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From: Archimedes Plutonium on 13 Jul 2010 04:55
After some rumbling fits and starts, I am confident these are proofs
of the Polignac Conjecture. Looking in Wikipedia this conjecture dates
to the early 1800s in France
and concerns the distribution of primes by a metric spacing of even
numbers.
So that Twin Primes are the N+2 primes and the Quad Primes are N+4
primes
and the N+6 Primes have a metric separation of 6 units.

Given the list of the first few primes:

2, 3, 5, 7, 11, 13, 17, 19, . . .

The first Twin Primes is 3 and 5
The first Quad Primes is 3 and 7
The first N+6 Primes is 5 and 11

Polignac's Conjecture is that each of these sets of primes are
infinite sets, such
that N+2 is an infinite set and N+4 is an infinite set etc etc.

Now the proof of Polignac follows from one format, the proof of the
infinitude of Twin Primes.
That proof is this:

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 repeat the above with minor modifications for that of Quad
Primes N+4

Infinitude of Quad 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+4 is the complete list of all the
primes with
p_n and p_n+4 the last two primes and they are quad primes.
(3) Form Euclid's numbers of W+2 = (3x5x 7x 11x . .x p_n x p_n+2) +2
and W -2 = (3x5x 7x 11x . .x p_n x p_n+2) -2 with the proviso of
deleting
the 2 prime.
(4) Both W+2 and W -2 are necessarily prime because when divided by
all the primes that exist including 2 into W+2 and W-2 they leave a
remainder, so
they are necessarily prime from (1) and (2)
(5) Contradiction to (2) that W+2 and W-2 are larger quad primes.
(6) Quad Primes are an infinite set.

The same format goes for N+6 primes with a deletion of the 3 prime

So in turn all the primes of form N +2k are proven to be infinite sets
by
the Indirect Method.

A few passing thoughts by the Author:
Everyone in math knows that to understand these number theory
conjectures is
easily understandable to everyone, especially those not even in
mathematics
can digest the problem in a few minutes of time. So the wonder is why
such
easy problems yet never any proof. May I suggest the reason that easy
problems of the
Twin Prime conjecture and Polignac conjecture is that noone looked to
see if there are
flaws in the Symbolic Logic Structure when putting together Euclid's
Infinitude of Primes
Indirect Method. Not that these math problems were hard and difficult,
to the contrary, they
are simple and easy proofs. What made them unproveable is a lack of
understanding that
Euclid's Numbers are necessarily prime in the Indirect Method.

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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