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From: Archimedes Plutonium on 12 Aug 2010 05:32

I wrote earlier:
>
> Sorry about all that thrashing around, but am settled, although tired.
> It is not a proof by
> Mathematical Induction nor Fermat's Infinite Descent, although it
> appears as such.
>
> I figured out what makes it work, or what forces it to become a
> summand of two primes.
>
> It is not by contradiction.
>
> Let me use the example of 100 again and pretend as if at 100, the
> Goldbach breaks down
> and has no two prime summands. So I haul in the Goldbach Universal
> Repair Kit. It is very simple kit and looks like this (K-2, 2) and the
> K is 100 where Goldbach broke down, so I
> drop down to the previous Goldbach that worked-- 98. So the (K-2, 2)
> is (98,2)
>
> Now I look for all the primes that satisfy Goldbach at 98 and they are
> (67,31) and (37, 61)
> and (79,19).
>
> Now how does the Goldbach Repair Kit become so Universal in fixing any
> even number that
> is broken down with only one prime summand? It is easy to see how it
> is Universal, for watch how it repairs 100.
>
> I take 67 and 31 and immediately try adding 2 to either one to see if
> I can end up with two
> prime summands, and nope, it does not work. So now I play the second
> round trick and add 4
> to one while subracting 2 from the other to see if I can come up with
> two prime summands? And the answer is yes for I can get (29,71).
>
> But let me continue with round three where I add 6 and subtract 4 from
> one to the other. With the summands of (67,31) how many rounds can I
> play with adding and subtracting? Well I can
> play this rounds to the limit of subtracting 28 and adding 30.
>
> And what forces it to always work and repair the damaged Goldbach even
> number? It is easy to see why it must work universally. If it did not
> work means there are no primes in the interval 0 to 100 that are
> separated by a metric length of 2, by a length of 4, of 6 of 8 of 10
> all the way to 50.
>
> As I wrote the first twenty five primes above:
> 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
> 67, 71, 73, 79, 83, 89, 97
>
> The primes 3 and 97 are of length 94, the primes 5 and 97 are of
> length 92 etc etc.
>
> If Goldbach was false at 100, means there are no prime pairs between 0
> and 100 of lengths
> 30 to 60.
>

I am not actually sure if it is of lengths 30 to 60. Am too tired and
will check
tomorrow.

The idea is basic, though. The idea is that if a even number >2 fails
Goldbach
at let us say K, then form (K-2, 2) and the repair kit then goes
through a large band
of numbers, adding and subtracting in increments of 2. So if Goldbach
is false, then that leaves a wide band of no prime pairs of a length
separation. In other words a huge gap.



> So we see why Goldbach has to be true, otherwise we would have the
> primes from 0 to
> 100 be only this set: 31, 37, 41, 43, 47, 53, 59, 61,
> 67, 71, 73, 79, 83, 89, 97
>

I am not sure if it would be that set. But a large number of the
primes in an interval
would be gone.


> So here I have given a proof of Goldbach without the referral to the
> Algebra that every Even
> Natural >2 has at minimum two prime factors. The above Proof works
> because the repair kit is universal, and it must work for the reason
> that there are no large holes in the primes given a metric length. So
> there is something new in mathematics in this proof. The idea of a
> repair kit has never been used before.
>

It is a new kind of proof because I am not aware of "repair kits" ever
being used
in a proof.

I still think there is a proof using Galois Algebra or Projective
Geometry where multiplication has the even numbers >2 all having at
minimum 2 prime factors which becomes Goldbach
in addition, so that as we interchange between multiplication and
addition we retain the 2 prime
factors or summands.

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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