c

_{1}x

^{2}+ c

_{2}xy + c

_{3}y

^{2}= c

_{4}+ c

_{5}x + c

_{6}y

which is called a binary quadratic Diophantine as x and y are the unknowns, to a general form like:

u

^{2}- Dv

^{2}= C

where u and v are the new unknowns and in solving them you find x and y with the original equation.

It's a simplification of number theory as the traditional ways to reduce involve three different ways depending on the values of the c's, while I found there is one way available.

I used a mathematical tool I call tautological spaces for that discovery, which also gave me a very useful relation on the simplified form, where I'll go back to x and y for the unknowns:

The equation x

^{2}- Dy

^{2}= F

requires that

(x+Dy)

^{2}- D(x+y)

^{2}= -F(D-1).

Remarkably that alone allowed me to connect equations when F=1, with Pythagorean Triples when D-1 is a square, which was the first indication for me, of the importance of D-1, where since then I've been able to explain the size of fundamental solutions, with factors of D-1 being part of it.

That covers research mostly completed in 2008. I've updated some of it recently, for better exposition.

These results were I believed intriguing and they helped my confidence as I could just play with actual numbers and watch them behave as the equations required. But I still was looking for social validation from mathematical society.

Recently I found that I could do even more with the simplified form and solve for y modularly:

Instead of x

^{2}- Dy

^{2}= F, let x = z-ky or x = -(z-ky), so:

(z-ky)

^{2}- Dy

^{2}= F.

Then it can be shown that if integer solutions for x

^{2}- Dy

^{2}= F exist, it must be true that:

2ky = z - Fz

^{-1}mod D-k

^{2}

(z-ky)

^{2}= k

^{2}y

^{2}+ F mod D-k

^{2}

Here there is the additional requirement that 2kyz + F = -1 or 1 mod 8, or 0 mod 4, if D-k

^{2}is a square.

And now it was REALLY cool watching numbers behave as expected including quadratic residues where their spacing now made mathematical sense as being governed by those equations.

James Harris