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From: Archimedes Plutonium on 12 Aug 2010 00:30


Archimedes Plutonium wrote:
> Archimedes Plutonium wrote:
> (others snipped)
> >
> > Proof of Goldbach: Every even number >2 is the sum of at least two
> > primes. Every
> > even number >2 is the product of at least 2 primes, for example 4 is
> > 2x2, 8 is 2x2x2,
> > 6 is 2x3. Notice the symmetry, that all even numbers are at least the
> > product of two
> > primes translates into all even numbers >2 must be at least the sum of
> > two primes.
> > Now 8 is both 2+2+2+2 but also 3+5. So is that a detriment to the
> > proof? Not at all.
> > Because the key idea is that there is no even number >2 that is the
> > product of only one
> > single prime. So we see here, how Algebra of multiplication translates
> > into addition.
> >
>
> Now what is especially intriguing about the proof of Goldbach is that
> we see an locking
> together of Galois Algebra of addition and multiplication in one proof
> that has never
> before been seen in the history of mathematics. It has immense
> implications for other proofs
> such as Riemann Hypothesis for there we have another example of a
> series of addition equal to a (series) of multiplication.
>
> So that in the proof of Goldbach, it is true because every even
> Natural >2 has at minimum two
> prime number factors and so every even Natural >2 must have at least
> two prime numbers as
> sums. If there exists one Natural >2 that is the sum of a singlet
> prime with a composite and no two primes yields the sum, then there
> exists a Even Natural >2 whose prime decomposition has only a singlet
> prime factor.
>
> So what is the Galois Algebra that says addition is interchangeable
> with multiplication?
>

The more I think about this, the more I realize I do not need the
Algebra of interchange
between multiplication and addition, where both require a minimum of
two primes for every
Even Number >2.

Let me crudely write out the proof, continually improving it.

PROOF:
(1) Every Even Natural >2 has at minimum two prime factors in a
decomposition. For example 6 = 2 x 3

(2) Hypothetically assume there is a Even Natural >2, call it K, that
has no two prime summands which added together equals K.

(3) Now K has at least two prime factors in a decomposition of
multiplication

(4) Now let me use an example to guide this proof of that of 12 which
to Goldbach would
be 7 + 5. But as an example, say it only had one prime such as 10 + 2.

(5) Now can I achieve a contradiction

(6) I think I can, and maybe I do not even need the multiplication
lemma that every even number >2 has two prime divisors.

(7) Without loss to argument take 2 as the singlet prime in Goldbach
then we have the Goldbach pairs as (K-2, 2), such as the (10,2) for
12.

(8) But then 10 or K-2 has two prime summands. And in this case they
are (5,5)

(9) This is almost looking like a Ferrmat's Infinite Descent or
Mathematical Induction.

(10) So we have the decomposition of K into (K-2, 2) and the
decomposition of K-2
into (p_1, p_2)

(11) Now, all I need is the idea that if I add 2 to that of either p_1
or p_2, I end up with
two prime summands.

(12) looking good and shaping up good and nicely, because it looks
like a mathematical
induction for a Goldbach proof, where the idea is that if Goldbach
breaks down somewhere
it is a even number called K and we can then utilize K-2, and 2 as
summands and that we
know K-2 obeys Goldbach, that all I have to retrieve is the adding of
2 to either the p_1 or
p_2 yields two prime summands. Maybe, or maybe not, the multiplication
lemma comes in
handy. tired now and will continue later....

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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