Saturday, September 27, 2008


JSH: One of the great relations?

I found it quite profound that now I know that in the ring of integers, given

x^2 + Dy^2 = F

I also know that

z^2 + D(x+y)^2 = F*(D+1)

which is the start of what I call a Diophantine chain, where if the original equation has an infinite number of solutions for x and y, then you have an infinite chain, but otherwise you have a finite chain.

One poster has already replied derisively giving z explicitly in terms of x, y and D which kind of puzzled me as it just looks like such a beautiful result to me.

Like with D=-2, you have:

x^2 - 2y^2 = 1, followed by z^2 - 2(x+y)^2 = -1

and the next in the series is

w^2 - 2(x+y+z)^2 = 1

so you just get this flipping back and forth, and with one solution at the start you can get the solutions that follow, so with x=3, and y=2, you have next that

z^2 - 2(5)^2 = -1, so z^2 = 49, so z=7, and then you have

w^2 - 2(3+2+7)^2 = 1, so w^2 - 2(144) = 1, so w^2 = 289 and w=17.

And you can do that forever.

I play with equations now like

7^2 - 5(3)^2 = 4

so I know that

z^2 - 5(10)^2 = 4*(-4)

and can get z then is 22, and notice that 4 divides off so I have

11^2 - 5(5)^2 = -4, so

w^2 - 5(16)^2 = 16, so w=36, and you can divide off 16, and get

9^2 - 5(4)^2 = 1.

So is it one of the great relations?

x^2 + Dy^2 = F

requires that

z^2 + D(x+y)^2 = F*(D+1).

Or am I just in love with one more of my discoveries. More of my math?

I don't even want to read much of any more math texts any more, which I guess is wrong, but it's weird.

I can now look over my own mathematical research and puzzle over it as I figured it out months or years ago and the complexity at times escapes me, as I forget how I figured it out.

So I just kind of stare at it and wonder about that person I used to be. He cared so much too. Was so excited. Believed in so many things. What a sad pathetic character was he with his scribblings, his hopes and his dreams.

So faithful in his love of the human race. So believing in that "someone" out there who cared about the truth to whom he was always talking.

But now there is just me. Looking over these things that often I don't even care any more to understand. Pondering…

I won. I killed the other guy inside of me. So now there is only me.

And success is the only real reward.

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