JSH: Group theory undone

Some of you may know that I've found that Pell's Equation can be more simply solved by using alternates to it, like the negative Pell's Equation, as given x^2 - Dy^2 = 1, for instance, if you have a solution to

j^2 - Dk^2 = -1

then you have a solution for x, from x = 2j^2 + 1.

For instance, 16 - 17 = 1, so j=4, k=1, and I have x=2(16) + 1=33.

And of course, 33^2 - 17*8^2 = 1.

And you can see x tends to be the square of j so it's DUMB to work hard at finding x directly, if you know that the negative Pell's Equation is available, and it is for D a prime number 50% of the time. Two other alternates are available for the other prime cases, and one additional alternate for special composite cases, so all positive integer values for D are covered.

Now in the old view of the world you might have had, such a simple thing HAD to be part of the mathematical literature, for say, physicists working on the zeroes of the Racah coefficient who were looking for the best techniques, so they could just look it up!

And be confident that they were using the best techniques in the world for solving Pell's Equation.

Ok. That was the old view. It was wrong. Throw that notion out the window as check the literature and it's STILL not there.

Fun time!!! Check Wikipedia, as I've been talking about this thing for days now. See an interesting failing in terms of how up date it is.

See: http://en.wikipedia.org/wiki/Pell's_equation

The modern mathematical world has a glass ceiling. Being right is NOT enough, even when it's about something as historically huge as Pell's Equation which is an area where physicists need the best information if, say, they're solving for zeroes of Racah coefficients using Pell's Equation.

So what does any of all that have to do with Group theory?

Well it turns out that the way mathematicians teach to solve Pell's Equation has been superseded by my own research as I generally solved binary quadratic Diophantine equations MONTHS ago.

Having the best mathematical research available on the subject I can find interesting things!!!

I'm advanced. You are not. So yes, I can find dumb things you are doing using the old, obsolete methods!!!

You are primitives. I am the advanced one.

Now mathematicians continue to ignore my research which doesn't surprise me as I used the same research technique that I used to generally solve binary quadratic Diophantine equations previously to prove Fermat's Last Theorem, and in the course of arguing about that result I discovered a MASSIVE "core" error in number theory.

It's a core error as it is involved in the foundation area of number theory. So yeah that makes it very huge.

That core error among other things invalidates Galois Theory as a useful tool—doesn't prove it wrong just says it doesn't do what math people think it does. So Group theory as it's currently understood can't be right.

Wow. Nuts, right?

How can it not be right and work so well?

Um, I didn't say it's not right. I said as it's currently understood can't be right.

Which brings us back to Pell's Equation!!!

Turns out mathematicians teach this convoluted way to solve it that involves, yup, Galois Theory!!!

And it WORKS! You can use continued fractions to solve Pell's Equation and read all kinds of fun explanations about class number etc, and other things that do not show up at all in my own research!

There are multiple ways to explain the same thing, and multiple ways to solve it.

What's weird is that a mathematical reality probably has the keys to how in this case as, given

x^2 + Dy^2 = F

it is trivial to show (just multiply out and simplify if you aren't sure) that

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

So the D kind of sits there. This equation is key to solving Pell's Equation in my theory as you have the form:

u^2 + Dv^2 = F*(D+1)^n

where n is a natural number you can run up as high as you wish, and then do some cool things with residues to make looking for a solution easier, but I won't go into detail as the main point here is that there is this neat, little mathematical equation, which probably explains why there is an easier way to do things, and it gets better as finally I can talk about someone else's research!!!

Pell's equation without irrational numbers
Authors: N. J. Wildberger
http://arxiv.org/abs/0806.2490v1
(Submitted on 16 Jun 2008)

Wildberger shows that rational methods—not the class number irrational methods preferred by mathematicians—can be used to solve Pell's Equation, further putting the nail in the coffin of the notion that these approaches that fall into the arena of the "core error" I found are actually the main part of the real mathematical answer.

The implications are HUGE.

There may be a rational explanation for Group theory that has nothing to do with Galois Theory, at all.

And the actual explanation for how it works, may say something deep about our reality, especially how it relates to prime numbers.

But mathematicians don't LIKE the rational approaches as those aren't "sexy". They prefer teaching the old ways.

They refuse to acknowledge my general solution to binary quadratic Diophantine equations.

And they are letting those of you who try to relate number theory to physics use crap methods, crap math, even when it's so bad I can talk about how dumb it is, quickly in posts on freaking newsgroups.

The fringe world is beyond the mainstream. Top physicists in the world out-gunned, upstaged by what math people maintain are rantings of some nut on Usenet.

But I have the best mathematics.

You do not, as long as you listen to, and trust, mainstream mathematicians who have made it their business for you to be behind the curve.

Wake up. Some of you are supposed to be skeptics! Why won't you act like it?

Massive screw-ups have happened before. Trust has been betrayed before.

You are not experiencing anything new. Nothing new under the sun.

And what's wrong yesterday is wrong today and wrong tomorrow, so you theoretical physicists with the clock running on your careers are losing ground, daily. Losing time.

Time you will not get back.