Saturday, October 04, 2008


JSH: How can I top that?

I'm kicking back now kind of wondering to myself as I try to get some sense of the feeling of accomplishment with the discovery of my own infinite series, and the wild thing to me is the feeling that, hey, good thing I didn't find this thing years ago as I can't see how I'd have done anything else, as how can I top it?

And what's really wacky wild is that years ago I thought it'd be this great thing to find a simple proof of Fermat's Last Theorem, and got that, and it was like nothing as the math world wouldn't accept it, and it didn't feel satisfying anyway, and I had to use this weird new analysis technique I'd invented called tautological spaces, when I wanted something that felt CONCRETE.

So I kept at it. Found my prime counting function. Non-polynomial factorization. The object ring, and nothing really had that truly solid feeling to it, no matter what I could prove, and people kept ripping on me, so I kept going.

It's so ironic.

But it does go back to when I was a kid and I'd read about series like the Taylor series and wonder about that and it didn't sink in to me until now that I really wanted my own.

So, ok, proving FLT is cool. Discovering the object ring is cooler. Inventing my own analysis technique with tautological spaces is rather awesome.

But NOTHING beats having my own infinite series which is an infinite series of equations! So it's better yet!!!
  1. x^2 + Dy^2 = F

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

  3. ((1-D)x-2Dy)^2 + D(2x + (1-D)y)^2 = F*(D+1)^2

  4. ((1-3D)x + (2D^2 - 4D)y)^2 + D((3-D)x + (1-3D)y)^2 = F*(D+1)^3
and that goes out to infinity. To get successive terms in the series you use the algebraic result that given:

u^2 + Dv^2 = C

it must be true that

(u-Dv)^2 + D(u+v)^2 = C*(D+1).

And where whenever the exponent of (D+1) is even, you can have a case where you just have a multiple of x and y, so you can solve for D, which defines possible values for F in terms of x or y.

An infinite series of equations that I like to call a huge number theoretic super structure.

My own infinite series. Finally, something I really FEEL is concrete.

Can't see how I can top that. So I'm done.

Time to kick back and rest on my laurels, drink more and chase more women.

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