x

^{2}- Dy

^{2}= 1

And after some fun analysis with my own research I found the following:

D(x+y)

^{2}- D + 1 = S

^{2}

where back then I used capital S, and later started using a small one. But one thing I don't mention is that S can be calculated explicitly, as I find it sort of tedious. So yeah, with my method to reduce binary quadratic Diophantine equations, 's' actually has an explicit value as a function of x and y. But finding it I think is the kind of thing that is best done by a computer with the general equation. So the 'S' above actually is a function of x and y and can be calculated.

And for the first time ever I'm going to demonstrate that calculation where I'm picking a simple example as I like easy.

So let's continue then and calculate S, here with this simple example and start by expanding out the left side:

Dx

^{2}+2Dxy + Dy

^{2}- D + 1 = S

^{2}

And next, we do a smart thing of subtracting and adding--so not changing things really--D

^{2}y

^{2}

Dx

^{2}- D

^{2}y

^{2}+ D

^{2}y

^{2}+ 2Dxy + Dy

^{2}- D + 1 = S

^{2}

So, now we move things around to group with an eye on our original equation multiplied by D:

Dx

^{2}- D

^{2}y

^{2}- D + D

^{2}y

^{2}+ 2Dxy + Dy

^{2}+ 1 = S

^{2}

And now I need to add and subtract x

^{2}:

Dx

^{2}- D

^{2}y

^{2}- D + D

^{2}y

^{2}+ 2Dxy + x

^{2}- x

^{2}+ Dy

^{2}+ 1 = S

^{2}

And now I use my original equation twice! And I remind that equation is x

^{2}- Dy

^{2}= 1, and it makes a lot disappear so that I end up with a square:

D

^{2}y

^{2}

^{2}= S

^{2}

So I have finally that S = Dy + x, or -Dy - x.

Which is of course also: S = x + Dy, or -x - Dy

I like using S = x + Dy, and now can go back to my original finding and simplify to get:

(x+Dy)

^{2}- D(x+y) = -D + 1

And to me that is a pleasure. But of course it's my research and I also know that my find of this result was a first in human history. Later it would be key in my figuring out the 'why' of the fundamental solution to the main equation which is also a first in human history.

So why wouldn't I enjoy?

James Harris

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