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From: Archimedes Plutonium on 15 Jul 2010 06:57
I am just fresh off of proving infinitude of Mersenne primes by using
the Indirect Euclid Infinitude of Primes
proof method where W+1 and W-1 captures two new necessarily prime
Euclid Numbers. So, fresh off of that
experience led me tonight to think of Goldbach's Conjecture.

Now let me recite some of my past history on working
on the Goldbach Conjecture before I give this simple proof. I like to
cite or recite the history because it shows us that if we are in a
state of thinking about something, we can see what links were there to
make the new thoughts.

I thought I had proven the Goldbach Conjecture circa
1991 or thereabouts and set up partitions. Such as this partition for
the number 8.

0 8
1 7
2 6
3 5
4 4

Now I set up partitions like that for all even numbers
and the argument that I posed for a proof was that notice where the 3
and the 5 for 3+5 = 8. My proof argument was that one side of the
partition had to contain a prime number as well as the other side due
to a theorem that between n and 2n exists a prime number. Great so
far, now all I needed was some way of saying that every even number
had a prime lined up from one side that lines up with the prime on the
second side, just like the 3 lines up with the 5 above.

So I hunted around in 1991 and came up with the idea that I multiply
those lined up numbers and that led me
to the idea that if Goldbach was false, then composite numbers that
had only two prime factors such as 3x5 =
15 that if Goldbach was false, then at least one number that is
uniquely decomposed by the Fundamental theorem of Arithmetic into two
prime factors does not exist.

Well, that is alot of lining up and someone in sci.math
in the 2000s pointed out that this was no proof. And I argeed with
him.

But tonight I offer a brand new attack. It retains some bits and
pieces of my old attack.

Proof of Goldbach Conjecture that every even number larger than 2 is
the sum of two primes.

Proof: This is an Algebra Proof and requires the translation of
addition into multiplication. In other words, we replace addition by
multiplication from the
fancy Galois theory of Algebra of groups, rings, field etc. Every even
number beyond 2 is decomposable by multiplication by at least two
primes. So that 4 is 2 x 2,
and that 6 is 2x3 and that 8 is 2 x 2 x 2. So hold on a minute here.
The number 8 maybe strange looking but remember in Goldbach with
addition that 4 = 2 + 2 to satisfy Goldbach. For Multiplication that 8
is 2 x 2 x 2
is no encumbrance to the idea that every even number has at least two
prime factors, even though the number
8 has three prime factors. You see, I am switching into a Goldbach for
addition to a Goldbach for multiplication
and a Goldbach for multiplication would simply say that Every Even
Number has at Least Two Prime Factors, but it probably has more in
many cases. I am letting the Structure of Algebra to convert addition
to the operator of multiplication. So, now, all I need is to note that
Every Even Number must have at least two prime factors. That is
obviously true, because every even number has "2" as a prime factor so
then every even number must have another prime factor and thus must
have At Least Two Prime Factors. Now, get the Algebra Galois Machinery
or Framework and switch over from Multiplication to Addition. It is
known from Algebra that the operators are interchangeable. So the
proving mechanism boils down to this. If Goldbach is false, then there
exists an even number larger than 2 which does not have two prime
factors in multiplication.
QED

Comments: I have always felt that when a math conjecture is easy to
communicate and easy to understand by almost anyone, that the proof of
the conjecture must also be a simple idea proof. A conjecture such as
Riemann Hypothesis which is inaccessible to anyone not a
mathematician, would have a complex proof. But a conjecture that is
accessible to grade-school children, then the proof of it
in the end is as simple as it is accessible.

However, in the above, the proof relies on Algebra theory that
multiplication is interchangeable with addition, and that is a complex
idea and theory and proof.

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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