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From: Archimedes Plutonium on 14 Jul 2010 15:44


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

I mentioned this concept earlier of zugswang but the meaning I applied
to that
word is the idea that a math proof forces the conclusion upon us, not
that we
have to go searching in the proof to be convinced of the outcome.

So I went and looked up the word and to my surprize it is already used
in mathematics,
but I think the use should be vastly extended to mean that in a math
proof, it is often the
case that the proof is not compelling, that the proof is not batting
you over the head and
saying "you cannot think otherwise because these steps are forcing
you".

--- quoting Wikipedia ---
Zugzwang (German for "compulsion to move", pronounced [ˈtsuːktsvaŋ])
is a term originally used in chess which also applies to various other
games. The concept finds its formal definition in combinatorial game
theory.
--- end quoting ---

In the case of Mersenne primes, I need a zugzwang or zugswang of
convincing that
Euclid's Number in Indirect is of form (2^p)-1 and that is why I bring
into the battle the
Mathematical Induction.

Alot of modern day so called alleged proofs, it is easy to see they
have absolutely
no Zugswang in their alledgedry such as Wiles's FLT, Appel & Haken's 4
Color, the recent
Poincare conjecture.

Some math proofs are overflowing and overspilling with Zugswang such
as the proof of
the Pythagorean theorem that shows only pictures and no words of
squares on the sides
of a right triangle. That proof is full of Zugzwang.

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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