## JSH: Using tautological spaces

Maybe the best way to move things forward is to show how to use these things I call tautological spaces, so others can see their value, so I'll explain things in this post where the simplest thing to point out is that what I call a tautological space is a mathematical identity.

So what I do is get identities, subtract an equation to be analyzed from that identity, and probe the residue, where to maneuver with the identities I use "mod" from modular arithmetic, for instance:

x+y+vz = x+y+vz is the identity but I present it as

x+y+vz = 0(mod x+y+vz)

and then you can manipulate that identity easily with much of the complexity of what you're doing hidden away.

For instance, moving forward with the above:

x+y = -vz(mod x+y+vz), and I can square both sides to get

x^2 + 2xy + y^2 = v^2 z^2 (mod x+y+vz)

and now if I were analyzing x^2 + y^2 = z^2, I can just subtract it away to get

2xy = (v^2 - 1)z^2 mod (x+y+vz), which is

(v^2 - 1)z^2 - 2xy = 0 mod (x+y+vz)

and that is the residue. Notice that v is a free variable, so you can make it whatever you want, while x, y and z are constrained by the equation to be analyzed which I call the conditional.

I use x+y+vz because I invented this technique while tackling Fermat's Last Theorem, so I was always subtracting x^p + y^p = z^p, but eventually found I actually needed

x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)

which goes to the issue of the exponents of the variables in the identity and the answer is, they can be almost whatever you want (non- zero and positive). Since I invented this field I've thought about terminology, and I call the set of exponents the hyperdimensional set, and you can describe the tautological space completely by that set, so that last is:

{2,2,1,2}

so it's a 4 dimensional tautological space, and I call it a tautological space as I'm using identities--which are tautologies as well--and manipulations with them gives you a space where the identity holds.

When I introduced tautological spaces on the sci.math newsgroup back in December 1999, I got a lot of various skeptical reactions, where the most important was the belief that identities can't be useful for analysis.

It's pointless to lay claim to my disputed proof of Fermat's Last Theorem here as an example of the power of analyzing the residue from these identities though I will point out that I *could* tackle x^p + y^p = z^p by using a rather complicated identity, though I ended up subtracting out x^{2p} + 2x^p y^p + y^{2p} = z^{2p}.

But more recently, and more easily considered than FLT, I subtracted out:

c_1*x^2 + c_2*xy + c_3*y^2 = c_4*z^2 + c_5*zx + c_6*zy

from an identity in a tautological space with the hyperdimensional set {1,1,1,1}.

That is, I used x+y+vz=0(mod x+y+vz), got a more complex identity with a very few simple algebraic manipulations so that I could subtract out the conditional, and got a result.

With z=1, I found I had a way to simplify checking that expression and solving it as a Diophantine equation and though I've given it many times before in recent threads as I've tried to publicize this result, I'll give it now:

(2A(x+y) - B)^2 - 4AS^2 = B^2 - 4AC

where

A =(c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3)

B = 2(c_2 - 2c_1)*(c_6 - c_5) + 4c_5*(c_2 - c_1 - c_3)

and

C = (c_6 - c_5)^2 - 4c_4*(c_2 - c_1 - c_3).

So you now have to find some integer S and x+y, but then you can easily enough get x and y, so it's a vast simplification, where using classical methods, with

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

you'd do different things depending on the coefficients but would do some completion of the square and other manipulations to get to something of the form u^2 - Dy^2 = N, so using tautological space I found a way to do that all in one sweep.

That is more to demonstrate that using identities does not just lead in circles, though it can, which is one of the warnings for those who try to use this analysis technique.

Picking values for your free variable is a critical step which can requires some creativity and fumbling as you figure out how to probe your residue.

On a sidenote, one way of looking at what I've done is that I've simply extended from Gauss who pioneered use of "mod" by having the modulus be an equation, to use an identity, to get complicated identities with less effort and subtract some equation to be analyzed.

Why I got to invent this technique, I don't know.

Note that I use x, y and z with v the free variable for historical reasons but you can use as many variables as you like, and of course, call them what you want, while one requirement is that the free variable be multiplied times at least one of your other variables, so no, it's not an accident that I have x+y+vz, but a necessity learned when I was exploring this approach, as I tried several different forms at first before I settled on that one.

And that is a quick go over of tautological spaces.

Any equation you have from wherever you can put into a tautological space with an additional variable, so you can force a degree of freedom, which shows up in the residue.

It can take some thinking to figure out how to manipulate from the basic start, so for instance with a hyperdimensional set of {3,7,2,5}, you'd start with:

a^3 + b^7+v^2 c^5 = 0(mod a^3 + b^7+v^2 c^5)

where I like v for the free variable but changed other variable letters just to be a little creative.

I always use positive signs but don't see why that's a necessity. It's just been the way I do it.

Some may wonder, what good is that technique with fields? Is it only good with integers?

Answer is, you're using identities, so the result holds in a field, though you may have trouble finding a meaning then for the "mod".

I've used tautological spaces with Diophantine equations just because that's where the motivation for the invention has been, but once trying to get creative I found some funky yet useless trigonometric identities using them, like v^2-1 + 2vcos(θ) + 2cos(θ)^2 ≡ 0(mod sin(θ) +cos(θ)+v)

For more see my math blog: http://mymath.blogspot.com/2005/07/trigonometry-relations.html

But I really don't know what the full power of this technique is, though I'd suspect it'd be up there as I've done a few things with it, and, well, you can pull in ANY mathematical equation, so you can pull in all previously known mathematical equations into a tautological space to subtract them out and look at the residue, so, I like to say, tautological spaces encompass all prior human mathematical knowledge.

Hopefully I helped my case a bit with this post and maybe the physics community will be a better audience than the mathematical community was back in December 1999 and the months thereafter when I talked about these ideas relentlessly, until I stopped and focused more on my results from using them.

The naming conventions are of course my own as the inventor. But I think I did ok.

For more on tautological spaces as well as the research I've done with them, see my math blog.

## JSH: Any ideas what to do?

This latest result is great for those of you who think I'm facing rational resistance to understand what is really going on.

x^2 + Dy^2 = N

meaning

z^2 + D(x+y)^2 = N*(D+1)

is to me fun mathematics. Interesting number theory following from amateur research of my own, where I use this wild technique I got to name as I invented it, which is tautological spaces. So I figured all kinds of things out about quadratic Diophantine equations in 2 variables by subtracting from an identity made using

x+y+vz=0(mod x+y+vz)

and later using z=1.

So you have this remarkable analysis technique using identities. And I get complex identities, subtract equations from them, and analyze the residue where I can give you

x^2 + Dy^2 = N

requiring

z^2 + D(x+y)^2 = N*(D+1)

as just one of the discoveries from this path, and then let you think about the reaction of the mathematical community, which is disappointing.

I've had a paper published on other research which followed from my previous research using tautological spaces, where I'd chased Fermat's Last Theorem and found I could show a subtle error in number theory.

See: http://mymath.blogspot.com/2008/04/re-visiting-non-polynomial.html

So you see, ignoring the mathematics here is at least partly about hiding error, which is why the fight against the research is so political where there are so many smear tactics used against me.

They betrayed the discipline itself, and to escape they have to always distract from the truth, as mathematical proof is their enemy.

Think about it. Say some math professor writes complete garbage in a certain academic style, and a committee of others claim it is correct, who can come in and get the truth known if they all just keep agreeing?

You try. They'll smear you. Call you a crackpot. Question your sanity, and ultimately just never acknowledge no matter what you prove, as all they have to do is keep agreeing.

But why would anyone do such a thing?

Because that's how they get paid. If they tell the truth then they are no longer mathematicians.

But by talking complexities they can write complete garbage in a certain style that is said to be correct by others because they are writing complete garbage in a certain style and now all they have to do is keep out real checking, like by computers.

So computer checking of claims does not meaningfully exist in the math field.

And then they have to go after real discoverers like me, with the most powerful tactic being to do nothing.

And it's not just me, as consider Britney Gallivan.

See: http://en.wikipedia.org/wiki/Britney_Gallivan

I want you to understand that they will suffocate if they can ANY research that threatens their ability to put forward garbage in a particular style and claim it's valid mathematics, and they will not even properly acknowledge the work of a young teenage girl, as she was a teenager at the time (since graduated from college).

That could be you. It might be you if you're someone else who ran into this dark reality out there.

So what can be done? I'm really wondering.

I've talked to mathematicians by email and one in person at my alma mater. I've gotten feedback from notables like Barry Mazur. I've tried contacting the press about the situation. I even contacted a U.S. Attorney once about the situation.

From what I've seen, they've blockaded all the doors leaving me with posting on newsgroups, where other posters in a dedicated way shadow my posts to pump in negatives and distract often from the mathematics.

Or I can put things on my math blog and wait and hope. And years later…

One of the remarkable things the world has done, according to Google, is a vote for my research through search engine results, like my favorite: Google "definition of mathematical proof"

But that is scary as well as it implies also that the world doesn't know what to do either.

Who can win against them? How can the truth win against the agreement by committee?

For those who think this issue is still just about one person, consider that the current financial crisis in the world was partly driven by complex mathematical models sold to financial institutions.

Lying about mathematics is not just a minor thing.

Without the correct mathematical ideas in place humanity cannot move forward in science and technology.

Short-sighted people worrying about their paychecks or having a job (as what would they do if they had to do REAL research versus faking doing math?) are willing to destroy the future of the entire human species.

If you thought this issue was unimportant, look to the financial crisis playing out, and really consider that math people sold those financial ideas. Math people did it.

Your society helped break the world.

[A reply to someone who asked James who would he trust better than mathematicians to write the computer programs checking mathematical proofs.]

Expert systems are not written entirely by the "expert".

Your question is like wondering how medical diagnosis programs can get written if medical doctors only would write them.

Or do you naively believe that only medical doctors write such programs?

I'm curious as this issue repeatedly comes up, where math people seem bizarrely to believe that mathematicians would actually write the programs!!!

Do you all think that? Are you all so naive about modern computer science?

Who do you think write these programs?

I'll tell you: computer scientists.

Like mathematicians do mathematical research, computer scientists write programs for expert systems and can do so in areas where they themselves are NOT expert.

Sometimes the math community seems remarkably obtuse about the simplest things!!!

Or are you playing stupid?

Like mathematicians are experts with mathematics.

Computer scientists you, you, I find it hard to describe you people, are the experts with computer systems!!!

Duh.

## JSH: One of the great relations?

I found it quite profound that now I know that in the ring of integers, given

x^2 + Dy^2 = F

I also know that

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

which is the start of what I call a Diophantine chain, where if the original equation has an infinite number of solutions for x and y, then you have an infinite chain, but otherwise you have a finite chain.

One poster has already replied derisively giving z explicitly in terms of x, y and D which kind of puzzled me as it just looks like such a beautiful result to me.

Like with D=-2, you have:

x^2 - 2y^2 = 1, followed by z^2 - 2(x+y)^2 = -1

and the next in the series is

w^2 - 2(x+y+z)^2 = 1

so you just get this flipping back and forth, and with one solution at the start you can get the solutions that follow, so with x=3, and y=2, you have next that

z^2 - 2(5)^2 = -1, so z^2 = 49, so z=7, and then you have

w^2 - 2(3+2+7)^2 = 1, so w^2 - 2(144) = 1, so w^2 = 289 and w=17.

And you can do that forever.

I play with equations now like

7^2 - 5(3)^2 = 4

so I know that

z^2 - 5(10)^2 = 4*(-4)

and can get z then is 22, and notice that 4 divides off so I have

11^2 - 5(5)^2 = -4, so

w^2 - 5(16)^2 = 16, so w=36, and you can divide off 16, and get

9^2 - 5(4)^2 = 1.

So is it one of the great relations?

x^2 + Dy^2 = F

requires that

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

Or am I just in love with one more of my discoveries. More of my math?

I don't even want to read much of any more math texts any more, which I guess is wrong, but it's weird.

I can now look over my own mathematical research and puzzle over it as I figured it out months or years ago and the complexity at times escapes me, as I forget how I figured it out.

So I just kind of stare at it and wonder about that person I used to be. He cared so much too. Was so excited. Believed in so many things. What a sad pathetic character was he with his scribblings, his hopes and his dreams.

So faithful in his love of the human race. So believing in that "someone" out there who cared about the truth to whom he was always talking.

But now there is just me. Looking over these things that often I don't even care any more to understand. Pondering…

I won. I killed the other guy inside of me. So now there is only me.

And success is the only real reward.

## JSH: Pondering a shift

Looks like I have a mathematical result which should have big physics implications in terms of usefulness as well, which will defy any and all attempts to discredit or demean it as not important, which presents me with the question of how to shift from angry discoverer fighting an unfair political war against mostly angry sci.math'ers and do-nothing academic mathematicians, to someone who needs to move towards positives.

And, in case you missed the result, I found that given a quadratic Diophantine equation in 2 variables:

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

I could prove that solutions to it required solutions to

(2A(x+y) - B)^2 - 4Az^2 = B^2 - 4AC

where A, B and C are given by

A =(c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3)

B = 2(c_2 - 2c_1)*(c_6 - c_5) + 4c_5*(c_2 - c_1 - c_3)

and

C = (c_6 - c_5)^2 - 4c_4*(c_2 - c_1 - c_3)

so you still have two variables to determine but now they are x+y and z, with a simpler equation, and if you find solutions for x+y, you can substitute out x or y in the original equation with the guarantee of solutions there.

But I didn't stop there as I then found that given any Diophantine equation of the form (this research is the latest from the last few days):

u^2 + Dv^2 = F

you have a connected equation, which you can verify with the result above, of

w^2 + D*(u+v)^2 = F*(D+1)

which is a remarkable little result! And with it, you can go on to find a general method to solve all cases where solutions exist, except for D=-1, where factoring still is the answer.

To see a simple example of how this works, consider D=-2 and F=1, where I'll copy from a recent post tonight:

x^2 - 2y^2 = 1, is you'll note followed by

z^2 - 2(x+y)^2 = -1

and the next in the series is w^2 - 2(x+y+z)^2 = 1,

so you just get this flipping back and forth, and with one solution at the start you can get the solutions that follow, so with x=3, and y=2, you have next that

z^2 - 2(5)^2 = -1, so z^2 = 49, so z=7, and then you have

w^2 - 2(3+2+7)^2 = 1, so w^2 - 2(144) = 1, so w^2 = 289 and w=17.

So I got from knowing that 3,2 is a solution for x^2 - 2y^2 = 1, 17, 12 is also a solution and that 5,7 is a solution to x^2 - 2y^2 = -1.

And that is kind of to help you see that there is no doubt about the mathematics and if you want to make your own simple examples you might try

x^2 + y^2 = 25, or even x^2 + y^2 = 5, as you don't have to use a square.

As to why this result is important: well it discretizes in a complete theory all conic sections, except again for D=-1 which is just the factoring result, that is x^2 - y^2 = F.

Also there is 2000 years of other mathematical ways of doing what the little bit of mathematics I showed you can let you do, where I'll say that the general solution is about residues.

Also you can tell if there is an integer solution to

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

by checking if A is a quadratic residue modulo B^2 - 4AC, AND B^2 - 4AC is a quadratic residue modulo A.

If you boil all the mathematics and theory down to the nitty-gritty I think it's not a lot of pages, while my paper covering it all, including an example is 10 pages, but I used 12 point type.

Obviously for me there is gratification with a result I THINK math people who argue with me cannot deny, or lie about, or do all that other annoying stuff, but I have to still be concerned that they will try and, gasp, might get away with it!

So why? Why might they?

Well mathematics is another world in terms of how its people see their discipline and I'm an upstart, upsetting a lot of applecarts who proved Fermat's Last Theorem. Found a result key to resolving the Riemann Hypothesis, and did other things besides including giving a rather good definition of mathematical proof. Google it. I know, I say that a lot, but it's rather wild! No other human being in the world can tell you to do the same. No one.

And I am from the physics world. Not the math world. And I've said nasty things about their world and how it operates.

But that was then. I'm looking to be a much nicer guy unless the political fighting keeps going, and then I take back being nice and go back to the other way. Maybe not productive but it made me feel better.

After all, all joking aside, I have had major mathematical results now for over 6 years, where my reward from the world has been the label of crackpot and watching mathematicians either deliberately do nothing when I knew I'd proven to them what I'd accomplished, or call me nasty names and work to convince other people that I am crazy.

How would YOU behave if you were in my situation?

Think you'd even still be here?

Imagine you had some incredible physics result, did everything right, had a tremendous amount of proof, including replicated experimental results and rather than give you your Nobel prize—physics people mostly just ignored you or called you crazy, for year after year after year.

Think you'd survive it? Think carefully.

After all, what then would you believe in?

## JSH: Diophantine chains and x^2 - 2y^2 = 1

Given the general result that whenever you have

x^2 + Dy^2 = F

you have a connected equation

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

where finding a solution for it gives you a solution for the first, you can immediately figure out how to get an infinite number of solutions with D=-2 and F=1, the classical Pell's equation, as then you have

x^2 - 2y^2 = 1, followed by

z^2 - 2(x+y)^2 = -1

and the next in the series is

w^2 - 2(x+y+z)^2 = 1

so you just get this flipping back and forth, and with one solution at the start you can get the solutions that follow, so with x=3, and y=2, you have next that

z^2 - 2(5)^2 = -1, so z^2 = 49, so z=7, and then you have

w^2 - 2(3+2+7)^2 = 1, so w^2 - 2(144) = 1, so w^2 = 289 and w=17.

And you can do that forever.

It's kind of weird to look at an elementary result that gives all the answers where other techniques were used before, and consider how that relates to them.

But even weirder that I have the general result that is part of a theory that handles ALL Diophantine quadratic equations in 2 variables, except for D=-1, which just gives you a factorization result and no chain.

x^2 + Dy^2 = F

connecting to

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

is the underlying machinery which leads to an elementary solution, and now one can ponder how that simple answer relates to the more complicated ones in the area, like in explaining exactly what is really happening with continued fractions—regardless of the classical explanation.

After all, centuries ago men believed that the planets moved in circular orbits because they thought that was perfection and why would God do it any other way? But perfection turned out to be elliptical orbits and beyond.

Advancements in knowledge advance explanations—from the primitive to the new as our knowledge as a species, grows over time.

## JSH: Not how I thought it would be

I don't care any more if anyone believes me or not I just want to stop figuring out this stuff. I just want it to stop.

Every rationalization I have for why I keep working at this math crap is gone and I just wish I could stop.

They hate it. The math people. They hate the knowledge.

I was a kid reading about discoverers wondering to myself if I were in Newton's place or in Euclid's or any of the others, would I figure out the same things?

Now I wish they hadn't. I wish this sick, serpentine, twisted, nothing of a world could just die.

And take its things with it.

I want the human race to just quit lying, quit pretending and just die.

You do not deserve knowledge.

You deserve to just die and quit desecrating this earth.

I used to dream. I used to hope. I used to believe in something.

Now all I dream about is the extinction of the haters. The end of the liars. The death of all of you who hate knowledge and pretend you care. Pretend you give a damn.

Pretend you want to know.

Before God I pray, end this madness. They have prayed for their end my Lord.

Let them die.

## Diophantine chains, citation

I came up with the following:

http://rmmc.asu.edu/abstracts/rmj/vol13-1/owipag1.pdf

DIOPHANTINE CHAINS
JAMES C. OWINGS, JR.
ABSTRACT. We solve a large class of quadratic binary Diophantine equations without resort to the theory of quadratic number fields. Examples are given of equations amenable to our approach, including some which are intractable by classical methods. A corollary is drawn concerning the size of smallest possible solutions

The paper was written in 1983, and I only saw one page on the web so couldn't dig into it to see what technique they use.

But it does tell me that at least as early as 1983, people were still working on the problem.

I'm still researching around on this issue but it does look like a case where I can show a clear advantage with my techniques where math society is once again dragging its feet.

I already have 2 rejections on my paper with the general solution to 2 variable Diophantine quadratic equations.

The financial crisis in the US picked up steam after those rejections.

You do not understand reality as well as you think.

Your rejection is driving the world to the brink.

If humanity does not choose knowledge then it has no future.

The irony is that you don't get years to slowly decay and die.

The collapse of civilization upon my failure will be immediate.

And it is beginning now.

### Friday, September 19, 2008

Pondering the ongoing battle to get proper recognition for my research I am yet again making a post which is nothing about a particular mathematical proof but all about the social necessities that must be accepted.

I have a B.Sc. in physics from Vanderbilt University, where I went on a full scholarship.

For most of my life I have had the "gifted" label, and have belonged to a couple of high IQ societies, as I was in MENSA for a very brief time (very brief) and for longer was in an ultra high IQ group called the Ultranet where I was a provisional member because my IQ was only in the 1 in 1000 range by test, so I didn't have a high enough IQ for full membership. Some of the other members had the highest IQ's, by test, in the United States.

What else. Um, as a teenager I twice went to Duke University as part of their Talent Identification Program. (T.I.P.)

First time I learned geometry in a week. Second time I took structured C from an IBM researcher on loan to Duke for the program.

Having had most of my life to consider what "smart" means I'm mostly immune to insults telling me I'm stupid. For me, stupid is something I see every day. I know what it is. Stupid is, as stupid does, as Forrest Gump would say.

Part of the reason I turned to mathematical research was the notion that finally there was an area where all that mattered was the truth, and what you could prove.

For years though I was just a crackpot, for real, with various ideas I tried against Fermat's Last Theorem with little success, though often I deluded myself into thinking I had it, and I took the heat when I had to come back and admit failure after at times going on and on about corrupt people ignoring my research. That was back in the late 90's.

But one day in frustration, and pondering things like "degrees of freedom" I started working on a way to add more variables to the Fermat problem as I couldn't think of anything else to do with x, y and z, and eventually I figured out:

x+y+vz = x+y+vz

as a way to do it, with v my new variable and my new degree of freedom.

(Oh that STILL wasn't enough for that hard problem as I eventually turned to x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2, after a few years of playing with the simpler one.)

And three years later I started having correct results but noticed that people kept arguing with me, but hey, I could just get published, right?

After rejection after rejection after rejection I finally DID get published and some sci.math'ers broke the journal process with barely a sweat using some emails, betraying the reality that the academic journal process is WEAK, and you can Google SWJPAM to get more of the story.

Short of it though is, editors published my paper, sci.math'ers emailed the editors, editors pulled my paper, journal died a few months later, quietly.

I don't believe in a lot as I've studied a lot—that "gifted" thing, means I absorb information extremely rapidly and I like to study history—and I know what's going on in terms of the big picture but can't quite figure out why demonstrably from what is happening to me no one really cares about mathematics.

My academic credentials probably mean more to most than any mathematical proof I ever find, which is the way of the world.

What people think is nothing compared to mathematical proof, but while I do math research as a hobby, I can't spend as much time at it as I'd like, so I'm making concessions to social reality and considering as a real problem to solve, how do I get proper recognition for my research so people will leave me alone to do more of it?
My original purpose in starting this thread was to try and counter the harsh political war that has been I believed so successful in blocking acceptance of my research. I wanted people to know that yes, I do have a degree in physics, and I tossed in other things that I hoped my help people to understand that I am not just some person mouthing off without a record demonstrating my abilities.

I know the problem here: we rely on our institutions in a very complex world.

But you should also rely on proof, and common sense.

Academic journals do not just die. Google SWJPAM, the journal that keeled over after sci.math'ers broke your vaunted journal process.

No other human being on the planet can suggest you look up a definition of a phrase like "mathematical proof" and you get that man's personal one.

See if you don't see my math.

Some of you in academia may fear my rhetoric against the current status quo.

But whether it is from suggestions from someone like me, or collapsing funds available as the world gyrates through financial crisis, you will lose that money anyway.

The difference is, that as people like me are blocked and the answers are not correct, then you get events like those unfolding in the United States where the answers are ignored by people who think they can just, will their way through.

Our ancestors were better able to survive because of their intelligence, while today many think of it as a status object, to be waved around like a symbol, or to be put forward like a nice car, but it is about survival.

The meek will not inherit the earth, as if anyone does, it will be the intelligent as they did before as we are their children.

We as a species are the intelligent who inherited this earth.

And we as a species are the ones fighting to give it back to darkness.

I live in a country that prides itself on electing dumb people to office, and now the entire world is paying the price.

But if you were smart about it, how else could it have turned out?

Repudiate intelligence? Then why not just put chimpanzees in charge of the world?

The way things are going now, we'd probably be better off with a chimpanzee than George W. Bush.

I have mathematical proof. I even defined it.

The people arguing with me call me names. Gee, how smart do you have to be to do that?

They deride evidence. As if, anyone can get high in Google search results, right?

And they tell you they have the answers…

But where is THEIR proof?

Read this thread and see the people against me and understand what I face: politics in the face of evidence, denial in the face of proof, anger and hatred against the facts.

Recidivism back to where we came from, as if basic ape behavior matters more than the facts.

But FACTS rule the world.

## JSH: Degrees of freedom

Back in December 1999 I discovered a mathematical analysis technique that allows a researcher to add degrees of freedom with any mathematical equation. I found it from frustration.

The technique is to use something like

x+y+vz = x+y+vz

with "mod" so you have

x+y+vz = 0(mod x+y+vz)

where you want to analyze an equation in 3 variables, so to analyze x^2 + y^2 = z^2, you'd do the following.

x+y = -vz (mod x+y+vz), square both sides

x^2 + 2xy + y^2 = (vz)^2 (mod x+y+vz)

and subtract your equation to be analyzed, so you have

2xy = (v^2 - 1)z^2 (mod x+y+vz)

where now v is a free variable, so it is your extra degree of freedom.

The ever widening flame war against my research appears to be about number theorists suppressing this technique.
There have been a lot of criticisms leveled against this technique since December 1999 as I posted about it on math newsgroups back then.

I wrote a paper using it. The sci.math'ers managed to shoot it down with some emails and killed a mathematical journal in the process—oh, and brought into question the ENTIRE journal system as well in so doing.

Now I have a result with 2 variable Diophantine quadratics, handling in 6 pages what previous mathematical tools took books to cover, and the reason I'm pushing this point now is that

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

has relevance to physics so versus just talking about some esoteric mathematical issue I can tell you that the number theorists fighting to suppress this research are TAKING something from you, as a physicist.

And if you want to just let it all work itself out, consider that I first put these ideas forward in December 1999. I then utilized them with one result that went into a paper which killed a freaking mathematical journal, and showed a break in the journal system.

I've now covered 2 variable quadratic Diophantines in a couple of pages—but the political war from the math newsgroups at least clearly continues as you can see in replies to my recent threads!

Resistance to new ideas is one thing.

But I'm seeing an all-out war with an absolute position of denial, and fighting this research no matter what is discovered, as consider the recent discovery, with a very vicious political fight which is a lot about smearing and nothing about utility to researchers.

History has shown these type of battles happening before.

The difference now is, you're living through one.
And I keep coming back to that as part of the key to this puzzle.

No matter what you may think of the value of Pell's Equation to physics or any of my other number theory research this idea of making complex identities using a free variable to subtract equations from it to be analyzed is just a remarkable idea.

Even credit for that alone could put me in texts all over mathematics and physics, and the fight against that technique is already over eight years old as I first presented it on math newsgroups December 1999.

Betrayal is as old as humanity.

Denial is as well.

The analysis technique I pioneered in the late 20th century could be one of the keys to understanding in THIS century, if I can get it past the gatekeepers betraying the ideal of pursuit of knowledge.

None of you with even a modicum of intelligence can deny the intriguing nature of the concept, or the reality of the furor generated by my use of it in now two cases:
1. A paper published with research using this technique was retracted
after publication and the freaking mathematical journal DIED.

2. Turning the technique against 2 variable quadratic Diophantine equations I came up with a general theory including how to solve them, in less than a month.
People who hide from such huge truths are not the kind of people who should be considered great by any means, or even sort of good at what they do.

You lose all credibility as scientists as skepticism here gives only one answer.

Your mathematical colleagues betrayed you and the rest of humanity and are STILL in the process of committing their crime: fighting an advanced mathematical analysis technique.

And you know it. Doing nothing here is about who you are. How corrupt you are, inside.

## Complete theory for 2 variable Diophantine equations, paper now available

I have completed the basic theory for 2 variable Diophantine equations. That is, the mathematical theory covering equations of the form

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y.

That theory gives the equation for determining existence of integer solutions, as well as a method using what I call Diophantine chains to actually find solutions when they exist.

The paper does include some basic Pell's Equation results as well, like the result that for every solution to Pell's Equation of the form

x^2 - 2y^2 = 1

you have a solution to the negative Pell's equation z^2 - 2(x+y)^2 = -1.

I am publishing through Google Docs:

I did want to put the new mathematical tools forward but also I was curious to find out things like, can people actually access the document? And more importantly, will anyone reply with issues with the details of the theory itself?

After all I found a proof of Fermat's Last Theorem over 6 years ago. I no longer believe in the academic system. I read news reports of supposed findings, or listen to babbling about "String theory" or supposedly how old the universe is, and now kind of yawn, if I bother at all.

It is weird though. I discretized conic sections using my tautological spaces. And added to my math yet another remarkable result, which took me 8 days to do, with mathematical tools so powerful they can do that with little to no effort.

2000 years of mathematical research in that area. 8 days for me to cover with a more powerful and succinct mathematical theory.

So I had this one idea I call tautological spaces and it can do so much, but it doesn't seem like that big of a deal. But I remember being a kid playing with parabolas, graphing them over and over again, and being excited about their properties. Reading other people's math. Now I read my math.

And to my math that I read I now have a full discrete theory for conics in general, figured out within a few days, using a mathematical technique, I invented.

And the world calls me a crackpot.

I no longer believe in people, with good reasons. Over six years of good reasons with the knowledge of my accomplishments and the world's reaction to them. But at least I still like people, mostly. I just kind of see most as, primitive.

I believe in mathematics.

Long after all of you are dead. Long after the sun has cooled and the Milky Way has drifted apart, what I've discovered will still be true.

And I defined mathematical proof. So I know of what I speak.

## Latest idea, solving 2 variable Diophantine equations

After pondering the TSP situation for a while I decided to let that subject drop for a while, as it incubated, and wandered off to do other things, but one day found myself pondering the 3 variable Diophantine equation of the form

c_1*x^2 + c_2*xy + c_3*y^2 = c_4*z^2 + c_5*zx + c_6*zy

And I figured out this theorem about it, and noticed that with z=1, I had

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

and a way to simplify from that to an equation of the form

A(x+y)^2 - B(x+y) + C = w^2

where

A =(c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3)

B = 2(c_2 - 2c_1)*(c_6 - c_5) + 4c_5*(c_2 - c_1 - c_3)

and

C = (c_6 - c_5)^2 - 4c_4*(c_2 - c_1 - c_3)

so I found a way to simplify any 2 variable Diophantine equation to a simpler one, as if you find integers w and x+y, such that the second is true, you can then solve for x and y directly, so you solve the first, if it has solutions.

But if the second does not have solutions then neither does the first!

For those wondering about solutions, with Diophantine equations solutions must be integers only.

So, for instance, with x^2 - 2y^2 = 1, x=17 and y = 12 work because 17^2 - 2(12)^2 = 1.

I found other tidbits along the way like that given the Pell's Equation:

x^2 - 2y^2 = 1

you automatically have a solution to the negative Pell's Equation:

z^2 - 2(x+y)^2 = -1

so you also can immediately get that 29 is a solution for x+y, and then find that z=41, as

41^2 - 2(29)^2 = -1.

And yes, I'm talking about these things with math people but so far in arguments they are just saying I have nothing new!!!

So here we go again. I say I found something nifty and people jump out of the woodworks to claim it's not.

Maybe Patricia or that Cranmer guy have comments this time?

Ok, so what good is the result?

Well, for physics people it could mean some explanations for physics stuff, but I'm not totally sure.

I'm just a guy who has ideas and the professionals in these fields blow me off, so I end up posting about them.

If you program the mathematics above you might want to go to my math blog where I have a complete theory, which includes an idea for determining when solutions can exist and solving using what I call Diophantine chains.

And yes, it is frustrating to me that no matter what I can prove it seems that established people who I've seen time and time again betray their academic credentials just get to act like normal, go to class, teach their students, collect their paychecks and government grants—while I'm stuck begging for attention for mind-blowing, revolutionary research on newsgroups.

The system is broken. It is hostile to amateur researchers. And the gatekeepers have just locked the doors and thrown away the key.

So I get to deal with people who are often very wrong about the details of my research, but who know that the status quo is to disagree with me, so they do.

Proof is not enough. These class wars are pushing the limits as the people who are at the top feel comfortable with things as they are.

If it were up to them, humanity wouldn't need to learn anything new at all, as what more do they need anyway?

An amazing result that still absorbing the full implications of, as before, the state of the art in this area was the use of far more complicated techniques to simplify to a simpler equation and then to solve it!

To get a better picture of what I mean, here is an example, where if you wish you can go to other sources to see how it is done with the other techniques known. And yes, for me it is kind of wild to be playing with techniques I just invented about a week ago:

To keep things easy for me, I'll use

x^2 + 2xy + 3y^2 = 4 + 5x + 6y

so I have

c_1 = 1, c_2 = 2, c_3 = 3, c_4 = 4, c_5 = 5, and c_6 = 6

so next I need to calculate

A = (c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3) = -8

B = 2(c_2 - 2c_1)(c_6 - c_5) + 4c_5*(c_2 - c_1 - c_3) = -40

and

C = (c_6 - c_5)^2 - 4c_4*(c_2 - c_1 - c_3) = 33

and I have then the new quadratic Diophantine:

(2A(x+y) - B)^2 - 4A*S^2 = B^2 - 4A*C

which is

(-16(x+y) + 40)^2 + 32S^2 = 2656

and dividing off 16, I have

(-4(x+y) + 10)^2 + 2S^2 = 166.

Which has a solution at S=9, giving

-4(x+y) + 10 = +/- 2

and trying the positive first gives x+y = 2, while the negative gives x+y = 3.

Trying the first case, x = 2-y and plugging that into the equation gives

(2-y)^2 + 2(2-y)y + 3y^2 = 4 + 5(2-y) + 6y

which is

4 - 4y + y^2 + 4y - 2y^2 + 3y^2 = 4 + 10 - 5y + 6y

which is

2y^2 - y - 10 = 0,

so y = (1 +/- sqrt(1 + 80))/4 = (1+/-9)/2 = -2 as the other case is a fraction.

Then x=4, so I can try x=4, y=-2, with

x^2 + 2xy + 3y^2 = 4 + 5x + 6y

and get

16 + 2(4)(-2) + 3(-2)^2 = 4 + 5(4) + 6(-2)

which is 12 = 12, so they balance out as they must. I'll leave the second solution to the reader. Notice there are only two.

So the equation turned out to be easy to solve using these techniques.

## JSH: The mathematical Catch-22

Mathematicians reject my mathematical results, but rejected them even when I got a research result formally peer reviewed and published. I've mentioned it before but it's worth pointing out again that you should Google "SWJPAM", as those are the initials of the now dead mathematical journal that DID publish me, before retracting after some sci.math posters went after my paper with concerted emails.

They've rigged the system so that nothing I do is supposed to matter. Nothing.

Not mathematical proof. Not formal peer review. Not publication.

But the techniques I pioneered with that paper allowed me to figure out how to take ANY Diophantine quadratic equation in 2 variables, that is, an equation of the form

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

and directly relate it to an equation of the form

A(x+y)^2 - B(x+y) + C = z^2

where I give what A, B and C are in my post: Solving Quadratic Diophantine equations in 2 variables.

Tell me what mathematics you currently have which can do the same thing as simply?

Now you can solve these Diophantine equation in 2 variables without regard to their form by relating them to the simpler form of

A(x+y)^2 - B(x+y) + C = z^2

in one swoop, as once you find an x+y that will work, then trivially you have x or y from the first equation.

But why would any mathematicians fight a revolutionary mathematical analysis technique?

Because the field is corrupted. That technique also revealed a serious error with certain esoteric mathematical ideas in number theory which was a reason for them to want to kill my earlier paper.

Since number theorist who specialize in the area shown to be flawed are not really doing correct mathematics in this area those current practitioners are not the best in the world so they cannot do the best mathematics in the world, which is why they couldn't give you the simplification I just did.

And I started working on it last Friday. One week is what it took.

I have seen how math society ways of doing things from this group of people have been corrupting the physics field itself with "string theory" and the de-emphasis on experiment as if a mathematical explanation alone that cannot be tested can be trusted, but people can lie or just make mistakes and all nod their heads in agreement.

If you wanted a dramatic demonstration of the failures of the current math field then you have it now.

They fought me for over 6 years to hold back knowledge of mathematical techniques that I used to simplify

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

in only 4 days. 4 days traversing an area with 2000 years of mathematical research.

Progress is about progression. People today with modern problem solving techniques can do the seemingly impossible—if allowed.

You must let us be allowed and stop the gatekeepers who are fighting an endless psychological war against people like me.

Human error is as old as humanity itself. History is full of fights of one group or another against revolutionary ideas without which science and human knowledge in general could not progress. You know the history.

You know of the battles of discoverers against idiot and politicians, sycophants and fools, who were fighting for some old order or another.

I am not asking you to just trust me. I am giving you mathematical tools with which you can advance human knowledge. LEARN more. DO more.

The choice to continue with the status quo is the acceptance of the defeat of the pursuit of human knowledge, to make me the Galileo who failed.

If that is your wish, then if you want to make this about one person and feel some glow of warmth at my defeat—the first in human history if it occurs—then you need to know what else happens: this species dies soon thereafter.

The role of the discoverer has never been presented this clearly before because the danger has never been this great before. The drum-beat of annihilation beats louder and its the complacency that may kill as too many think global warming is just another thing, or that somehow there will be a replacement for oil in time.

Because you cannot solve the problems when you block the problem solvers!!!

I am so frustrated that the very people who fail are selling themselves as the answer and blocking out the discoverers who CAN get answers in time.

I may be the first major discoverer to fail in human history. Guess there would have to be one, eh?

But I'd also be the last, as after that failure, there is no future. No path to greater science and technology—as you will not have the mathematics, and you will not have the problem solving techniques, and you will not have the pursuit of truth as the highest ideal.

You'll have pretenders who get what they can until the end comes and there is nothing left for anyone.

## JSH: Awesome simplification

Sorry to add a talking about post to my other post giving a route to a general solution to 2 variable Diophantine equations but it's just so freaking cool. I achieved the dream of every decent physics student: stunning simplification.

Now whenever you have some complicated Diophantine in 2 variables, like

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

you can turn it into

A(x+y)^2 - B(x+y) + C = S^2

where I gave how you calculate A, B and C in my earlier post.

That is HUGE. As in awesomely huge, and it's just so cool to me that I was the one to discover the mathematics.

And some of the other results that have followed quickly have been just kind of cool, as in, who knew?

Who knew that for ever Pell's Equation solution, solutions to x^2 - Dy^2 = 1, you had an integer solution to a circle or an ellipse?

And you can just SEE it in action with x^2 - 2y^2 = 1, where scan through any list of Pythagorean Triplets—integer solutions to u^2 + v^2 = w^2—and the cases where x and y can be integers just jump out at you, for instance Wikipedia has such a list at

http://en.wikipedia.org/wiki/Pythagorean_triple

conveniently at the top of the page, and I can just look through it and see 3,4,5 and 20,21,29 as the ONLY cases where solutions for x^2 - 2y^2 = 1 can exist with x+y less than 100, as w=x+y, and v=u+1, for every solution to x^2 - 2y^2 = 1.

That is for EVERY solution of x^2 - 2y^2 = 1, there is a Pythagorean triplet where w=x+y, and v=u+1, where

u^2 + v^2 = w^2.

And did you know that for EVERY solution to x^2 - 2y^2 = 1, there is a corresponding solution to

z^2 - 2(x+y)^2 = -1?

There are these DEEP mathematical connections that have physics implications and I'm glad to be able to share them with you.

Hope you go to my post: Solving Quadratic Diophantine equations in 2 variables

The mathematics is a stunning simplification over a previously complex area. The implications are huge. And the Gold Rush is on!!!

What can you now prove in physics that you couldn't even touch before?

The prizes of knowledge and that other stuff await those who figure it out first.

## Solving Quadratic Diophantine equations in 2 variables

Last Friday I discovered the following theorem, which I call the Quadratic Diophantine Theorem:

In the ring of integers, given the quadratic expression

c_1*x^2 + c_2*xy + c_3*y^2 = c_4*z^2 + c_5*zx + c_6*zy

where the c's are constants, for solutions to exist it must be true that

((c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3))v^2 + (2(c_2 - 2c_1)*(c_6 - c_5) + 4c_5*(c_2 - c_1 - c_3))v + (c_6 - c_5)^2 - 4c_4*(c_2 - c_1 - c_3) = n^2 mod p

for some n, where p is any prime coprime to z for a given solution, when

v = -(x+y)z^{-1} mod p.

For example with x^2 + y^2 = z^2, I have

c_1 = 1, c_2 = 0, c_3 = 1, c_4 = 1, c_5 = 0, and c_6 = 0

which gives

-4v^2 + 8 = n^2 mod p

for every prime coprime to z, for some n (remember ring is ring of integers) when v = -(x+y)z^{-1} mod p.

Making the substitution for v gives

-4(-(x+y)z^{-1})^2 + 8 = n^2 mod p

so

-4(x+y)^2 + 8z^2 = n^2*z^2 mod p

and since x^2 + y^2 = z^2, I can substitute out z, to get

4(x-y)^2 = n^2*z^2 mod p

so the requirement is met, as of course, there are an infinity of integer solutions to x^2 + y^2 = z^2.

And a square was required here because p can be any prime coprime to a solution for z, so an infinite number of primes must work!

Notice that the result also applies to the general diophantine quadratic in 2 variables by making z=1.

The theorem is proven easily using what I call tautological spaces.

Intriguingly the theorem shows a route to generally solving any quadratic Diophantine in 2 variables by letting z=1, as then you have

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

and the result is that

((c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3))v^2 + (2(c_2 - 2c_1)*(c_6 - c_5) + 4c_5*(c_2 - c_1 - c_3))v + (c_6 - c_5)^2 - 4c_4*(c_2 - c_1 - c_3) = n^2 mod p

where v = -(x+y) mod p, so you can substitute out, and have a result true for all primes p:

((c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3))(x+y)^2 - (2(c_2 - 2c_1)*(c_6 - c_5) + 4c_5*(c_2 - c_1 - c_3))(x+y) + (c_6 - c_5)^2 - 4c_4*(c_2 - c_1 - c_3) = S^2

where S is some integer. Notice then using some additional variables to make the exposition easier that with

A =(c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3)

B = 2(c_2 - 2c_1)*(c_6 - c_5) + 4c_5*(c_2 - c_1 - c_3)

and

C = (c_6 - c_5)^2 - 4c_4*(c_2 - c_1 - c_3)

you have

A(x+y)^2 - B(x+y) + C = S^2

and by completing the square and simplifying you have that

(2A(x+y) - B)^2 + 4AC - B^2 = 4AS^2

which is just another Diophantine equation in 2 variables but much simplified, and when you solve for x+y and S, you immediately get a solution for

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

but also you can check solvability easily with the result that

(2A(x+y) - B)^2 = 4AS^2 mod (B^2 - 4AC)

so A must be a quadratic residue modulo (B^2 - 4AC).

The result is important enough for all of that alone, but I'll mention that I looked for preliminary results with the theorem by using it on Pell's Equation, and found several tidbit results:

With x^2 - Dy^2 = 1, I have that

S^2 - D(x+y)^2 = -D + 1

so you see the second Diophantine equation connected to the first!

With D=2, I get then that x^2 - 2y^2 = 1, is connected to

S^2 - 2(x+y)^2 = -1

so for every solution of the first there is a solution of the second.

For example with D=2, I have x=17 and y=12 as solutions, but to x^2 - 2y^2 = 1, but notice I can immediately get that S = 41 is a solution using 17+12=29, so I have for the second equation 41^2 - 29^2 = -1.

Another result of interest I found with Pell's Equation is that for every solution to x^2 - 2y^2 = 1, there is a Pythagorean Triplet—an integer solution to u^2 + v^2 = w^2—of a particular form, which is that v=u+1, and w = x+y.

So for the result I noted above where x=17, and y = 12, I have u = 20 and v=21, and w=29, as

20^2 + 21^2 = 29^2.

The further result for x^2 - Dy^2 = 1, is that for every case where D-1 is a square there is a relation with a Pythagorean Triplet, and if it's not a square, there is a relation to the ellipse:

(D-1)u^2 + v^2 = w^2

where again w=x+y.

So in some sense Pell's Equation may be elliptical which may explain to some extent how it comes up in the sciences.

I hope my fellow physics explorers will welcome these new additions to our mathematical arsenal and look to see what deeper answers may be found through this simplification of quadratics in 2 variables.

For further information including the proof of the theorem, go to my math blog.

## JSH: Pythagorean Triplets and Pell's Equation

Now that I have a general theory for all 2 variable quadratic Diophantine equations it's worth coming back to note again the weird connection I found between certain Pythagorean Triplets and Pell's Equation in the form

x^2 - Dy^2 = 1

when D-1 is a perfect square. For instance for D=2, I have that for every solution of Pell's Equation you have a Pythagorean Triplet!

But the triplets are special in that with u^2 + v^2 = w^2, v = u+1. The connection is that w is x+y from Pell's Equation.

The more general result is that u = sqrt(D-1)j, and v = j+1, while w still equals x+y.

Intriguingly that means that proof that there are an infinite number of solutions for certain Pell's Equations is proof that there are an infinity of Pythagorean Triplets of a certain form!

An easy example with D=2, is x=17, y=12, where notice you are paired with the triplet 20, 21, 29.

That is just some low-hanging fruit that I thought I'd mention. Kind of been a whirlwind of results flowing from playing with my Diophantine Quadratic Theorem.
New argument now I'm starting to see is that I've found nothing new, though I will add that for me the Pell's Equation result is just a fun tidbit which is nothing compared to the main result of generally solving the 2 variable Diophantine equation.

A succinct example of the tidbit result claimed to not be new is the easy to show case that for EVERY solution to

x^2 - 2y^2 = 1

you have a solution to the negative Pell's Equation:

z^2 - 2(x+y)^2 = -1.

For instance, x=17, y = 12 is a solution to the first as

17^2 - 2(12)^2 = 1

and with x+y=29, you get z=41 for the second, as

41^2 - 2(29)^2 = -1.

To me that it's easy to explain so I have to wonder why no one it seems has said it in that way in human history before…

2000 years of mathematical history with Pell's Equation.

The will to lie about a subject that old is a powerful and demonic one, and for those of you who have wondered how I could be right, if so many people are arguing with me, here you can see.

They argue with me because these battles are supposed to be hard.

If it were easy then there wouldn't be a choice, now would there?

I'm set. It's you who has a fate in the balance.

It's your life that is being decided now. Not mine.

Who are you really?

In a sense, me and the others here are just agents to test your mettle.

God's way of testing your worth as human beings.

## JSH: More with Quadratic Diophantine Theorem

I introduced a nifty new theorem in another thread which has become bogged down with replies so here's a new thread where I can bring focus back to the theorem itself.

In the ring of integers, given the quadratic expression

c_1*x^2 + c_2*xy + c_3*y^2 = c_4*z^2 + c_5*zx + c_6*zy

where the c's are constants, for solutions to exist it must be true that

((c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3))v^2 + (2(c_2 - 2c_1)*(c_6 - c_5) + 4c_5*(c_2 - c_1 - c_3))v + (c_6 - c_5)^2 - 4c_4*(c_2 - c_1 - c_3) = n^2 mod p

for some n, where p is any prime coprime to z for a given solution, when

v = -(x+y)z^{-1} mod p.

I've focused in other places on solutions true for all primes, but notice you can also just pick a small prime and simply loop through all residues modulo that prime for x, y and z to see if any will work to give you a quadratic residue modulo that prime, which is accomplished by just looping v through all possible residues. One problem with that approach is that you're then assuming that z is coprime to that particular prime. There may be other ways to use the theorem. I just discovered it a few days ago so I'm still figuring things out.

I'll go ahead and give a simple example of it in use, which is for a case which is true for all primes.

For example with x^2 + y^2 = z^2, I have

c_1 = 1, c_2 = 0, c_3 = 1, c_4 = 1, c_5 = 0, and c_6 = 0

which gives

-4v^2 + 8 = n^2 mod p

for every prime coprime to z, for some n, when v = -(x+y)z^{-1} mod p.

Making the substitution for v gives

-4(-(x+y)z^{-1})^2 + 8 = n^2 mod p

so

-4(x+y)^2 + 8z^2 = n^2*z^2 mod p

and since x^2 + y^2 = z^2, I can substitute out z, to get

4(x-y)^2 = n^2*z^2 mod p

so the requirement is met, as of course, there are an infinity of integer solutions to x^2 + y^2 = z^2.

I'll also give the solution for Pell's Equation:

x^2 - Dy^2 = 1

so

c_1 = 1, c_2=0, c_3 = -D, c_4 = 1, c_5 = 0, c_6 = 0, and z=1

which gives

4Dv^2 - 4D + 4 = n^2 mod p

and v = -(x+y) mod p, so I have

4D(x+y)^2 - 4D + 4 = n^2 mod p

and since that must be true for all primes p, since z=1, I have in general that the left hand side must be a perfect square so it must be true then that

D(x+y)^2 - D + 1 = S^2

where S is some integer, and I have in general that

x+y = sqrt((S^2 + D - 1)/D).

The utility of this result was questioned in other threads though my primary interest in putting it forward was to make sure the theorem didn't just give circular results and clearly it does not.

But also that result can be used to relate certain values for D to select Pythagorean Triples as notice that if you let S = jD +/- 1, you have

x+y = sqrt(Dj^2 +/- 2j + 1)

which is

x+y = sqrt((D-1)j^2 + (j +/- 1)^2)

and I have the existence of solutions related to another Diophantine relation of the form

(D-1)u^2 + v^2 = w^2

with the condition that u = j and v = j+/-1.

For instance with D=2, I have that I need solutions to

u^2 + v^2 = w^2

with u=j, and v=j+/-1, and j=20 works as 20^2 + 21^2 = 29^2, and gives x+y = 29, and again x=17, y=12 is a known solution to x^2 - 2y^2 = 1.

So you can find solutions for D=2 just by scanning through Pythagorean Triples.

These first examples do not necessarily give a sense of the full reach of the Quadratic Diophantine Theorem as it was just discovered by me Friday, and I'm just casting about now looking for quick and easy examples.

The full theorem works to put conditions on

c_1*x^2 + c_2*xy + c_3*y^2 = c_4*z^2 + c_5*zx + c_6*zy

and it's not clear to me how much you can do with it at this time, but it's fun to wonder about it.

Deriving the theorem was trivial using what I call tautological spaces, which are mathematical regions where a particular truth holds.

The simplest presentation I know of, of a tautological space is

x+y+vz = 0(mod x+y+vz)

which is equivalent to

x+y+vz = x+y+vz

so the truth of the space is that x+y+vz is a factor of everything within it as shown.

I do basic manipulations on tautological spaces and then subtract an equation to be analyzed from that space and then analyze the residue.

I have pioneered this technique which I discovered back in December of 1999, when I was desperately looking for more analytical power in pursuit of a proof of Fermat's Last Theorem.

To me it is a natural extension of Gauss's work on congruence relations.

## JSH: Now that's funny

The posters "Rotwang" and "Angus Rodgers" were trying to cast doubt on the value of my Quadratic Diophantine Theorem by showing easy derivations that followed in hindsight from a result I found using that theorem with Pell's Equation:

x^2 - Dy^2 = 1

with D a natural number for integer solutions to exist there must exist an integer S such that

x+y = sqrt((S^2 + D - 1)/D).

The posters tried to dismiss that new result as trivial by showing it's easily proven.

But what had me laughing was the clear fear in the later postings by Rodgers when I noted the value of the result HE found, as he tried to back-track from it!!!

As I've noted, some of you despise mathematics.

His fear is clear: he realizes that a huge result could validate my research, and he's terrified of that implication.

It makes me wonder though, why would such a person even bother to do anything in the math field at all?

What does he get out of it?

Dismissal of knowledge seems too easy for him. What kind of person must he be?

I titled this post "Now that's funny" but it's actually kind of sad.

As I've feared, I think it clear that some of you despise human knowledge and would rather block growth and learning than accept a shift in your social order.

The Math Wars are class wars.

Some of you DECIDED I was beneath your class and you have fought me dearly to hold what you see as your class protection, and the human species, and its growth in knowledge be damned as you don't care about anything else but your class positions.

It is clear to me that some of you are actually anti-mathematicians, hiding out in the field, pretending to be people who give a damn about the truth and discovery, when you are in the field for something else.

[A reply to someone who noted that James had said that he had “no intentions of picking up a renewed posting frenzy” but then he posted five times more often than before.]

Situation changed.

I thought I was done with discovery. Hoped I was done.

Then I had this idea to try tautological spaces against

c_1*x^2 + c_2*xy + c_3*y^2 = c_4*z^2 + c_5*zx + c_6*zy

and next thing I knew I had this beautiful theorem and the results just came flying out!

It's overwhelming you know.

I don't feel exactly like I control this process, but am being controlled.

Which is why I know that people like you are fated to lose.

You see yourselves as fighting one man, and you think you know what man is, and what you can say or do to break me down.

But you are fighting mathematical discovery itself, and thousands of years of human history when yes, there was competition before, so the competition I'm facing from men like you, fighting for the status quo to maintain your hold on your mates and to help you breed is not new.

So you know then that every time in past history you have been defeated or we would not have computers, airplanes, or medical science.

The discoverer has always won.

Only now maybe, just possibly your competitive male brain is considering that maybe, just maybe you were never just fighting one man.

You were fighting the spirit of the human race itself—the destiny of our entire species—acting through me.

You were fighting for the annihilation of the human species.

Which is why you lost.

## JSH: Kind of like zombies

Ok I did the one post noting the obvious--that protecting your professors is taking away your own opportunities. But now I think it worth noting that you people have been AMAZING in terms of your faithfulness against your own interests.

It's like, of course if there are new research techniques over-turning older results that is an opportunity for young people.

Why do you think I keep posting to alt.math.undergrad?

Supposedly your own self-interest would have made you push for new ideas but for years now, you've been like mindless zombies. Following along when that means that for most of you, you will never even have CAREERS in mathematics.

All these older mathematicians are blocking your places as they have them.

Oh, and yeah, mathematics is a fascinating field. The person with the more powerful techniques can get the huge results. Duh.

I have the most powerful mathematical research techniques. I can find the huge results.

So those of you mindlessly trotting behind your professors like good little math students not only are against your own self-interest, you're running out of time, as THEY had their time, but for you there is the near certainty that as time goes by, I'm likely to take out entire areas of research possibilities.

So you could work hard. Follow the system. And wake up one morning to hear that I've nailed yet another huge result, like researchers who do anything with Pell's Equation could be hearing today, if they bother caring.

And think about it. Those of you who know about the new research might be seeing in amazement your society acting as if it doesn't exist!

Texts continuing to trot out the old info, and people working hard, putting in their mental sweat and effort. Trying as hard as they can do figure out more about…about the trivially solved.

The social reality I face IS formidable.

For years now I've seen how willfully many of you will ignore major mathematical techniques to trot along behind your professors and be good little math students, so yes, I understand you may continue.

Just for social crap, to feel good, to think that, hey, those are the rules and you are following them, so does it really matter if your mathematical knowledge is actually primitive? If history will overturn any validation you receive from today's society?

I think for most of you, it does not, as that is some vague hypothetical.

But I hope some of you DO care about correctness. And DO care about mathematics as discovery and truth versus just agreeing with some professors.

But for years now, most of you have been, well, more like zombies than math students.

I have proven Fermat's Last Theorem. And in so doing found a math error in "core" that is over one hundred years old. I have found THE prime counting function. I have results across prime number. I am now coming back to Diophantine equations with immediate success.

You stand against me and you stand against mathematical history for people who will lose in the judgment of history.

And some will go down very badly, more than happy to take down as many of you with them as they can.

The puzzle for me for so many years has been: why do so many of you hate yourselves?

Why do so many of you despise your own futures?

A question I have often asked myself is, why don't any of you seem to have a sense of self-preservation?

I ponder that question now.

It is with great joy that I note there is a huge result on the generalization from Pell's Equation that posters figured out quickly in hindsight from my initial post giving

x^2 - Dy^2 = 1

with a natural number D, which requires that there exist some integer S such that

x+y = sqrt((S^2 + D - 1)/D)

and also I guess I may as well add that

x^2 + Dy^2 = 1

requires that

x+y = sqrt((D - S^2 + 1)/D)

immediately explaining a finite number of solution.

One poster should have a place in mathematical history for being the first to give the general equation for

Ax^2 - Dy^2 = c

with A and D natural numbers and c a non-zero integer and that could be you.

I quote his proof:

Define S = Ax + Dy. Then:
S^2 + (D - A)c
= A^2x^2 + 2ADxy + D^2y^2 + Dc - Ac
= A(Ax^2 - c) + 2ADxy + D(Dy^2 + c)
= A(Dy^2) + 2ADxy + D(Ax^2)
therefore
x + y = sqrt((S^2 + (D - A)c)/(AD))

He has his place in mathematical history now.

That could have been yours.

Some of you seem to think that protecting your math professors is protecting your own careers.

You are very wrong.

And blocking your place in history.

Search in Google on my Quadratic Diophantine Theorem, learn the proof, and the techniques I used.

Study my mathematical approach and you can re-write the textbooks across number theory.

You can be the great mathematicians of the future.

Or you can try to ignore my growing body of research results so you can muddle along for years to try to maybe make it as post-grads if you can even manage to work in the mathematical field, to wait for decades for the older mathematicians to finally decide to retire, grubbing for research dollars.

The choice is yours.

Make mathematical history.

Or labor in the shadows for the rest of your life for older men who had their time.

Now can be yours: History. Greatness.

Discovery.
I'm seeing indications that the value of the result isn't jumping out

The result is that given

Ax^2 - Dy^2 = c, there must exist integer S such that

x + y = sqrt((S^2 + (D - A)c)/(AD))

so now, you can explain existence for

Ax^2 - Dy^2 = c

where A and D are natural numbers and c is a non-zero integer, as you have immediately that

S^2 = Ac mod D and S^2 = -Dc mod A

so, for instance, Ac = 6 with D=7 cannot give integer solutions.

i.e. 3x^2 - 7y^2 = 2 has no integer solutions.

Further, when solutions are available, you can search for S modulo D and A.

It is a remarkably powerful little result for its utility and ease of proof, where Mr. Rodgers it appears used hindsight with a result I originally gave about Pell's Equation which is that given

x^2 - Dy^2 = 1

there must exist an integer S such that

x+y = sqrt((S^2 + D - 1)/D)

which I found using my Quadratic Diophantine Theorem.

So far though he seems uninterested in either credit or appreciation for the result, which I guess is kind of sad.

It shows where some in the current mathematical community are today.

My guess is he is worried that he has inadvertently supported my research when he was trying to give evidence dismissing it by showing how easy the derivation is.

So you can see the political behavior I decry with a man trying to block his OWN place in mathematical history.

I suggest that mathematicians allow him to do so, as the result follow easily enough from my Quadratic Diophantine Theorem anyway, and maybe there then is a lesson here, and a question:

What could drive a man to try and forgo a place in mathematical history out of fear that he might support my research?

## JSH: Now we'll see

What I like about the Diophantine Equation Theorem, which I just worked out today, is that now we can all see how long it takes for the mathematical community to recognize the result.

There are already posters insulting me in threads, how long will that continue?

Will there be yet another battle from the math community over this result?

Posters casting doubt on the theorem?

A refusal by any established mathematicians to recognize it? Possibly claiming it's not of interest?

Time will tell. So now we wait.
So far, more of what I've seen before. I did get excited too quickly about the Pell's Equation result, but it is also clear that certain posters are doing their best to trivialize everything.

At least I managed to find a result relating certain Pythagorean triples to cases of Pell's Equation where D-1 is a square, which is I'd think a somewhat interesting result.

Still the reality remains that I was checking the theorem, so the weekend was a huge success as the best news for me is that the Quadratic Diophantine Theorem doesn't just go in a BFC—Big Freaking Circle—as it clearly gives answers.

That posters have shied away from the reality of the full theorem to focus on specific issues with my assertions is consistent with what I've seen before.

Time will tell what the math community does next.

So, short answer is, more of the same from certain posters. It's still a wait and see on what the rest of the mathematical community does.

But without a doubt, I have yet another tool.

## JSH: Nifty little result on quadratic diophantines

Oh hey, after I came up with the Quadratic Diophantine Theorem, I started looking over research on quadratic diophantine equations and it's kind of interesting because hey, looks like my research can help!

Like it will give criteria on Pell's equation, and may even offer a route to generally solving a 2 variable diophantine quadratic as you can just let x=1 to go from the 3 variable expression in the primary theorem.

Such a powerful little result it looks like after a little research, and I was wondering if it would be a big deal as I debated about putting it on my math blog. Oh yeah, proof of the theorem is on my math blog. Turns out it's easy to prove with tautological spaces.

Cool. Ok, going to read up more on quadratic diophantine equation stuff that was already done before my research.

In the ring of integers, given the quadratic expression

c_1*x^2 + c_2*xy + c_3*y^2 = c_4*z^2 + c_5*zx + c_6*zy

where the c's are constants, for solutions to exist it must be true that

((c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3))v^2 + (2(c_2 - 2c_1)*(c_6 - c_5) + 4c_5*(c_2 - c_1 - c_3))v + (c_6 - c_5)^2 - 4c_4*(c_2 - c_1 - c_3) = n^2 mod p

for some n, where p is any prime coprime to z for a given solution, when

v = -(x+y)z^{-1} mod p.

For example with x^2 + y^2 = z^2, I have

c_1 = 1, c_2 = 0, c_3 = 1, c_4 = 1, c_5 = 0, and c_6 = 0

which gives

-4v^2 + 8 = n^2 mod p

for every prime coprime to z, for some n (remember ring is ring of integers) when v = -(x+y)z^{-1} mod p.

Making the substitution for v gives

-4(-(x+y)z^{-1})^2 + 8 = n^2 mod p

so

-4(x+y)^2 + 8z^2 = n^2*z^2 mod p

and since x^2 + y^2 = z^2, I can substitute out z, to get

4(x-y)^2 = n^2*z^2 mod p

so the requirement is met, as of course, there are an infinity of integer solutions to x^2 + y^2 = z^2.

And a square was required here because p can be any prime coprime to a solution for z, so an infinite number of primes must work!

Notice that the result also applies to the general diophantine quadratic in 2 variables by making c_1 = 0 and x=1.

The theorem is proven easily using what I call tautological spaces.

## JSH: Explaining the huge math error

For years now I've tried to raise the alarm about a huge problem in an esoteric branch of number theory where there may be now a huge case of academic fraud, where the problem for me is that the mathematical community itself is so impacted by the size of the error that I haven't found a way to get mathematical proof of it accepted.

Luckily it's easy to explain and crucially depends on the distributive property so my hope is that if I can convince mathematically experienced members of the physics community of the existence of the error they can help with the daunting task of handling the non-response from the math community and the issue of possible widespread academic fraud.

It's simple to explain the error.

Trivially if I have a polynomial factorization like x^2 + 3x + 2 = (x+1)*(x+2), I can multiply both sides by 7, like

7*(x^2 + 3x + 2) = (7x+7)*(x+2)

and divide that 7 off, just as easily, without a trace, as I could multiply by anything.

But over six years ago I came up with a technique where you have a polynomial multiplied by a constant but factored into non-linear functions:

P(x) = 175x^2 - 15x + 2

and

7*P(x) = (5a_1(x) + 7)*(5a_2(x)+ 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0,

So now you have something more complicated, where I've enmeshed the quadratic P(x) with a quadratic generator, and it turns out that now in an areas where mathematicians routinely operate to try and prove things, you cannot divide the 7 off, when with an integer x:

a^2 - (7x-1)a + (49x^2 - 14x) = 0

has non-rational roots.

For instance, with x=1, you have the a's are the roots of

a^2 - 6a + 35 = 0

and if you try to use the quadratic formula you have a = (6 +/- sqrt(-104))/2, and if 7 divides just one of those there is a problem in SEEING it because sqrt(-104) is imaginary, and in fact you can rigorously prove that in something mathematicians call the ring of algebraic integers it is IMPOSSIBLE for either of the roots to have 7 as a factor.

So I can prove that one of the roots has 7 as a factor by the distributive property AND prove that it cannot have 7 as a factor in the ring of algebraic integers. So there is an apparent contradiction.

But the ring of algebraic integers is the ring mathematicians have used for over a hundred years for arguments they think are proofs in number theory and it just betrayed a huge problem because think back to

7*(x^2 + 3x + 2) = (7x+7)*(x+2).

I can use functions f_1(x) = 7x and f_2(x) = x - 5, and have

7*(x^2 + 3x + 2) = (f_1(x) + 7)*(f_2(x) + 7)

and that doesn't change how things work so what's changed with

7*P(x) = (5a_1(x) + 7)*(5a_2(x)+ 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0?

The TYPE of function has changed. I've gone from linear function to non-linear ones that are the roots of a quadratic generator.

But if

a*(f(x) + b) = a*f(x) + a*b

by the distributive property, how can the TYPE of function change the behavior?

a*f(x)

does not say, yeah, only if f(x) is linear, now does it?

Sure, maybe if one of the functions has 1/7 in it, the value can change but the distributive property remains the same. But the ring of algebraic integers cannot have 1/7 in it. It does not.

So something weirder is happening.

Turns out that no matter how trivial all this sounds the problem is huge enough to invalidate mathematical arguments thought to be proofs over the entire 20th century until now, as the ring of algebraic integers became widely used by mathematicians in the late 1800's.

I have gone to the mathematical community with this problem, and even got a paper bringing attention to the problem published in the now dead math journal Southwest Journal of Pure and Applied Mathematics, or SWJPAM for short.

Members of the sci.math newsgroup mounted an email campaign against the paper.

The journal pulled it after publication.

It managed one more edition and then quietly shut down.

See:

http://www.emis.de/journals/SWJPAM/

and

http://www.emis.de/journals/SWJPAM/vol2-03.html

If the simple mathematical argument I gave you above is correct, then it implies that ALL number theorists today who use the ring of algebraic integers may have flawed results which are not correct because it has a completeness problem, so the very people who are tasked with accepting this error can be completely invalidated by it.

It may remove some of the arguments considered great works over a period of a hundred years, and make useless a tremendous amount of mathematical knowledge which can be shown to lack value by it.

At this point in time, rather than acknowledge the error, mathematicians are continuing to TEACH the flawed mathematical ideas, and have shut all doors—like publication.

The Catch-22 here is that the professors who would be tasked with acknowledging this error are the very people who would find tremendous invalidation from it, so much so in fact, that many of them may have no real mathematical accomplishments at all: from their doctoral theses to their latest research.

Admitting the truth would remove the very basis for their positions.

So then, what is the solution here? How do you handle an error of this magnitude?