Friday, September 06, 2013

Leveraging asymmetrical forms

One of my key ideas involves using an asymmetrical form in an identity for analysis.

Explicitly the identity is: x+y+vz = x+y+vz

Where I use it modularly with: x+y+vz = 0(mod x+y+vz) which is the equivalent.

I call that a tautological space.

The reason it is asymmetrical should be apparent as the asymmetry is with the variable 'v' attached to the 'z', in contrast with symmetrical forms, like, for instance, the quadratic form.

Intriguingly the asymmetrical form allows you to in a sense "break" other forms which lead me to a shocking result, which follows from a key factorization:

7(175x2 - 15x + 2) = (5a1(x) + 7)(5a2(x)+ 7)

where the a's are roots of

a2 - (7x-1)a + (49x2 - 14x) = 0

It also lead me to a better way to reduce binary quadratic Diophantine equations. Which gave me the following really cool relation:

u2 + Dv2 = F


(u-Dv)2 + D(u+v)2 = F(D+1)

Using that I was among other things able to find for the first time a set of rules governing the fundamental solution to x2 - Dy2 = 1.

My guess is that mathematicians rely too much on symmetrical forms.

Shifting to a simple asymmetrical form allowed me to figure out things rather quickly and improve on existing techniques with little effort.

It suggests that asymmetrical forms may be the key to a much greater advancement of mathematical understanding.

James Harris