JSH: A simple error

Posters in attacking my proofs showing inconsistency with the ring of
algebraic integers routinely move outside the ring. Here is a post
meant to show you how they do it.

In the ring of integers, consider x^2 + 3x + 2 = 0, which of course
factors as

x^2 + 3x + 2 = (x+2)(x+1)

and now solve for it using the quadratic formula, but kind of weird by
NOT resolving the square root then

x = (-3 +/- sqrt(1))/2

and now make the substitution, x=2y, so you get

4y^2 + 6y + 2 = 0, so you can divide by 2 to get

2y^2 + 3y + 1 = 0,

and your solution now becomes

y = (-3+/- sqrt(1))/4

which is two solutions where one is not an integer.

So you moved outside the ring of integers.

So what's the trick?

Well, with integer solutions you can resolve the square root and throw
away one solution, which is what most people routinely do, so they say
that sqrt(4) = 2.

When you do not resolve the square root--or cannot when it is non-
rational--then you cannot throw away the other solution, so it gets
dragged along, and if you do what posters typically do in replies
against my research, and blanket divide a variable like x above so
that you divide MORE THAN ONE SOLUTION you end up pushed out of the
ring of algebraic integers.

When I've pressed them on the reality that the sqrt() returns more
than one value, posters have replied with derision noting that
mathematicians have DEFINED it to have one value, so that they can
continue their trick unabated, as if it were a legitimate criticism
against my research.

But as I've noted repeatedly, the ring of algebraic integers is
inconsistent, and you cannot prove that is is from within the ring!

It is too weak as a ring, to allow you to prove that certain results
are not within it, so posters are forced to go outside the ring to
try and make their objections.

As a reminder, the updated paper--I had to clear out some errors noted
by Rick Decker--is linked to at my Extreme Mathematics group:

http://groups.google.com/group/extrememathematics/web/non-polynomial-factorization-paper

One crucial addition to the paper besides error fixing is the noting
that I use identities mostly, and one equation that is not an
identity, so that equation MUST drive the conditions, and it can be
placed easily enough in the ring of algebraic integers.

This result is one of the biggest in mathematical history
demonstrating an actual inconsistency with a well-known mathematical
object, which mathematicians have unknowingly used for over a hundred
years without understanding how it can lead to false arguments that
appear to be proofs when they are not.

Readers should note that I have multiple mathematical discoveries at
this time where all have been vigorously attacked by posters who
clearly have a need to deny any mathematical result if they feel it
will give credence to my research.

They are dogmatic in their resistance, which is part of the reason I
call these continuing arguments against mathematical proof--and even
publication in a peer reviewed mathematical journal--the Math Wars.

I have rebutted the sci.math newsgroup which killed a mathematical
journal with false claims, and bears a responsibility to accept
accountability.


James Harris

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