclid Number is a Mersenne formed
> prime
>
> All that is needed is to see that p_N+1 Euclid Number is a Mersenned
> formed prime
>
> Looks like I need not worry about whether p_N+1 Euclid Number is prime
> for the
> method gives me it as necessarily prime, what matters is whether it is
> a Mersenne
> form. That means it is a exponent of 2.
>
> That means it is it is from the stock of 2, 4, 8, 16, 32, 64,
> 128, . . .
>
> I do not have to worry about the -1 in (2^p)-1 for the Euclid Number
> can achieve that.
>
> All I have to worry about is that the Euclid Number has a rootstock
> from 2^p
>
> So let me do the Indirect Method on just the first initial Mersenne
> Primes to get a feeling
> for the flow of how the mechanics works.
>
> Definition of Prime
> Suppose all primes finite with 3 the last and largest Mersenne prime
> {2,3} are all the primes that exist yields Euclid Number (2x3)+1 = 7
> Contradiction and formed a new and larger Mersenne prime of 7
> Hence Mersenne primes infinite
>
> Again to see the flow
> Definition of Prime
> Suppose all primes finite with 5 the largest prime in set {2,3,5} are
> all the primes that exist yields Euclid Number (2x3x5)+1 = 31
> Contradiction and formed a new and larger prime with is Mersenned
> prime
> Hence set of all primes infinite
>
> So in the one proof case I can say Mersenne primes infinite but in the
> second case I can
> only say all primes infinite.
>
> So, beginning to see some daylight in this:
>
> In the Supposition step of the Math Induction what I was supposing
> true is that
> the Euclid Number for p_N as that of (2x3x5x7x. . .xp_N) is of the
> form of (2^p)
>
> So then, all I need show is that the p_N+1 continues with the
> 2,4,8,16,32, . . . string.
>

I usually find that talking something to death achieves my goal.

Okay, so I supposed in Math Induction that p_N was true, and so
(2x3x5x7x. . .xp_N) was of the form 2^p

Now when in p_N+1 I include that into the new Euclid Number as such:

(2x3x5x7x. . .xp_N x p_N+1)

what I am doing is squaring in the series 2,4,8, 16, 32, 64,
128, . . .

So that we can see that 8 x 8 is


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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