Wednesday, May 23, 2007

 

JSH: Understanding the rebuttal

Some of you may know that I had a number theory paper published in a now defunct electronic math journal where several sci.math posters emailed the journal claiming errors in my paper and the editors pulled my paper after publication. More on that subject is best found by doing Google searches on "SWJPAM", the initials of the journal.

I have completed a thorough rebuttal of the objections raised against my research in the area covered by the paper, but that rebuttal may be hard for some of you to understand so here is a short explanation.

The paper is a meta proof which shows a problem with the ring of algebraic integers by proceeding only in that ring with expressions valid in that ring, until a conclusion is reached which is outside the ring.

Like if you were in the ring of integers, using equations valid in the ring of integers, and found that suddenly you had fractions like 1/2. It turns out that you cannot do that with the ring of integers, nor can you do that with any other major ring or field.

The paper begins in an integral domain with a factorization of a polynomial:

175x^2 - 15 x + 2 = 2(f(x)+1)(g(x)+1)

I then proceed with some simple algebra and some substitutions to find

175 x^2 - 15 x + 2 = (5a_1(x) + 7)(5a_2(x) + 7)

where I use 2f(x) = 5a_1(x) + 5, and 7g(x) = 5a_2(x), where the purpose is to get to a result valid in the ring of algebraic integers, as the a's are solvable, and are easily shown to be roots of

a^2 + (3x + 1 + 5Q(x))a + (49x^2 + 7Q(x)) = 0

where Q(x) is a function of x, which allows me to traverse through the full set of possible solutions for f(x) and g(x), and notably you have a monic, so that I know that with algebraic integer x, and Q(x), the a's must be algebraic integers.

That part of the meta proof I have had for some months, but recently I realized that I could derive a key expression by use of identities, and the paper shows that you can subtract

r^2 + rs - (2 + 2xt + tQ(x))st + s^2 = (2x+Q(x))t^2

from an identity found using r+s+vt = r+s+vt, where v=1+7x, and letting s=7, and t=5, it is possible to derive the same key expression used to get the a's.

But the key equation that is subtracted from the identity is only generally valid in the ring of algebraic integers, when Q(x) has specific values as only then is it monic.

Using one of those values I demonstrate a result valid in the ring of algebraic integers, with x=1, and then find another result not valid in that ring with x=2, showing that you can use only equations valid in the ring of algebraic integers, with ring operations, yet be pushed out of the ring.

Understanding meta proofs can test the limits of your mathematical abilities. In this case the meta proof has to cover an infinite ring, and show how it is flawed, where that flaw specifically is that you can be pushed out of the ring of algebraic integers using only expressions valid in the ring and operations valid within the ring.

No other major ring has the same flaw, not the ring of integers, not the ring of gaussian integers, and the major fields do not have it either.





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