Wednesday, April 29, 2009


JSH: Referral to NY Times "End the University as We Know It"

The New York Times has an excellent editorial. Quoting from the Source:

End the University as We Know It
Published: April 27, 2009
If higher education is to thrive, colleges and universities, like Wall Street and Detroit, must be rigorously regulated and completely restructured.

Not surprisingly one of my favorite pieces of the excellent article discusses tenure:
6. Impose mandatory retirement and abolish tenure. Initially intended to protect academic freedom, tenure has resulted in institutions with little turnover and professors impervious to change…
Your academic world WILL change. It's not about if, and it's not about an ability by any of you by sheer force of denial to stop it.

As the years have gone by I have challenged many of you to re-think how you look at reality itself, but education eventually leads to a need to act in reality, or, in other words, school is almost out.

For those who refuse to change the same situation exists as it was for those who hated automobiles. Who loved their horses and saw no reason for the horse and buggy to be replaced.

Before them, there were people who liked the Ptolemaic system. It had its problems but worked well enough to plan crops. How can you argue with such a thing?

Change runs into the reality of recalcitrance to change but on a planet with 4.6 billion years of history, the length of time of your resistance doesn't even amount to an eye blink.

For some time I have considered that some of you deep down know you are wrong but hope to simply get your careers done before change throws out everything you've done, but our times are faster than in the past.

Some of you who are undergrads my yet not reach graduation before the changes are set. Those of you who are grads may find that you can't get through your doctoral program in time. And those of you tenured professors may find your retirement plans are torn asunder before you ever get to them.

Our world demands progress.

You will change because at the end of the day, even if you resist it, reality is bigger than us all. We will all change.

Oh yeah, and as I've said, tenure will be abolished and the academic world worldwide WILL be revolutionized.

As I go mainstream, you will be dragged along—whether you wish it, or not.

Saturday, April 11, 2009


JSH: A hundred years plus of error

Maybe at least some of the mathematicians fail to do the right thing in the face of proof because of the enormity of their loss.

A subtle but devastating error entered into number theory over a hundred years ago.

For a field like mathematics it is unbelievably devastating in terms of its impact, quite simply, reversing the story for the last one hundred years.

Where they thought there was great brilliance in many cases there was complex failure.

I fear, some of them just simply kind of cut-off in the face of the reality, and then they pick themselves up, and go back to using the broken ideas, and teaching them to new students.

I had a Cornell math grad who contacted me by email offering to help, start replying back with stories about wondering around late at night studying the sky as he worked through the mathematics himself, which was his own suggestion to do!

After a few months he replied again claiming the mathematics was out of his area.

But, of course, there may be implications for physics, but luckily, physics has to work in the real world.

The primary impact will probably be explanation, along with gaining some incredibly powerful mathematical tools.

Make no mistake, what I call tautological spaces allow you to encompass all prior knowledge of a set of equations—and extend beyond it.

Which is how I quickly and easily generally solved binary quadratic Diophantine equations—finding these Pell Equation results which have opened a window to getting you to face the real situation—and years ago, found a proof of Fermat's Last Theorem, and found the devastating error itself.

You can use Google—a harbinger of the new—to get some grasp of how my influence is being presaged.

Google: definition of mathematical proof

I think I also may come up highly in Yahoo!, and consider who gets to define mathematical proof for the world?

I am the latest major mathematical discoverer. That's why I get that right.

Google: solving binary quadratic Diophantine equations

Google: tautological spaces

Google: spherical packing problem

The mathematical community in number theory has a hundred plus years of error hidden by "pure math", and there is a physics impact, as the proper explanation for group theory will probably be interesting. At a minimum prime numbers appear to be key in fascinating ways, and they offer bigger clues about random in the real world.

Oh yeah, and my prime counting function appears to be a number theory version of a damped oscillator, which may underpin fundamental reality.

Quite simply, you either get with the more powerful knowledge, or wait until humanity completes moving beyond you, and new students will arrive, and new methodologies will build.

Science will adapt.

The world will not wait for you to make the right decision.

The Google searches by themselves show that yes, you can be superseded by more powerful technologies, better thinking, and the reality that Progress is a force far beyond any of you to permanently stop.


JSH: Why it's white collar welfare

Merit is supposed to matter.

I can mathematically prove but if the people who are in charge of recognizing proofs decide not to do so, what then?

The reason I call the system white collar welfare is that these people are paid to do nothing, but it's worse as they have mistakes easily shown and explained, but they continue to teach the mistakes and fight against knowledge of the errors, so we pay people to teach their students false information.

So it's worse than welfare.

I'm hammering this subject about Pell's Equation because I think those of you taught like I was, would believe that academics would accept a clear demonstration of a better technique and simply acknowledge it.

But mathematicians may feel they can't give any ground with me because they have a glass ceiling in place.

It is an absolute. If they give ground they shatter the glass ceiling.

It's not clear how long this situation can last though.

I find it hard to believe that physicists will simply accept not doing the best methods, or give in to requests to quietly shift to better techniques without acknowledging the source—if that occurs.

The glass ceiling is in place in mathematical society today.

It is against merit.

I've found a consequence for physicists with one particular equation that is easily demonstrated, but there are bigger issues.

The point is to open the door to skepticism. And force you to look at your mathematical colleagues with the eyes of physicists, and not just trust.

Given x^2 - Dy^2 = 1, for instance, if you have a solution to the negative Pell's Equation,

j^2 - Dk^2 = -1

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

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

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

That's just information.

I have other information, including a paper that was published in a formally peer reviewed mathematical journal and then yanked after publication, and later the journal quietly died:

EMIS, a European agency, is keeping alive the archives of an American journal tossed aside like so much trash by its American owners.

10 years of papers in that journal from mathematicians who trusted their society I guess.

Denial here does not help physics any more than denial in the past did.

The truth will win, so it's about time. But each day you give the class behavior reward, is another day that people who betrayed the trust of society are paid to do wrong. It's a reward for academics who betray the public trust.

Each day is a validation of their system, and evidence that working against merit can still work, when intelligent people allow it to happen.

You saw that with the financial crisis that has gripped our world.

Now you have an intellectual crisis, where inaction is the worst thing you can do.

Inaction allows the white collar welfare system to continue, gleefully.

Each day is a win for them. Another day to teach students wrong information. Another day to spend more funds given for wrong research. Another day to teach you bad math techniques, and refuse to update to better ones.

Another day for you to pretend that the world will just be ok.

I'm sure you hate such behavior when you read about it in others. Now it's you.

What world are you trying to give to your children?

One where they can learn bogus information masquerading as best knowledge?

Math society is teaching you wrong NOW. What makes you think they will stop with anyone else?

I've seen no evidence they will. That math journal died over 5 years ago.

Any of your kids majoring in mathematics?

If so, make no mistake, some of what they're learning was proven false over 5 years ago.

That dead journal is just another sign.

Occam's Razor: What's the simplest explanation for academics who refuse to update when given even the most basic of explanation of a much simpler, better approach?

Ans. Fraud.


JSH: Physics relevance of Pell's Equation?

I'm curious about how much this bit of number theory comes up in physics.

Mostly I see things on zeroes of Racah coefficients. Here's one:

W. A. Beyer1, J. D. Louck1 and P. R. Stein1

(1) Los Alamos National Laboratory, 87545 Los Alamos, NM, U.S.A.
Received: 11 February 1986 Revised: 22 August 1986

Abstract The interface between Racah coefficients and mathematics is reviewed and several unsolved problems pointed out. The specific goal of this investigation is to determine zeros of these coefficients. The general polynomial is given whose set of zeros contains all nontrivial zeros of Racah (6j) coefficients [this polynomial is also given for the Wigner-Clebsch-Gordan (3j) coefficients]. Zeros of weight 1 3j- and 6j-coefficients are known to be related to the solutions of classic Diophantine equations. Here it is shown how solutions of the quadratic Diophantine equation known as Pell's equation are related to weight 2 zeros of 3j- and 6j-coefficients. This relation involves transformations of quadratic forms over the integers, the orbit classification of zeros of Pell's equation, and an algorithm for determining numerically the fundamental solutions of Pell's equation. The symbol manipulation program MACSYMA was used extensively to effect various factorings and transformations and to give a proof.


I'm curious about how big of a deal it is to know an easier way to solve them, as I've noted the following:

Given x^2 - Dy^2 = 1, for instance, if you have a solution to the negative Pell's Equation,

j^2 - Dk^2 = -1

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

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

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

It's rather trivial number theory to show that all positive integer values for D can be handled by alternates like the negative Pell's Equation, where you get the square improvement with all.

There is a bit of a puzzle I think as to why the better way to approach Pell's Equation is not commonly taught.

Unless someone comes up with a reference that mentions it, so far what I've seen on-line is just a direct tackling of Pell's Equation itself using various methodologies, which then, is mathematically naive (sorry but that is I think the best description).

I have been pushing this issue for a few days, but now I'm more curious about possible explanations different from what I have.

So then, what is the physics usefulness of Pell's Equation? Is the zeroes of the Racah coefficient it?

Do techniques for solving it already somehow encompass the simple result I've found?

If not, what then? How can the literature be updated?

IF so, where is that shown?

Thursday, April 09, 2009


JSH: Understanding pain and why they destroyed the journal

So people insult me on newsgroups to make it painful for me to continue in the hopes that I will stop so mathematicians can keep being fakes and doing dumb math that is in many areas, painfully stupid.

That's why you insult people.

I like my analogy of Tiger Woods going after amateur golfers to make the point.

Top people don't do that. Top mathematical knowledge does not require you to insult people into silence.

Notice how much fun I'm having now. THAT is mathematics. When you're right, and get something simple enough that math people around the world have more trouble lying about it—as they lie ALL the time—you can get somewhere.

And yes, they killed that poor math journal and now you know why. Google: SWJPAM

I found a massive "core" error. Proving it requires mostly simple algebra but it completely overturns the modern math world in number theory.

Upsets a lot of applecarts as they say.

Their math is either wrong, or primitive in the area of the error.

The continued success of modern mathematics is primarily with research found around the time of Gauss and before.

After that begins the junk period. Over a hundred years of it.

To handle the failure of the research, math people started calling it "pure math" and claimed someday it might be useful.

But it will never be useful, because it's wrong. And THAT is what they're protecting.

THAT is why there is the insult war, and THAT is why they killed that math journal.


JSH: Group theory undone

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

j^2 - Dk^2 = -1

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

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

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

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

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

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

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

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


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

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

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

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

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

You are primitives. I am the advanced one.

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

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

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

Wow. Nuts, right?

How can it not be right and work so well?

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

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

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

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

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

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

x^2 + Dy^2 = F

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

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

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

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

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

Pell's equation without irrational numbers
Authors: N. J. Wildberger
(Submitted on 16 Jun 2008)

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

The implications are HUGE.

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

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

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

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

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

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

But I have the best mathematics.

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

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

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

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

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

Time you will not get back.

Tuesday, April 07, 2009


JSH: And they will wait to see

I have little doubt that the mathematical community not only will not acknowledge my finds of really just trivial stuff with Pell's Equation that makes it a lot easier to solve, but, I believe, they will keep teaching it as they have.

So yeah, you could take a course from a "top" mathematician at your pick of the supposedly best schools in the world, and have some guy, confidently and arrogantly teach you Pell's Equation without mentioning trivial solutions for D = n^2 - 2, and not telling you that it's best to solve alternates to Pell's Equation that have solutions that are roughly the square root of the Pell's Equation solution.

And God forbid if this paragon of mathematical society would ever sink to the low of daring to talk about RATIONALS with Pell's Equation, a Diophantine equation mind you so rationals are verboten, right?, and give you the rational parameterization for circles, ellipses and hyperbolas:

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

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

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

A poster on a math newsgroup after railing at me with lots of insults for daring to note this area as a massive failing of math society then said that result wasn't worth mentioning, or noting.

They don't wish to record it. They as in math society, which is still quiet.

In THEIR society they will keep getting math prizes including the big prizes. They can still get acclaim, and accolades, advance their careers and have wonderful mathematician lives with junk results as they do not have Mother Nature looking over their shoulders forcing them to be right.

But you do.

Mathematicians will happily not teach you common knowledge now because I'm making it such, because they are waiting to see if you will let them. And they will keep getting funding, keep getting awards, keep getting prestige no matter how wrong they are as they have a committee society.

A group of mathematicians says "great!", and you get a prize.

But YOU need experiments that shows theories work. YOU need mathematics that is the best knowledge available.

Because YOU have real competition, and need real results, not a committee to hand you a prize or grants or whatever.

I shot down standard usage of Galois Theory years ago with fairly trivial algebra.

Mathematicians have still gotten prizes using it since I did.

They destroyed an entire mathematical journal versus face the result. Google: SWJPAM

Mathematics professors have still advanced their careers in mathematics. Grad students in mathematics have received their Ph.D's for bogus work.

Their society has gone on just fine as it is completely disconnected from reality.

So they will wait to see what you do.

Maybe the end of this impasse will be the day when one of you is sitting in some prestigious course from some distinguished mathematician teaching you Pell's Equation without any of the easy knowledge that I've pointed out and you will flip out, and run from the place screaming in agony, and when you regain some of your sanity, you will finally say: THIS MUST STOP.

Until that day. We all wait. With them.


JSH: I feel sorry for them

I don't care about math society as I've been arguing with you people and sending papers to journals and emails to mathematicians, like Lenstra, to whom I sent these latest results, long enough to know you don't give a damn about anything but yourselves, and maybe your academic careers, if you even have that, which a lot of you on these newsgroups do NOT have but you're just nasty people in general.

So I don't feel sorry for you.

My take on it is you hate math anyway, but some of you found a niche where you could learn just enough to fool enough people that you could pretend to like it.

But those physicists working on things like zeroes of Racah coefficients using Pell's Equation, or trying to find uses out of Galois Theory, are just poor pathetic souls who believed in you, and wasted their lives as a result.

They can't get that time back either.

So yeah, show your true colors yet again as you make fun of me. Insult like always while we wait for world governments to wake up to the reality, and ponder how much junk they have as "math", from people who have a contempt for knowledge.

And then they can take your money away.

But no one can give back to those physicists, or others who trusted you, the time you took from them.

And no one can give them back the hopes and dreams they lost as a result.

Physics isn't like mathematics. They can't get prizes for crap, unlike you in the math field. There will be no Nobel prizes for those people. No big win with a gain of knowledge for humanity.

Just an empty and lost feeling when the enormity of their loss hits them.

They were victims, but who to blame?


JSH: sqrt(n^2 - 2) and Pell's Equation

One of the weirder blind spots with Pell's Equation is that if D = n^2 - 2, then the first solution for x is given by D+1, so you can trivially solve x^2 - Dy^2 = 1 for all such cases:

(n^2 - 1)^2 - (n^2 - 2)n^2 = 1

and notice n^4 - 2n^2 + 1 - n^4 + 2n^2 = 1.

Since solutions to Pell's Equation are good approximations for square roots, sqrt(n^2 - 2) is approximately (n^2 - 1)/(n^2 - 2) which makes sense so it's an easy result.

e.g. D=7, then x=8, and y=3, as 8^2 - 7*3^2 = 1, and sqrt(7)*(3/8) approximately equals 0.992 to three significant digits.

One more example.

D = 119, then x = 120, and y = 11, as 120^2 - 119*11^2 = 1, and sqrt(119)*(11/120) approx. equals 0.99996 to five significant digits.

To me that is the weirdest blind spot with Pell's Equation where my guess is that Fermat and Euler were quite aware of it, but probably considered it too trivial to mention.

The other weird blind spots though are with the alternates to Pell's Equation which I've also discussed.

Turns out that in general you should NEVER try to directly solve Pell's Equation except for trivial case above, as x tends to be the square of the solution I call j with the alternates, so it's not like you're doing twice the effort or something small, you're doing a squared times!!!

Sunday, April 05, 2009


JSH: Considering my binary quadratic Diophantine solution

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

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

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

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

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

My take on what mathematicians do is that they specialize.

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

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

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


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

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

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

That D number though may hold secrets to our reality.

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

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

Mathematics is a tool.

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

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

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

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

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

You people don't deserve the best knowledge.

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

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

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

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

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

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


JSH: Math society realities

This wacky story about the one stop equation for 3 of the 4 conic sections in rationals should be more than enough to convince those of you who haven't completely gone over to the Dark Side that math people follow their own wacky rules, where a lot of what they do is anti-thesis to physicists, but it can be kind of appealing to lazy people.

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

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


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

showing it more traditionally versus the way that results from my own re-derivation. (Some of you may remember I screwed up my own derivation and for a while thought I'd solved the factoring problem partly as a result!)

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

But to math people x^2 - Dy^2 = 1 is just "Pell's Equation" so who cares if you can rationally solve ellipses, circles and hyperbolas with it, and categorize 3 of the 4 conic section with one number? The D number?

To THEM it's just a Diophantine equation inappropriately attributed to this Pell guy by Euler, and that is that.

But even more wackily, even as a Diophantine equation they missed reporting on very obvious stuff, like that if D = n^2 - 2, where n is a natural number greater than 1, that is positive integers greater than 1, then D+1 is the first solution for x in Pell's Equation:

Which also means—for various reasons I won't go into for this post—that n^2 - 2 tends to be prime, or the product of few prime numbers.

I mean, that's just cool! And it even has prime numbers!!! It's sexy even by math society "pure math" standards.

In my own battles with that community, where I discovered a HUGE error at the base of what they call number theory which blows Galois Theory out of the water as a useful methodology, I've seen a worship of past mathematicians which the rest of the world refuses to accept.

These people will not accept corrections to their own story about their history.

Mathematics for them is a religion.

Modern mathematicians today are a priesthood.

I have no doubt that they will happily ignore everything I've discovered or re-discovered about "Pell's Equation", and that they could teach as they have for thousands of years without feeling at all that they are missing anything—just like people in any religion.

But physicists give in to this behavior at their peril.

Who knows how many theoretical physicists have blown their futures or any chance of decent research by working with Galois Theory.

And the rational solutions for ellipses and hyperbolas could at a minimum allow, say, a categorization of orbits by a SINGLE NUMBER, the D number, that might yield crucial clues about things like the stability of our own solar system.

Math people, I fear, and I know some of you may think I'm being too hard but hey I found a deep ERROR IN THEIR FIELD AND THEY LIED ABOUT IT TO LIVE IN ERROR—do not care.

They live in error like people in any religion because they do not see it as error.

Did Jesus walk on water? Come on. The laws of physics say, no.

But if you're a Christian, do you go with what you know MUST be true by reason, or do you go with your BELIEFS, by faith?

Mathematicians live by faith, not reason. Or find another explanation for their errors and omissions with "Pell's Equation" as I assure you these postings will have little impact on that wack group of religious freaks.

And tomorrow they will still be with you, asking your to give them full prestige as researchers and academics, despite their ability to have fascinating amnesia on certain issues.

They live in their own little world.

I don't. But they say I do because I'm out here "preaching heresy" to them.

I live in the real world where many of you waste your lives, your intellectual energy, and your careers because the task in front of you seems too hard—facing mathematicians as they are.

Real physics though is your victim.

Saturday, April 04, 2009


Conic sections and D number

Remarkably then, a parametric rational solution for circles, ellipses and hyperbolas has been known since antiquity as according to at least on web source the following result was known to Fermat:

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

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


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

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

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

So for D>0, you get hyperbolas. For D<0, you get ellipses, including the circle at the D=-1 value.

Here's a link for those wishing to see an established source give the circle one:
See: eqns. 16 & 17

So you can categorize those conic sections entirely with one number: the D number.

I'd guess it's related to the eccentricity.

For practical purposes I'd think that these rational solutions could lead to some rather fast computations, like for astronomical orbits, as well as maybe show some physics from the D numbers that result in nature.

The weird thing is the math people had these equations for hundreds of years, but they look at x^2 - Dy^2 = 1 as a Diophantine equation—integers only.

Fermat himself according to the source I read dismissed the parameterization as of no interest to him as he could generate solutions so easily with it!!! He was looking for integer solutions only. But hey, he WAS a lawyer who just did math for fun!!!

Isn't history great?


JSH: Weird blind spots with Pell's Equation

Looking over the literature over Pell's Equation available on the web, it appears that there are weird blind spots in what is commonly reported related to results having to do with:

(x-1)(x+1) = Dy^2

leading to some simple relations for x, which give you what I call the alternates to Pell's Equation:

j^2 - Dk^2 = -1, j^2 - Dk^2 = -2, and j^2 - Dk^2 = 2

The alternates are easier to solve in general because j is roughly sqrt(x), and gives you x, where the first j that works gives you the first x, so there is NO good reason to work on Pell's Equation directly in general. But things are weirder than that!!!

There are some easy solutions for special cases of D. For instance, if D = n^2 - 2, then the first x = D+1.

So there's no point in records with Pell's Equation as you can just take any integer n, square it, subtract 2, and for that D, you have the first x from D+1.

But also you have a result about n^2 - 2, as it will tend to be prime as it must be a prime or the product of primes which match by quadratic residues for -1, -2, or 2.

That is, all prime factors of n^2 - 2, must ALL have either -1, -2, or 2 as a quadratic residue!

My guess on what may have occurred is that with the continuing fractions solution in hand, math people don't just play with the full equation, and are taught to just solve Pell's Equation using continuing fractions or something.

Euler and Fermat may not have cared to talk as much about the alternates for their own weird reasons.

Remember they were involved in challenges and stuff. It was an intellectual activity for them which was somewhat about passing the time which could be a big deal for members of their class.

And Fermat was a lawyer. For him math was just that thing he did out of boredom (or he was driven for reasons he didn't fully understand).

Like can you imagine? He was shown the rational parameterization of Pell's Equation but couldn't be bothered with it. Yet conics are a HUGE area, and rational parameterizations have practical application.

The oddities of some ego games by some great men long dead may have skewed history in a fascinating way.

Friday, April 03, 2009


JSH: Symbolism is telling

That my research now encompasses a classical result on the circle, by the emergence of its rational parametric solution from my rational parametric solution to Pell's Equation is I think a symbolic passing of the torch to the next generation and type of mathematician in a way that connects back to the rich, history of the mathematical field in a wonderful way.

That a new type of mathematician was needed is probably the result of new technologies, like Usenet, and the ability for so many people to talk out mathematics, with the possibility of so much more of a muddled mess as a result.

I have to wonder how my predecessors would have handled Usenet and the Internet.

Can you imagine Gauss arguing things out on newsgroups, or Sir Isaac? I can't.

The new age has required a new type.

Some of you may finally begin to puzzle through the mathematics that shows the flaw that corrupted number theory which has rendered useless results over more than a hundred years, during a period when many cried rigor.

Rigor is not enough.

Modern problem solving techniques that have been ignored by the modern mathematical community are the future. Brainstorming and others tactics part of what I call extreme mathematics are what the best will be doing. The rest can muddle along, possibly mostly as mathematical scholars and teachers.

I have no doubt that many of you will be very slow in picking up on the changes that must take place now, and this post will probably be met with yet another rain of insults, but that is important in and of itself.

As a new era of super math evolves, and as mathematics grows up as a field, insults will mean nothing to the best, and I'm sure the rain of them will not stop.

After all, the ones of you with the ability to be the next generation of mathematician, will move so fast, and reach such heights that the others, will have no chance to keep up.

Insults may be the best they can manage.


Parametric Pell's Equation and circle

Intriguingly, in rationals, my parametric solution for Pell's Equation, can be used to give the familiar parametric equation result for the circle:

With x^2 - Dy^2 = 1

I have proven:

y = 2[f_2*v - 1]/[D - (f_2*v - 1)^2]


x = [D + (f_2*v - 1)^2]/[D - (f_2*v - 1)^2]

where f_2 is an integer factor of D-1, so with D=-1, you have x^2 + y^2 = 1, and I'll give the circle result in its familiar form:

x = (1 - t^2)/(1 + t^2)

y = 2t/(1 + t^2)

See: eqns. 16 & 17

Cool! Isn't mathematical research wonderful?

The parametric form for Pell's Equation was clearly previously unknown or a lot of textbooks are hiding a beautiful result.

Also I'll remind that I've proven that Pell's Equation itself is more easily solved using alternates to it, like the negative Pell's Equation:

j^2 - Dk^2 = -1, where x = 2Dk^2 - 1 = 2j^2 + 1


j^2 - Dk^2 = -2, where x = Dk^2 - 1 = j^2 + 1


j^2 - Dk^2 = 2, where x = Dk^2 + 1 = j^2 - 1

and, of course, x^2 - Dy^2 = 1.

One of the alternates is always true if D is a prime, where the negative Pell's Equation is the dominant one, and is true if D is a prime that is 1 mod 4 or is the product of primes that all are 1 mod 4, while the other alternates are true for cases when D is -1 mod 4, or is the product of primes where all either have 2 as a quadratic residue or have -2 as a quadratic residue, else there is yet another alternate equation which is just not as pretty, so I'm not giving it here.

Because the first x that solves Pell's Equation is roughly j^2, it is mathematically naive to solve Pell's Equation directly, rather than use the alternates which are also solvable by the same techniques available for Pell's Equation, for instance, a continued fraction solution must exist.

It is amazing that a parametric equation for Pell's Equation which gives the circle parametric equation was previously unknown, but I think it is a beautiful "pure" result which maybe just didn't want to be born into this world until the 21st century.

Wednesday, April 01, 2009


JSH: Continuing mystery?

So the result I've been trumpeting showing alternates to Pell's Equation are easier is something that should be well-known.

Turns out you don't need any general solution to binary quadratic Diophantine equations.

You can figure it out just from using x^2 - 1 = Dy^2, by noting how D must divide through x+1 and x-1.

The sad story here is I think the answer is about random.

Mathematicians avoid random with prime numbers because random can answer questions supposedly of research interest, like the Twin Primes conjecture and Goldbach's conjecture. (First: True. Second: False.)

THAT to me is the best explanation.

The behavior of primes with Pell's Equation is random, and there is no way to hide that fact with the usual obfuscations used with prime numbers, so the equations I've been so excited about, are simply ignored.

Their role left undiscussed, as once you pull the thread…random with primes is what you get.

Your society is choked by money issues. Funding is more important than answers.

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