Solving for S, with a simple example

One of the more comforting things for me is to play with a simple and ancient equation, so yeah, I keep trotting it out there! And why not? My ideas gave new information on something wrongly believed to have delivered all it had. The ancient equation of course is called by mathematicians Pell's Equation:

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:

D²y² + 2Dxy + x² = S²

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