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From: Archimedes Plutonium on 16 Jul 2010 16:07
As far as I know, it has no name, so I am going to give it a name of
the Galois Algebra
Principle. Galois Algebra was discovered to fight the quintic problem
in the 1800s. Now,
I am using it to conquer the Goldbach Conjecture and also to show that
Fermat's Last
Theorem FLT is false unless a boundary between finite and infinite is
marked.

Galois-Algebra-Principle: In mathematics, addition is multiplication
and interchangeable
within Galois Group, Ring, Field theory.


The idea is rather fascinating and perhaps it has its own name, but
few have recognized
its worth or value. It is what has happened in Projective Geometry, in
that a line and point
are interchangeable. Now I can call that idea in Projective Geometry
the Projective Geometry
Principle, that theorems about a line are replaceable with stating the
features with a "point".

In Algebra we all know that fundamentally, addition is multiplication
and multiplication is addition. We know that three groups of two is
either the addition of 2 + 2 + 2 or is multiplication
of 3 x 2. So in a sentence 2 + 2 +2 = 3x2 is the Galois Algebra
Principle. Now where exactly in Galois theory can we point to the
principle and say that Addition is interchangeable with
Multiplication? Is it in finite groups? Is it in rings? Is it in field
theory? Or is it all of Galois theory?

So this principle is crucial and key to the proof of Goldbach and to
the pointing out that FLT
is false unless you precision define the boundary between finite and
infinite-numbers.

Review of Goldbach Proof: Every even number beyond 2 has 2 as a prime
factor and due
to the Unique Prime Factorization theorem, every even number beyond 2
has at Least two
prime factors where for example 4 is 2*2 and 6 is 2*3. Because of
Unique Prime Factorization
theorem every even number beyond 2 has two prime factors at a minimum.
Now apply Galois
Algebra Principle and interchange multiplication with addition. Every
even number beyond 2,
must have two primes as additive sums, because, at minimum they must
have two primes
as multiplicative products.

Now as for FLT, mathematics as a subject is very symmetrical. Physics
which has math as
a tiny subset is highly symmetrical. Yes we note some asymmetry in
physics and math when we examine local regions of the Universe but
overall the Universe is symmetrical, and so
math is symmetrical. This Projective Geometry Principle that you can
interchange line with point or point with line, bespeaks of the
universal symmetry of geometry and the Galois Algebra Principle where
you can interchange addition with multiplication bespeaks of the
universal symmetry of Algebra.

In math, in algebra there are oceans full of solutions to an equation
of this form:

a^n(b^n) = c^n

10^3(100^3) = 1000^3

Just like Goldbach, we interchange to

a^n + b^n = c^n and there should be solutions to when n = 3 or 4 etc
etc.
And there are, when we simply acknowledge that mathematics never
defined
a boundary between finite-number and infinite-number.


        The expression a^n+b^n=c^n is true for all n,
given the following values.
    a= ...9977392256259918212890625
    b= ...0022607743740081787109376
    c= ...0000000000000000000000001

So FLT was a false conjecture ever since Fermat penned it into a
corner of
a book or notes.

However, if mathematicians do their job properly, do their job which
is required
of them-- make the precision definition of what it means to be a
finite-number
and recognizes when a number has transitioned into being an infinite-
number.
Then can you prove that FLT is true for all those numbers called
finite-numbers
and also prove the Riemann Hypothesis. But as long as mathematicians
are
derelict of their job, of their duty, FLT and Riemann Hypothesis RH
and a large slew
of unproven problems will remain as such.

I like the boundary that Physics gives, because if there is no more
Physics going
on at 10^500, well, there is no biology and no intelligent life to do
any mathematics.
So, is FLT and RH true from 0 to 10^500. I believe so. And the added
benefit is that
mathematics proof changes altogether. Every proof in mathematics
becomes a Direct
Constructive Method of proving. We simply verify all the numbers from
0 to 10^500.


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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