Sunday, April 05, 2009


JSH: Considering my binary quadratic Diophantine solution

One of the things I did months ago was use what I call tautological spaces—invented to tackle Fermat's Last Theorem by me back December 1999—against binary quadratic Diophantine equations—kind of accidentally as I was going after a three variable expression originally.

And I freaking generally solved binary quadratic Diophantine equations, getting among other things, the result that every such equation for a circle, ellipse or hyperbola can instead be solved for integers by solving an equation of the form u^2 + Dy^2 = F.

I too at that time was focused on Diophantine only, and didn't care about rational or other solutions, but I thought that was a nifty result, tossed it on my math blog, wrote a paper, and submitted it to a journal or two and it got rejected by them.

But now having moved on to considering rational cases it looks like a lot of people missed the reality that certain conics in rationals are best contemplated by what mathematicians called the "Pell's Equation" and my result now looks like an exclamation point on that reality.

Mathematically, u^2 + Dy^2 = F, is the granddaddy equation, and its special case of u^2 - Dy^2 = 1, is more than up to the task of completely encompassing all that is mathematically of interest with circles, ellipses and hyperbolas.

My take on what mathematicians do is that they specialize.

So a rational parameterization of "Pell's Equation" is not of interest to the number theorists as they focus on integers. While the algebraic geometrists, don't do number theory.

Given x^2 - Dy^2 = 1, in rationals:

y = 2t/(D - t^2)


x = (D + t^2)/(D - t^2)

showing it more traditionally versus the way that results from my own re-derivation.

And you get a parameterization for hyperbolas with integer D>0, for circles with D=-1, and for ellipses with D<-1.

That D number though may hold secrets to our reality.

I see mathematics as a tool. But we have a weird system where some people specialize on tools—but do not really use them.

It's like if carpenters had to deal with hammer specialists, versus saw specialists, where saw people can't be bothered with areas that they think involve the hammer people, and God help you if you're really concerned about nails, as the hammer people know that hammers need nails, but they can't be bothered in their "pure" research with such practical matters!!!

Mathematics is a tool.

We let some people specialize in "pure math" and these people screw us over, in some of the most bizarre ways imaginable, and then can't be trusted to tell the truth about their own areas.

People who do only "pure math" are like tool people who only make tools but don't think they should be bothered with how the tools are used. Can you imagine what weird stuff such people would build?

How the hammer people might add weird gew-gaws to hammers and proclaim it was "pure"? Or how the nail people might decide that nails need two heads and call you insane if you argued that they were stupid?

These people have ignored my solution to binary quadratic Diophantine equations, and wouldn't you know, using it, I'm finding all this weird crap that they've also managed to miss or ignore in even their own prized "pure math" areas, and you know what?

More than likely I'm just talking to myself here for the good it will do.

You people don't deserve the best knowledge.

You deserve patting each other on the back for bottom level results when you could have truly magnificent and huge results, and my only hope then is that history will despise you and the crap you call research as much as I do.

If humanity survives, your tremendous volume of research will be on the crap heap, worthless.

Transcended by real research done by people who care about knowledge, are willing to do the work, who know how to be real skeptics who question.

You in contrast, are useless to your world. But you pretend otherwise and God help us, we're screwed until enough people figure that out, and toss you out.

Then maybe our students can get taught how to do real research, instead of playing pretend.

Your society is like Hollywood science. You people don't know the meaning of doing real research. Just a fantasy.

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