x2 - Dy2 = 1
And after some fun analysis with my own research I found the following:
D(x+y)2 - D + 1 = S2
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:
Dx2 +2Dxy + Dy2 - D + 1 = S2
And next, we do a smart thing of subtracting and adding--so not changing things really--D2y2
Dx2- D2y2 + D2y2 + 2Dxy + Dy2 - D + 1 = S2
So, now we move things around to group with an eye on our original equation multiplied by D:
Dx2- D2y2 - D + D2y2 + 2Dxy + Dy2 + 1 = S2
And now I need to add and subtract x2:
Dx2- D2y2 - D + D2y2 + 2Dxy + x2 - x2 + Dy2 + 1 = S2
And now I use my original equation twice! And I remind that equation is x2 - Dy2 = 1, and it makes a lot disappear so that I end up with a square:
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?