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From: Archimedes Plutonium on 14 Jul 2010 15:10
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.

P.S. I am in no rush and had better make it clear rather than obfuse.


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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