Thursday, October 02, 2008


Huge number theoretic structure and number crunching

I noted in a previous post the find of this HUGE freaking number theoretic structure that starts out simply enough:

x^2 + Dy^2 = F

followed by

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

followed by

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

which simplifies to

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

so you just plug in for x and y from the first equation into the second, and to get the next in the series you'd plug in again from the prior, so you build recursively:

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

and that goes ad infinitum. (That last is complicated enough I don't feel like simplifying.)

So it's this massively recursive thing where you feed in from the prior equation to get the next one and what makes it interesting is that of course whenever you have (D+1) with an even exponent on the side then you may have a solution with x and y multiplied by some power of D+1.

But checking that idea you find it sets D, and also sets F relative to D, so this freaking massive infinite super structure of equations is the source of all behavior for Pell Equations and more generally equations of the form x^2 + Dy^2 = F.

If that didn't make sense, imagine that with the third equation:

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

you check to see if it's possible that

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

So you could check

(x-Dy)-D(x+y) = +/-(D+1)x or +/-(D+1)y

and it turns out that if you do that you'll set possible values for D. So you're setting D at each level. And setting F relative to D, so it tells you what D and F can give solutions if that's true.

But playing with the thing requires some massive number crunching and symbolic manipulation which is best done by computer, I'd think.

Once mapped out at any level, then that's it. It's set perfect and static. Unchanging.

Actually this structure has always existed but I guess no one noticed it was there even though it's FREAKING HUGE and completely controls rational solutions to x^2 + Dy^2 = F.

I'm not even going to try to play with the thing beyond noting its existence leaving that for some person or team who wants to start mapping the super structure.

As a side benefit, any equations that fit within mapped areas can be immediately solved in integers.

Obviously with 2000 years of history with research in this area there should be some interest in this massive super number theoretic structure but I'm not holding my breath.

But hey I can post! And see what happens.

But it's so cool that I have to note again that there is this HUGE, MASSIVE number theoretic super structure made up of equations recursively built, layer by freaking layer, which completely controls Pell's equation along with others and it's just such a wild thing.

An infinite structure of equations just out there, running everything, mappable, though incredibly complex, and did I say it's freaking huge? And it's a super number theoretic structure?

I'm glad I found it. Kind of cheers me up a bit. You know?

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