Now after my success with a general method to reduce binary quadratic Diophantine equations, I see a need to advance the methodology of tautological spaces where that is simple enough.

To handle quadratic Diophantines in general I see a need for a more general tautological space:

a

_{1}

^{h}+ a

_{2}

^{h}+v

_{1}a

_{3}

^{h}+...+v

_{n-2}a

_{n}

^{h}= 0(mod a

_{1}

^{h}+ a

_{2}

^{h}+v

_{1}a

_{3}

^{h}+...+v

_{n-2}a

_{n}

^{h})

to handle conditionals with n variables, where I call h, the hyperdimension.

As an example consider n=4, where all the conditional variables are raised to the 1st power so h=1, as then the tautological space is described by:

T_S(1,1,{1,1},{1,1})

So it is a 6-dimensional tautological space a hyperdimension of 1, so there are two free variables which by my type of usage would be v

_{1}and v

_{2}, so that looks like:

x + y + v

_{1}z + v

_{2}w = 0(mod x + y + v

_{1}z + v

_{2}w)

The v's give additional degrees of freedom to allow reducing out to a general solution like with my Quadratic Diophantine Theorem.

Actually I realized the need to extend tautological space terminology by wondering about considering solutions where there are more conditional variables than say, x, y and z.

James Harris

Edited 6/24/12 to correct leaving off hyperdimension which IS necessary.