## JSH: Negative Pell's Equation existence conditions

The existence conditions for non-zero integer solution to j^2 - Dk^2 = -1, are:
1. For every odd prime factor p of D, p = 1 mod 4.

2. If D-1 has 8 as a factor it must have 16 as a factor as well.

3. D cannot equal 1 mod 5.

4. If D is divisible by 2 it cannot be divisible by 4.
Those are the first set of conditions required for the existence of the negative Pell's Equation.

There may be more, but probably not.

## JSH: Advancing theory of continued fractions

There is a reason for the discussions I've raised about Pell's Equation and alternates like the negative Pell's Equation besides demonstrating a woeful disdain for mathematical knowledge that some of you have, as the real prize is advancing the methodology of continued fractions. And that is a HUGE prize. Unlike any other available, as then we can see further on the shoulders of the giants before us, like Gauss and Lagrange.

You see, for the mathematically astute of you, there is a problem with Pell's Equation solutions leading to solutions for the alternates as the alternates ALL have solutions that are smaller than their corresponding Pell's Equation solution by roughly a square.

For instance, for j^2 - Dk^2 = -1, any solution MUST give a solution to x^2 - Dy^2 = 1, with x = 2j^2 + 1.

So you can see the square with j^2, and now we get to continued fractions!!!

Because you can solve Pell's Equation using continued fractions. But you can solve the negative Pell's Equation—when it exists—using continued fractions as well!!!

But it's SMALLER by a square!

I've hypothesized that people normally do continued fractions in the least efficient way, using all positives, when you should use negatives.

If so, then you can find a convergent much faster, meaning that you can advance the use of continued fractions by a huge margin.

I've been puzzling over this area for a while and thought recently, hey, why not toss it out there.

There are some dark forces among you though, so I played it out for a few days for you to understand who they are, and what they are.

They are, quite simply, anti-mathematicians.

They pretend to be mathematicians, but show their true colors when discovery is about, as their role is to block human progress!!!

They were put on this earth to make a challenge of it! For the great ones.

They were put on this earth to trip YOU up, as only the truly great among you can get past them.

## JSH: Negative Pell's Equation, consequences

A telling discussion erupted when I challenged these newsgroups on a simple result about the negative Pell's Equation and its connection to Pell's Equation, where posters claimed the result was previously widely known.

That result is that given j^2 - Dk^2 = -1, the negative Pell's Equation, x, for x^2 - Dy^2 = 1, is given by x = 2j^2 + 1.

I've noted that is not stated in mainstream mathematical literature. Posters have argued with me—often insultingly—claiming it is.

In response to a request for citations giving the result, they gave, if anything, citations to equations that could LEAD to that result, if you knew where to look!!! All math is derived from prior equations, except the basic axioms.

Now here are some obvious consequences of that result to further test your credibility in believing it is already part of mainstream literature.

Given non-zero integer solutions to x^2 - Dy^2 = 1:
1. x must be odd, when a solution to the negative Pell's Equation exists.

2. x = -1 mod D is required when a solution to the negative Pell's Equation exists.
The issue is whether that is mainstream knowledge in MODERN number theory, so whether or not you think you know number theory and know these trivial results is an issue because I am trying to teach here.

The people arguing with me are, in my opinion, hoodlums disrupting a class discussion.

Knowledge just is. But when some people hate knowledge because they think it hurts their class positions or their personal views of themselves, they can behave very badly in their attempts to hide knowledge.

I think Pell's Equation is kind of fun. There are interesting little results around it easily available to people who enjoy mathematics.

It's not my fault if these results are not taught by mainstream mathematicians.

And I should not be attacked for giving them now.

## JSH: What's remarkable

Years ago I discovered that posters would routinely lie to me in reply and they'd lie about even the most basic mathematics.

But what was remarkable to me was, other readers would side with them, even on trivial errors.

And the denial about the publication, withdrawal and destruction of the journal SWJPAM was telling as well.

It's just basic human psychology though.

It's a group effect.

A human being can be convinced of ANYTHING. And groups of human beings are more easily convinced than a single person!!!

So your knowledge of mathematics can be completely flawed, but if you have a group of people telling you otherwise, you believe.

But not if you're a true mathematician.

THAT is the test. It always was.

And most of you failed, long ago.

All that has remained is the clean-up.

## JSH: Learning from the negative Pell's Equation

For me the chilling proof that math society itself willfully lies can be seen with some really trivial algebra, Pell's Equation and the negative Pell's Equation which is why I keep mentioning it, as I can beat up on math society worldwide with this result indefinitely.

Given ANY set of non-zero integer solutions to the negative Pell's equation

j^2 - Dk^2 = -1

you will ALWAYS have a solution to Pell's Equation

x^2 - Dy^2 = 1

from x = 2j^2 + 1.

That is a mathematical absolute. Now go try to find it in a contemporary mathematical textbook.

What I like about this result is how clearly it shows the political nature of the modern field of number theory.

Number theorists, quite simply, lie. I dare them to keep ignoring this result! I like beating up on them.

## JSH: EMIS has my old paper back up?

Looks like EMIS has my paper back:

http://www.emis.de/journals/Annals/SWJPAM/Vol2_2003/2.ps.gz

I find that curious enough that I will read replies in this thread. I haven't been reading replies to my previous postings, as, what's the point?

Crazed and obsessive people I at times call the "angry idiots" dominate those threads and don't say anything new.

Here I'm curious as to whether or not anyone knows why it'd be back up now.

Oh yeah, those wondering how crappy this paper really is given the drama around it can now see it for themselves, if that link works. (It worked just now when I noticed it.)

Some sci.math'ers thought it worth upending the journal system to attack that paper with an email assault. The editors pulled it from the journal after publication and managed just one more edition before folding, shutting down.

The American journal had its archives kept up by EMIS, a European agency. Europeans saving the records of an American journal. But maybe they care about knowledge in a way people in this country demonstrably don't.

Makes me wonder how much my own country, the United States, truly values mathematical research versus Europe.

People show what they truly value by what they do, not by what they say.

[A reply to someone who wrote that James' lack of attention to typographical detail in his paper surely goes all the way from the care on the text to the mathematical reasoning.]

That's a non sequitur. Minor typographical errors do not prove a mathematical line of reasoning to be flawed.

It is interesting that you think so though, but readers considering this issue now years later know that extraordinary things happened after that publication: sci.math'ers mounted an email assault against it, the editors caved and yanked the paper after publication, they managed one more edition of the journal and then quietly shut down, but even weirder, its hosting university, Cameron University then scrubbed ALL MENTION of the journal from its websites!!!

They trashed a decades worth of papers, which should be disquieting to people who publish in electronic only journals.

EMIS saved its archives though, and now it seems saw fit to save my paper as well.

Discounting all of that over typos indicates you are very much, um, not rational.

That paper historically may be considered one of the biggest in mathematical history.

Deference to it by the editors will not seem strange to historians.

Leaving in minor errors might have seemed like the best thing to do to the editors, kind of like, how dare you correct something that huge? But then they went into shock at the reaction of their community, and fell apart.

Your mathematical community betrayed their trust, and destroyed a journal in the process.

[A reply to someone who told James what he should do in order to publish his article.]

I gave up on the math world. I'm working now on financial problems—more money in it.

Besides, I have the satisfaction of correctness. And can puzzle on the world's inability to see a massive error that may be stopping potential future progress.

Newton had calculus over two hundred years ago, and Archimedes had a lot of calculus thousands of years ago.

But calculus is most important today.

No one may ever know what human beings will not be able to do, if they're still around, in a couple of hundred years ago because of this block, but then again, that's why history gets to be interesting.

Who knows, if major discoveries weren't often lost human beings might be routinely doing interstellar travel by now.

But instead we're muddling around here burning up our resources and over populating—stupid species.

My guess is that the fate of humanity has already been decided. Against that backdrop my little tale is just something amusing. Not worth worrying about in the big picture.

## JSH: The Simple Lie

The fact that with integers x^2 - Dy^2 = 1, can always be solved by integer solutions to j^2 - Dk^2 = -1, using x = 2j^2 + 1, is just a mathematical reality.

The belief that modern mathematicians give a damn about their field and value knowledge over basic human politics is a simple lie.

So I can show 9^2 - 5*4^2 = 1, is given by x = 2*2^2 + 1, as 2^2 - 5*1^2 = -1, but you can ignore that because you wish.

And modern mathematicians who pathetically don't give a damn about anything but their grants so they can pay their mortgages I guess, can get away with a simple lie.

Check online sources on Pell's Equation:

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

http://mathworld.wolfram.com/PellEquation.html

See if you can see something so simple mentioned. I selected that result as it reduces to saying that given an integer solution to the equation often called the negative Pell's Equation you ALWAYS have a solution to Pell's Equation in integers.

Always.

A mathematical absolute.

Now that result probably was known to Fermat and Euler but incompetence entered the math field in the late 1800's and it kind of slipped through the cracks.

But if modern mathematicians were what they claim to be, so what? They'd just kind of laugh it off, note the obvious result and give me credit…oh.

Now you see where it's politics.

Can you imagine physicists stopping with a result in an area with two thousand years of research interest to prevent one man from getting credit?

Why do you admire mathematicians so much? When they lie to you in return?

Maybe some of you need to value yourselves and your knowledge better as that is nothing compared to the bigger lies that cover Galois Theory and shift what you think you know about Group Theory.

Your loss is just about history. When humanity adjusts and corrects, and much of your research is tossed on the heap, students later may read about you in a paragraph and imagine there is no way they could have been like you.

But you are you. History is waiting to happen. The destruction of your research is today.

## JSH: Pell's equation and question of math fraud

I've been pondering the situation where I found that what is commonly called Pell's Equation, has this parametric rational solution that math people don't talk about, and also has this connection to these alternate equations that make solving them easy, or vice versa, you can solve Pell's Equation with them, and now I can get some resolution to troubling questions.

Like, I have various math discoveries, which other people say are not math discoveries, and I've considered that hey, maybe I'm evolutionarily more advanced, so I can understand things that less advanced people can't, which is more simply captured by noting that dogs can't learn calculus. Their brains make that impossible. If I were evolutionarily advanced then I could have math discoveries simple to me that others could no more understand than a dog could understand calculus.

That was a scary scenario, and I'm happy to put it to rest, which is what Pell's Equation has allowed.

Pell's Equation is just x^2 - Dy^2 = 1, where math people are looking for integers that fit into all those boxes, so D is a positive integer, and x and y are integers. 9^2 - 5*4^2 = 1, is an example of such an outcome.

The equation has been known for thousands of years or something.

Well in our greatly advanced times, it seems mainstream math people have not bothered to note that if D is a prime number such that D = 1 mod 4, then another equation often called the negative Pell's Equation is solved by Pell's Equation:

j^2 - Dk^2 = -1, j = sqrt((x-1)/2)

So with my example above with D=5, notice that j = sqrt((9-1)/2) = 2, works, as 2^2 - 5*1^2 = -1.

And you could go the OTHER way, and find the solution to Pell's Equation, as j = sqrt((x-1)/2) means that

x = 2j^2 + 1, and x = 2*2^2 + 1 = 9, as required.

That always works. It's actually a rather trivial result mathematically which you can figure out from the main Pell's Equation as

x^2 - Dy^2 = 1, means that x^2 - 1 = Dy^2, so (x-1)(x+1) = Dy^2,

and if D is prime then it can only be a factor of x-1 or x+1, meaning that one of them must be either a square or 2 times a square as if x is odd then they both are even. You can work out all the alternates by noting how D can divide across. Trivial algebra.

That's not hard to understand, but if you go on the web and look at math texts on Pell's Equation you will not find that result:

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

or

http://mathworld.wolfram.com/PellEquation.html

You may think there is a trick in D being prime, where D = 1 mod 4, but turns out there are two more equations like the negative Pell's Equation if D does not equal 1 mod 4, and one more on top of those if D is a composite in a special way.

So there are 4 other alternates to Pell's Equation, where any of those will solve Pell's Equation if they exist, and the solution is smaller.

Here's a fun example to show the size difference:

1766319049^2 - 61*226153980^2 = 1

but, consider

j^2 - Dk^2 = -1

as you get a solution to x, when x^2 - Dy^2 = 1, with x = 2j^2 + 1, and with D=61, notice an astounding difference in the size of the solution—

29718^2 - 61*3805^2 = -1

and a quick check with your computer's calculator will show that, yes:

1766319049 = 2*297182 + 1.

You can use continued fractions to solve in either direction which is neat because the way math people teach continued fractions you may naively think there is only one answer they give, but math people traditionally go ONE WAY, using all positives.

The reality of the simpler solution for the negative Pell's Equation indicates they're going the wrong way, so using continued fractions properly—using negatives—could greatly simplify solving Pell's Equation by working to give one of the alternates sooner, and then allowing one to pull its answer from them.

Ok. That's not rocket science. It's relatively simple algebra that I assume you can all understand. I'm not trying to teach calculus to dogs here. I'm not some super advanced mutant freak who can understand things you cannot, which means, fraud is now on the table, as I've talked about the above for a while now.

The Wikipedia has not been updated. Wolfram hasn't updated. Far as I know math people are still teaching Pell's Equation like they always have, and emails on this subject to people like Arjen Lenstra, have not been answered. He's an expert in this area.

Near as I can tell math people have no intentions of ever acknowledging the above. Ever.

Now then, are they dogs who can't learn calculus, or are they academics who are playing a dangerous game, certain they can win, and damn humanity in the process?

You try and figure it out. It's your world as well as mine. If there is fraud on this level, you can be very certain it's very bad, and the reasons are terrifying and not at all good for the future of the entire human race.

After all, they're being rather daring at this point.