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From: Archimedes Plutonium on 14 Jul 2010 20:57


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.


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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