Wednesday, January 28, 2009


JSH: Real world is complicated

I am the most influential person who posts here.

It's that simple. And if you care you can find out I am, but of course, many of you believe that's a delusion or some narcissistic fantasy no matter what Google search results I can show you.

Now most of you have no clue about what influence is because you THINK you know what it is, and it's some naive Hollywood notion of fame, or celebrity, as the two can be different.

Right now, there is probably not a single mathematician living on the planet who is even close to my level of influence.

But as a GROUP mathematical society can still block me, so there is still the impasse.

It is requiring every single one of you worldwide.

THAT should give you some perspective on my real influence.

No single mathematician has the power to hold. Not even groups of you.

But ALL of you, worldwide are currently holding the line, and just barely.

Soon enough, you will no longer be able to do so, and I will be bigger than the entire mathematical field, worldwide.

It is just a matter of time.

Tuesday, January 27, 2009


JSH: Why many of you hate simple

Complicated and abstruse mathematical texts that take a lot of effort to decipher give many of you in "pure math" at least the feeling that with all that effort, you must have learned something important, but over a hundred years ago a subtle error entered your field. But it is a devastating error that means that over that last hundred years, much of number theory that people have championed, celebrated, and prided themselves on, is wrong.

A culture HAD to build up around the error, as on some level, especially for the more gifted mathematicians there must have been some sense that something was wrong. Some nagging feeling that to be denied required social structures, group pressures, and all the things that allow people to go on with false things, indefinitely.

There are simple answers for problems you have learned to answer with far more complicated ones.

You have been taught wrong. Many of you are being taught wrong.

But your defense now is only your society, as I have the proofs.

Therefore, the only path now is to take away your last defense.

There is no other option.

You have given none.

I tell you now, most of you are not even close to being a mathematician. And you never have been.

The only thing left now, is proving that to the world.
I was disappointed before, and I guess maybe I'm a little disappointed now, but it seems to me that it's pointless to get excited about how stupid the world is.

Why not instead, enjoy it?

Mathematics is key to our science, but "pure math" has existed for over a hundred years and things have been just fine.

So what if humanity has no future? From what I've seen, the human species is not worth crying over.

Maybe some aliens down the line will think this history was of some interest. Some scientists from a truly sentient species.

And none of you have a clue what I'm talking about, but think it's just some babbling.

Which is why you're so much fun! You have no future without the correct mathematics and you're too stupid to know it!!!

You are the end of the line for the human species. So there's no point in getting excited over ANYTHING!

Party time!!!


Monday, January 26, 2009


JSH: Probably right, but kind of depressing, you know?

Several months ago I found a general solution to binary quadratic Diophantine equations. In retrospect it kind of makes sense that with that first in human history, a solution to the factoring problem might kind of spontaneously emerge, but this thing I'm considering now is so freaking simple, and I kind of didn't quite figure it out myself anyway as others suggested simple things I just hadn't noticed, that it boggles the mind.

I hope is wrong, but it is, well, probably right.

Turns out you can solve for x^2 - Ty^2 = c^2, using some easy algebra. Trivial freaking algebra.

I posted it in one thread just a second ago, but I'll copy and paste it in here to explain some freaking things.

I'm going to switch variables here.

Given x^2 - Dy^2 = 1

you have solutions for an ellipse or Pythagorean triples with

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

where j = ((x+Dy) -/+1)/D, and j can be a fraction.

That's the result I started trumpeting just a little bit ago, though I first figured out that you could connect Pell's Equation to Pythagorean triples—when D-1 is a square—or discrete ellipses back in September of last year.

I just kind of wandered off from the result at that time.

Recently I thought about letting D-1 be a target to be factored, but figured out that D-1 always factors out, so that was a no-go, to my relief. Then a poster pointed out that you might factor with x^2 - Dy^2 = 1, using

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

And here now is the answer I worked out for THAT approach.

Where you can easily replace D with T, and x is a fraction. Solving for the variables is done trivially:

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

and it's just a matter of choosing a factorization of the left side. One possibility is:

x+y + j +/- 1 = (D-1)j
x + y - j -/+ 1 = j

but notice you can also factor D-1, for instance, assuming D is an odd number, I know it is even, so I could factor as:

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

So there are as many ways to choose factorizations are there are factorizations of D-1.

That's key. As you are taking D-1 apart, so the question is, does the math put it back together on the other side to force a trivial factorization? Maybe. That's the open question. If it does then this approach doesn't work.

It's as simple as that.

So you solve for (x+y) and j, given the 2 equations, using whichever factorization you pick.

Once you have them you find x and y, using that first solution for (x+y) and j, with

j = ((x+Dy) -/+1)/D

where now you have two equations again. In one of you have x+y with a number, in the second it is x+Dy, with a number, so it's easy to solve for x and y exactly. Trivial algebra.

Now you can substitute into:

x^2 - Dy^2 = 1

But I keep talking about x^2 - Dy^2 = c^2, because the x and y you have are likely to be fractions, but it doesn't matter as you just multiply out the denominators.

What makes this thing so damn scary is that you have to solve for variables TWICE using simultaneous equations, which kind of scrambles things in a way that one would think would make it difficult for the algebra to de-scramble on the other side to force trivial factorizations.

If the algebra does not care, then factoring was never a hard problem.

People just didn't know how to do it, so they thought it was hard.

But hey, the mathematics may save your butts still. It may de-scramble everything on the other side and force a trivial factorization.

Otherwise, it is the end of RSA encryption. Overnight.
Damn thing is actually right. I don't know how it's possible, though there STILL is some minor possibility that it may not work well, but theoretically, it IS the solution to the factoring problem.

What amazes me though is that it can probably sit out indefinitely.

What other explanation for that except the human species is worthless?

So far I have none. If I get some other explanation I will consider something else to do.

But right now, a simple mathematical method for factoring numbers of arbitrary size—it should factor an RSA number faster than you can encrypt with the damn thing—seems to be sitting out in public view—with no impact.

No impact. No acknowledgment.

I feel like maybe I should hold my breath, except that would be useless I think, as it appears, there is no reason to think that will change!

Humanity is, well, proved to be stupid in aggregate. The full evidence is that despite billions of people on the planet, the human species is, dumb.

I live on the planet of dumb and dumber.

Was I cursed or something? Maybe in a former life?

I'm on the planet of the idiots.

[A reply to someone who told James that his method does not work and that if he wanted to know why then he should try to actually factor some numbers.]

No need. You still don't know my modus operandi? I rarely need to test things, as I brainstorm on newsgroups and then put them on my math blog.

I also use Google search rankings to see when something is wrong.

If, as you say, the idea fails, it will drop in Google search rankings.

Right now the primary thread connected to it is RISING, so the Google search engine indicator says you are wrong.

I do not rely on single human beings unless you give me facts. Simply stating something without support does nothing.

Being someone who moves Google search engine results, I can use the Internet as a kind of global computer, seeing a continual vote that pulls in data worldwide.

Based on stats from my math blog that pull is over 2500 cities in 100 plus countries.

You as a single human being are nothing in comparison.

Sunday, January 25, 2009


Pell's Equation and the factoring problem

One of the weirder things to emerge recently from my latest research on solving quadratic Diophantine equations is a route to factoring using Pell's Equation where I'll admit a lot of details did not occur to me, but were pointed out by others.

What I found was that given x^2 - Dy^2 = 1

you have solutions for an ellipse or Pythagorean triples with

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

where j = ((x+Dy) -/+1)/D, and j can be a fraction.

So it's this trivial little equation—always been there—which unites Pell's Equation to Diophantine ellipses which can be Pythagorean triples when D-1 is a square.

Now factoring comes into the picture because you can conceivably set D equal to your target composite to be factored, and then

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

might non-trivially factor D, which was noted by the poster Colin Barker, but I'd guess that approach hasn't been considered viable because it's thought hard to get non-zero solutions for x and y BUT it IS easy to get non-zero solutions to Pythagorean triples or Diophantine ellipses, and the equations I found go in BOTH directions. Here's a demonstration where I worked from a Pythagorean triple.

Consider, 24^2 + 7^2 = 25^2.

If 7 is j+/-1, I can try j = 6 or 8, and find D=17 or D=10.

With j=6, D=17, I have j = ((x+Dy) -/+1)/D, so

17(6) = x+17y -/+ 1, so x+17y = 17(6)+/-1, and x+y = 25, so

16y = 17(6)+/-1 - 25, which doesn't give an integer y, which is ok, but I want an integer for this demonstration.

And trying j=8, D=10:

x+10y = 10(8)+/-1, so 9y = 10(8)+/-1 - 25, gives y = 6 as a solution, so x = 19, and

19^2 - 10(6)^2 = 1.

So you can go in either direction, though you may get a fractional j, giving the more general form u^2 - Dy^2 = C^2.

But that is ok for factoring as then you just have: (u-C)(u+C) = Dy^2

The simplicity of solving Diophantine ellipses was noted by the poster Achava. That is equations of the form:

nu^2 + v^2 = w^2

as you can use w+v = nu, w-v = u.

Then the latest research indicates that it is not at all difficult to find non-zero solutions to

x^2 - Dy^2 = C^2

where D is picked and C is determined by the method.

It occurs to me that no one knew before that you could directly connect Pell's Equation to Pythagorean triples and Diophantine ellipses, as if they did, then this solution would have been known.

To me that is further indication of a disturbing reality I've often noted on the newsgroups where posters make bold claims about research being "not new", no matter what the territory. So it is clear that they make things up.

Given that you can go in either direction from Pythagorean triples to a general form of the Pell's Equation, or from a Pell's Equation to Pythagorean triples or Diophantine ellipses, it seems very unlikely that factoring would have been considered a hard problem if this information had been previously known.

It also seems extremely unlikely that the world's Internet security system would be based on it, or that RSA encryption would exist.

Therefore, it is clear that the information must be in many ways new. Factoring was never a hard problem.

People simply were wrongly convinced it was because some simple algebra was, I guess, unknown, though it's not clear why it was not known. It is easy algebra, after all.

Trivially easy algebra.

Tuesday, January 20, 2009


JSH: Factoring with Pell's Equation

The result I noticed previously can also be a factoring result, intriguingly enough, which deserves its own thread.

Given x^2 - Dy^2 = 1

you have solutions for an ellipse or Pythagorean triples with

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

where j = ((x+Dy) -/+1)/D, and j can be a fraction.

So if a target composite N to be factored is given by D-1, you have that finding a non-trivial solution to

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

will give a difference of squares with the target composite i.e.

Nj^2 + (j+/-1)^2 = (x+y)^2, where j = ((x+(N+1)y) -/+1)/(N+1).

But it's even better, as given one non-trivial solution to a Pell's Equation you can trivially generate as many more as you like, so you can continually generate tries at non-trivially factoring N.

So solving Pell's Equation is directly connected to integer factorization.


JSH: Pell's Equation, circle, ellipses, nifty little result

Oh yeah, a while back I found I could get Pythagorean triples from x^2 - Dy^2 = 1, which I thought was neat, but I ended up moving to other things. Recently, however, a poster brought the subject up in a somewhat different context, which got me to thinking about it again, so here's the result:

Given x^2 - Dy^2 = 1

you have solutions for an ellipse or Pythagorean triples with

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

where j = ((x+Dy) -/+1)/D.

Notice that an integer j will always exist as x = +/-1 mod D, from Pell's Equation.

And that's it. I think it's nifty. Tiny. Concise. Does the job.

You can also at times use it to go BACKWARDS and get a solution to Pell's Equation from Pythagorean triples.

And it gives one other result which is that in general, with the Diophantine equation:

ax^2 + y^2 = z^2

where 'a' is a natural number there are always integer solutions and always an infinity of them, driven, I think intriguingly, by solutions to Pell's Equation.

So it's like, discrete ellipses and circles connected to discrete hyperbolas. You could kind of like call it, discrete conic love.

Friday, January 16, 2009


JSH: Research speaks for itself

Through the years I've been certain that useful research will be used. Seems like it makes sense but hey, it often seems like a crazy world, and I have experience with failure with my research. For years I actually was a math crackpot, with numerous failed attempts at proving Fermat's Last Theorem. Lots of failure. Years of it. But I believed that research speaks for itself so that if I were right, it wouldn't be about me and failure, it would only be about the success.

If I could ever succeed.

The more astute of you may have noticed a sharp dichotomy in recent exchanges on the newsgroups:I could keep going after failure because I found I just liked fiddling around with equations, as it occupied my time, and in accepting that a good reason for what I was doing was enjoyment and not the end itself, as there is no end, I started getting real results, and found this massive problem in abstract number theory which has tested my belief that research speaks for itself.

In the past I'd attack my own credibility considering it useless in considering mathematical proofs.

Now I have to convince that experts in the field of mathematics are for some reason or other ignoring quite a few major discoveries including a find of a massive error in some abstract number theory and the credibility issue has taken a new twist.

What is more credible? People or their results?

I and my opponents have taken two different paths in answering those questions.

Our two paths are revealed through Google searches:
  1. I've emphasized research. Explaining, explaining, explaining. I've worked at simplifying my prior results and worked at making more, and even focused on their usefulness. My focus has been on research.

    Results: Google search results linking to my research in key areas. e.g. Google: define mathematical proof

    Or, Google: solving binary quadratic Diophantine

  2. They've emphasized me. From personal attacks, questioning my sanity to repeating over and over again that I'm wrong, or that even if I'm right nothing I have is important, it has been a non-stop attack on my worth. Their focus: personality.

    Results: Google searches on my name bring up a webpage against me in the top 20 on major search engines, including Google and Yahoo! which I just checked. On Google the flame page came up #20, on Yahoo! it came up #18.
So the search engine results reveal two competing strategies.

I have yet to find searches on my name bringing up ANY of my research. Not any. So no, I can't just search on my name and come up with my research on solving binary quadratic Diophantine equations, or defining mathematical proof.

Their strategy of personal attacks worked in that sense. Their continued use of that strategy can be seen in reply after reply after reply on these newsgroups.

And my disdain for that strategy can be seen in my responses: I believed and believe that research speaks for itself.

They clearly believe in the usefulness of personal attacks. Attack the person to attack the research.

But what kind of scientists would engage in such behavior?

Oh, oops! My mistake. Talking about members of the mathematical community, not scientists.

Behavior speaks for itself.

My situation is about time. I see Google search results as leading indicators of world interest. I see the focus of search engine results on my research and not on me as a brilliant demonstration that it's not the person—it's their work.

I am very tempted to just kick back and relax, turning my problem solving skills to more natural things, like pursuing extremely beautiful women (now that takes a great deal of genius, a problem worthy of my skills).

However, I cannot simply walk away from the reality of some people abusing their position to maintain their ability to teach wrong mathematics, nor deny my responsibility to help in the end of the use of failed ideas, so that correct ones can be used, which can further the advancement of the entire human species.

To the world—They are your children: both the ones being taught false information now, and the ones yet to be born, who may need the continuing pursuit of knowledge.

Don't allow a world that wakes up in some distant future when the reality of the math error and its importance are finally realized, belatedly, only to find it is too far behind to catch up to the solutions it needs for problems we cannot imagine today any more than Galileo or Newton could imagine ours.

The fate of the world is not about personalities. Or who can be more creative in slamming the other guy.

It's about the research that speaks for itself. If you listen.

The future of our world depends on people asking questions.

Thursday, January 15, 2009


JSH: Relevance of the denial

Being someone in the difficult position of trying to inform the world about a big problem in abstract number theory where the experts in that area refuse to acknowledge it, I'm very much aware of the importance of showing non-rational behavior from people who are working so very hard to convince you that there is no support for my research.

The relevance for physicists is they have sold various mathematical techniques which increasingly I am certain do not work.

My leverage in getting heard is my own research, which has growing influence around the world.

New ideas take time to gain hold, especially when there is entrenched hostility from people already established in an area, but there has been some time that has already passed.

With the years comes use.

Google searches are a fun and easy way for me to map growing interest in my research and to see what pulls more interest most rapidly and not surprisingly to me, my least "pure math" results are the biggest drivers.

But I do Google searches all the time and have other data on a continual basis. POSTING about Google searches allows you to see what happens with the math people when you give them information they cannot stand.

So these posts are all about the replies they garner.

Nothing like seeing the behavior up close and personal.

Oh yeah, I still am amazed though by the take-over of the Google search: devastating error

And I'm also looking more into the question of just how bad is the weird math error for Galois Theory, as, of course, there is group theory which is so successful.

Trouble is, I already know of a case where Galois Theory is said to be relevant, where it turns out there are just two ways mathematically to look at the problem, which is with binary quadratic Diophantine equations.

There an odd bit of coincidental mathematics is key. It is so weird once you understand it, and then understand how much can pivot on such simple things:

Here intriguingly I can also cite someone! I rarely get to cite the research of anyone else as most of my own research is from scratch:

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

Abstract: We solve Pell's equation in a simple way without continued fractions or irrational numbers, and relate the algorithm to the Stern Brocot tree.

Problem for Professor Wildberger though is that solving Pell's equation is considered one of the plums of the theory using irrational numbers so math people don't want to know it can all be handled with rationals only. I have yet to see evidence that his paper has garnered much support or interest (if there is evidence please give it).

Oh yeah, of course, if I my research is correct then it stands to reason—if you believe in humanity AT ALL—that it would be picked up, and used. Our species is supposedly kind of efficient at such things, right?

So to some extent, it's all about time.


JSH: Leading authority check

If you go by Google search results it appears I'm becoming the world's leading authority in two key areas, which is a statement meant to be challenged, so here are the two searches, web engine must be Google and no quotes:

solving quadratic residues

solving binary quadratic Diophantine

I should come up in the top 10 for both searches, if not then consider the statement above rebutted for your particular area, as it is probably, at least for the moment, country specific, but that will probably change as my mathematical research continues to rapidly take over.

Sunday, January 11, 2009


JSH: Algebraic integer problem, another try at explaining

I find it hard to believe that even physicists would not get very interested in this result if they believed it true, even though it seems on the surface to only matter about some abstract number theory, so here's yet another try at explaining this esoteric math issue that takes out Galois Theory.

On the complex plane consider an unknown factorization:

175x^2 - 15x + 2 = (5c_1(x) + 1)(5c_2(x)+ 2)

Notice with just that information there are an infinity of functions that will work for the c's.

If you're wondering why THAT particular quadratic which I keep using over and over again, it's for historical reasons.

Now I also multiply by 7 for historical reasons but if it would help to multiply times some other number like say, 27, because it's not prime I can do that as well, and then the quadratic will change as well. But for now on with 7:

7*(175x^2 - 15x + 2) = 7*(5c_1(x) + 1)(5c_2(x)+ 2)

That is the OPPOSITE of what's normally done of course, as why multiply on an extraneous factor?

Answer is, because by doing this major "out of the box" step, I can reveal a massive error.

But next I have to make a choice!

7*(175x^2 - 15x + 2) = (5*7*c_1(x) + 7)(5c_2(x)+ 2)

and now I need to hide things a bit, which I can do with new functions:

b_1(x) = 7*c_1(x), and c_2(x) = b_2(x)


7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2)

with still as of yet undetermined functions, but I have a problem now with the 7 hidden, so I need a condition to start determining my functions:

Let b_1(0) = b_2(0) = 0.

But there are still an infinite number of possible functions at this point.

And now I need symmetry, as the issue is actually a symmetry problem, so I need one last set of new functions:

a_1(x) = b_1(x), and a_2(x) = b_2(x) - 1

and making those substitutions gives me:

7*(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

The symmetry with regards to the 7's is required so that the a's can be roots of the same quadratic, which is NOT possible until that fix is made, so it's not an option here.

Now, of course, I have that ONE SOLUTION—as it is one solution out of infinity—for the a's is, as roots of

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

and now you can solve for the a's using the quadratic formula:

a_1(x) = ((7x-1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x-1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

I need to emphasize at this point that the entire point of this exercise was to get to what I call a quadratic generator, which is just an expression that gives you quadratics, so for example, with x=1, you have

a^2 - 6a + 35 = 0

which is, of course, a quadratic. But also I needed a monic quadratic, which just means the leading coefficient is 1, which is satisfied, because then the a's will always be things called algebraic integers, when x is an integer.

But so what? Why bother?

Well, because mathematicians do not teach and prior to my research did not believe (I guess mostly they still don't) that given

a^2 - 6a + 35 = 0

only one of the roots is a product of 7, but my substitutions above mean that

a_1(x) = 7c_1(x) and a_2(x) = c_2(x) - 1

so you can SEE the 7 multiplying times the function c_1(x), where I remind that the start was:

175x^2 - 15x + 2 = (5c_1(x) + 1)(5c_2(x)+ 2)

Now the preferred ring for considering factors in such a situation for mathematicians is something called the ring of algebraic integers, and given all of the above you might suppose then that

a^2 - 6a + 35 = 0

has 7 as a factor for just one root, but instead you can mathematically prove the OPPOSITE result!!!

Well, you may wonder, maybe there is something weird about the algebraic operations I used to get to the a's, so I suggest that you go back through this post and consider each mathematical step carefully.

You will find that NONE of them are invalid on the complex plane, or apparently even on the ring of algebraic integers!!!

But that CANNOT be true, as if all the steps are valid on the ring of algebraic integers, then the result would hold in that ring, but we already have at x=1 that it DOES NOT HOLD IN THAT RING.

The mystery should resolve with a leap for some of you that of course the one thing that must be invalid on the ring of algebraic integers, as all the steps look like ok algebra is the start!!!

175x^2 - 15x + 2 = (5c_1(x) + 1)(5c_2(x)+ 2)

That is the only conclusion left for you. If that start is ok, then everything else follows on the ring of algebraic integers and the conclusion logically should hold, so it must not be possible for the c's to exist on the ring of algebraic integers for x=1, where you can get to the a's that are roots of a quadratic generator as given.

The more general conclusion though is that the c's cannot exist for ANY situation where you can get to a's that are roots of a quadratic generator!!!

So my using

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

is not just some special pick that matters as you have the conclusion that NO quadratic generator will work because all the algebraic steps I used are ok, and the conclusion for any that could would be that

a_1(x) = 7c_1(x)

which is the factor result being blocked by the ring of algebraic integers for certain cases.

So the ring of algebraic integers is BLOCKING the existence of

175x^2 - 15x + 2 = (5c_1(x) + 1)(5c_2(x)+ 2)

whereas, of course, the complex plane does not care, so everything was valid on the complex plane!

The ring of algebraic integer is saying, no way to a particular factorization. Now isn't that kind of odd? I think so.

So you have two conclusions:
  1. a_1(x) = 7c_1(x) found on the complex plane.

  2. Neither root of a^2 - 6a + 35 = 0 can have 7 as a factor in the ring of algebraic integers.
So I have a fight between the results, which is kind of neat!

The math appears to be fighting itself.

But why even care about what the result is on the ring of algebraic integers? Why should that matter?

Why can't I just say, screw that ring, I like the original result—as it makes sense—and I believe now that

a^2 - 6a + 35 = 0

has 7 as a factor for one root?

Turns out, you can.

And THAT is the reason math people get up in arms as traditionally they trust the ring of algebraic integers, but cannot in this instance tell you why you can't just say, forget it!!! It's bonkers!!!

Pull the thread though, and things get kind of nasty if you wish to toss out the ring of algebraic integers, as down the line you also lose Galois Theory as a useful tool.

Now this post is still an overview. I hope that maybe you now have a better grasp of what all the arguing is about, and maybe have some sense of why I say the issue is so huge, as I have shown you a first in mathematics:

direct demonstration of apparent contradiction

There is no other case in known mathematics where you can get something like the field of complex numbers disagreeing with something like the ring of algebraic integers.

Posters claiming that factors don't matter on the complex plane at all are being disingenuous as that is not the issue.

Like with y = 7x. On the complex plane y has 7 as a factor, but it also has 11, so what?

The equation however is valid in the ring of integers, where it is a factor result.


175x^2 - 15x + 2 = (5c_1(x) + 1)(5c_2(x)+ 2)

were valid on the ring of algebraic integers then a_1(x) = 7c_1(x), would be a factor result, in the same way.

Now to me it's kind of interesting that the c's CANNOT EXIST in the ring of algebraic integers if they can lead to roots of the same quadratic, for certain special cases, but I'm a person with a lot of curiosity.

And I've been taught at one of the best universities in the world to pull at threads, to see where that leads.

Math people fighting me on this issue are always coming to one conclusion: nothing to see here, not important, don't worry about it, just go with what you were taught, trust the textbooks.

Who to you sounds like the scientist, and who like people fighting to hold on to dogma?

Like I said, this post is an overview. But I have worked out the mathematical issues in much greater depth. I can delve very deeply into the esoteric aspects of the problem. Explain in extraordinary detail exactly how all the mathematics operates, bring in the wrapper theorem, talk about the ring of objects and show you how just advanced the mathematics is once this problem is properly considered.

It gets simpler but far more powerful.

But the first hurdle is your BELIEF.

And I have years of seeing how little I can get done when you just don't believe.

I cannot make progress if you do not have curiosity to care about the world's first case of apparent contradiction in established mathematics.

6 years have already passed here…

You being curious enough to force an answer from the mathematical community is the only hope that 6 more or more don't pass, and what then?

What if decades from now some physicists come to grips with this issue? Pull the thread and go where curious minds go?

And figure it out that I am right?

Consider the decades lost to the human species just because you and others like you were not curious enough to just find out the answers.

Real researchers wouldn't tell you to just let it all go and forget any of what I've shown here as nothing but trivial musings of a deranged mind.

What kind of people would? Think about it.

Saturday, January 10, 2009


Direct proof of algebraic integer problem

When you stop accepting mathematical proof then you are not mathematicians, no matter if you keep claiming you are.

In what follows I walk through a simple proof on the complex plane, and then show how the result on the complex plane gives a conclusion in other rings, and how in one ring, the ring of algebraic integers, there is a problem shown.

The argument is as easy as a, b, c so the issue is not correctness, but whether or not you and people like you really believe in mathematics.

On the complex plane given the expression

7*(175x^2 - 15x + 2) = 7*(5c_1(x) + 1)(5c_2(x)+ 2)

where the c's are not yet determined function of x, there is NOTHING in algebra that prevents me from choosing:

7*(175x^2 - 15x + 2) = (5*7*c_1(x) + 7)(5c_2(x)+ 2).

That distribution IS ALLOWED. It is one distribution but it is allowed as a valid step.

If you accept that one point you are 99% of the way to being able to cut through the noise when others work desperately to challenge the conclusion of this proof. Now I introduce new functions:

b_1(x) = 7*c_1(x), and b_2(x) = c_2(x)

and now require c_1(0) = 0, and c_2(0) = 0, so

b_1(0) = b_2(0) = 0, and substitute to get

7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2).

and again there is NOTHING that says I cannot do that on the complex plane.

The b's are still not determined functions of x.

I will introduce a final set of functions:

a_1(x) = b_1(x), and a_2(x) = b_1(x) - 1

Now I'll make those substitutions and get:

7*(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

and now note that a solution—no I'm not saying it's the only solution!!!—for the a's is, as roots of

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

You can now solve for the a's easily using the quadratic formula:

a_1(x) = ((7x-1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x-1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

and that is a valid solution set as can be verified easily enough by substitution.

But now you have the conclusion that only one of the a's was actually a product of 7, as you know the start:

7*(175x^2 - 15x + 2) = (5*7*c_1(x) + 7)(5c_2(x)+ 2).

The result that only one of the roots of

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

was actually multiplied by 7 becomes a factor result in other rings, but is contradicted by the ring of algebraic integers in specific cases.

THAT brings into question the naive use of that ring, as unfortunately it can be shown that the problem allows you to appear to prove things that are not true, and it also brings into question the usefulness of Galois Theory.


JSH: Duty to report

I want it clear to academics in the math field who read the sci.math newsgroup that they do not have a choice in this matter.

They have a duty to report the problem with the ring of algebraic integers.

It is also an ethical duty but can be a legal duty for those who receive government funding where there is legal binding within the contract.

This issue is not minor. For some of you proof of failure to report I believe based on my non-expert understanding of the law could be used in federal cases against you in the United States of America.

It would be the United States of America as the plaintiff against you, and possibly your university.

This notice is meant to make you fully informed that failure to act in the best interests of your country may be more than of academic interest and for some of you may be the worst decision of your lives.

My concern is that I've seen at times from academics a belief that they can choose to ignore results that do not promote their own self-interests.

While that may be true in other matters, in this case, it can be construed as fraud, and you may find yourself facing a US Attorney prosecuting you for at best fraudulent behavior though they may find other criminal acts with which you can also be charged.

I am not acting as a representative of the US Government or any other country. But as a private citizen hoping that by making you fully informed you may be more likely to do the right thing, but also by attempting to make you better informed it will be easier for prosecutions around the world, if you do not.

I am not a legal expert. If you have legal questions then you should consult a legal expert on this matter to find out what your duties actually are under the laws of your respective countries.

Friday, January 09, 2009


JSH: But their emotion is understandable

It's one thing for me to easily prove a massive error in abstract "pure math" number theory, but it's another for people in the field to accept it as it is a humbling reality, and the emotion is understandable.

These people have tremendous prestige. The phrase "beautiful minds" is often used to describe at least some of them.

They work in prestigious institutions where they have a lot of respect. Young people who are their students look up to them, listen to them, and obey them as they assign study, homework, and research tasks.

They ARE somebody.

But the error for number theorists at least can mean that in a sense it was all a big lie.

The concept of "pure math" became popular and some number theorists may see themselves as pure mathematicians only, and thus, sadly, may have no valid research results no matter how long their careers as the problem has been in the field for over a hundred years.

Their denial is very human. And very sad despite being understandable.

For physicists acknowledging the error will lift a huge burden, be exciting, and potentially open up wide new avenues for explanation of our physical world.

For mathematicians, it's embarrassing and raises serious questions about how they will rank themselves in the future.

For physicists, not such a bad thing, just an upheaval like ones that have come before.

For mathematicians, disaster, unlike any in their history, and something they have proudly proclaimed was impossible in their field.

The lesson here is that no field is immune from upheaval.

Their emotion is understandable, but they cannot be allowed to live in error for the sake of their feelings.

The world eventually will have to come to grips with the reality—yes, you are going to hurt their feelings.

Perspective though, there are much bigger tragedies in this world than humbled academics.


JSH: Critical argument, short demonstration

Math people are fighting to defend against accepting they have over one hundred years of error in a "pure" math area of abstract number theory, but I can give you a short proof to cut through their distracting rhetoric:

On the complex plane given the expression

7*(175x^2 - 15x + 2) = 7*(5c_1(x) + 1)(5c_2(x)+ 2)

where the c's are not yet determined function of x, there is NOTHING in algebra that prevents me from choosing:

7*(175x^2 - 15x + 2) = (5*7*c_1(x) + 7)(5c_2(x)+ 2).

If you accept that one point you are 99% of the way to being able to cut through the noise when the math people work desperately to hide their error. Now I introduce new functions:

b_1(x) = 7*c_1(x), and b_2(x) = c_2(x)

and b_1(0) = b_2(0) = 0, and substitute to get

7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2).

and there NOTHING that says I cannot do that on the complex plane.

The b's are still not determined functions of x.

I will introduce a final set of functions:

a_1(x) = b_1(x), and a_2(x) = b_1(x) - 1

Now I'll make those substitutions and get:

7*(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

and now note that a solution—no I'm not saying it's the only solution!!!—for the a's is, as roots of

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

You can now solve for the a's easily using the quadratic formula:

a_1(x) = ((7x-1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x-1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

and that is a valid solution as can be verified easily enough by substitution.

Now then, if you believe in mathematical proof, where in that chain of mathematical statements did I do something invalid or blocked by algebra?

But now you have the conclusion that only one of the a's was actually a product of 7, as you know the start:

7*(175x^2 - 15x + 2) = (5*7*c_1(x) + 7)(5c_2(x)+ 2).

The result that only one of the roots of

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

was actually multiplied by 7 becomes a factor result in other rings, but is contradicted by the ring of algebraic integers in specific cases.

THAT brings into question the naive use of that ring, as unfortunately it can be shown that the problem allows you to appear to prove things that are not true, and it also brings into question the usefulness of Galois Theory.

Yes, I know, it can massively hurt huge egos to find out that a ring has been naively used, but that is just reality. Huge egos in the math field may not wish to accept the truth, but they shouldn't be allowed to continue in error.

But they are trying to continue in error.

Which is why posters are so dedicated in arguing with me.

They are working desperately to hide one of the hugest errors in the history of the human species.

If they accept what is mathematically correct, then a house of cards tumbles down.


JSH: Bottom line, academic fraud in math field

You may have noticed a lot of threads arguing about some rather abstruse mathematical issues, and may believe it has nothing to do with physics while wondering why, even if I'm right, it's a really big deal, so here's a post to explain from a big picture view.

What I did was figure out a way to do something differently than before, but first note that on the complex plane with

7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2)

where b_1(0) = 0, and b_2(0) = 0, I deliberately have a construction where 7 has multiplied across just one factor.

So you have something like

7*(175x^2 - 15x + 2) = 7*(5c_1(x) + 1)(5c_2(x)+ 2) = (5*7*c_1(x) + 7)(5c_2(x)+ 2)

where c_1(0) = 0, and c_2(0) =0.

And if you understand ANY algebra then you realize I CAN choose to do that!

There is nothing on the complex plane that says I cannot!!!

It's ludicrous to claim that there is something mathematical that is blocking me from multiplying times 7 in that way, but that is exactly what posters are doing with the functions I have chosen because they are non-linear functions that are roots of a particular type of expression I call a quadratic generator.

In a sense that tells you all you need to know to realize they are wrong and I am right.

Unless you believe there is actually some way algebra can stop you from doing:

7*(175x^2 - 15x + 2) = 7*(5c_1(x) + 1)(5c_2(x)+ 2) = (5*7*c_1(x) + 7)(5c_2(x)+ 2)

If it can that's news too!!!

What the people arguing with me say instead is that in certain particular cases the 7—no matter what you want—is forced to split up into functions, like

7 = w_1(x)*w_2(x), so

7*(175x^2 - 15x + 2) = 7*(5c_1(x) + 1)(5c_2(x)+ 2) = (5*w_1(x)*c_1(x)+ w_1(x))(5w_2(x)*c_2(x)+ 2w_2(x))

where w_1(0) = 7, and w_2(0) = 1, but the w's split the 7 up at ANY integer values of x, where the non-linear functions I use would be non-rational because of an esoteric result in algebraic number theory about something called the ring of algebraic integers.

Now that is just stupid. There is no mathematical reason why I can't choose to multiply with 7 like

7*(175x^2 - 15x + 2) = 7*(5c_1(x) + 1)(5c_2(x)+ 2) = (5*7*c_1(x) + 7)(5c_2(x)+ 2)

and if there are people saying the algebra forces 7 to split up as a function of x, then they are quite wrong, but that is what posters have argued and that argument is what they used against my paper.

So what is the big deal? Well, turns out that a lot of algebraic number theory relies on results in that ring of algebraic integers as it was the basis for a lot of it back in the late 1800's, so my result blows all that up, and invalidates, oh, about a hundred years of "pure" mathematical research!!!

I can PROVE that easily but you can see the threads where math people have argued with me endlessly about the result I am explaining here which just says that yes, I can multiply the 7 in this particular way:

7*(175x^2 - 15x + 2) = 7*(5c_1(x) + 1)(5c_2(x)+ 2) = (5*7*c_1(x) + 7)(5c_2(x)+ 2)

And they say I can't, IF the functions are these particular ones that give their established theories problems, but they don't mind if the functions are something else. So it's all about the functions!!!

Now you may wonder how I even found this thing, and why I factor polynomials in weird ways, and the answer is, I came across it accidentally while trying to prove Fermat's Last Theorem, where there is a key point in the argument that relies on doing this very thing with a far more complex expression.

So if you believe that in general you can:

7*(175x^2 - 15x + 2) = 7*(5c_1(x) + 1)(5c_2(x)+ 2) = (5*7*c_1(x) + 7)(5c_2(x)+ 2)

then you also have that the techniques the math people are so desperately fighting are key in proving Fermat's Last Theorem, which I did back in 2002.

Number theorists then have been rogue for at least 6 years now.

So what they're doing is academic fraud.

Note, basic ethics would require ANY mathematician aware of this problem to ring the alarm and make certain that the issue is noticed and dealt with, but instead they've managed to stay quiet for six years despite my efforts to get them to do the right thing.

That indicates that their field is corrupted. They are willfully accepting money for fraudulent research. And willfully teaching errors to their students.

The mathematical proof is easy to the point of trivial. Review the discussions recently with the notion that I'm just saying you can multiply by 7 like

7*(175x^2 - 15x + 2) = 7*(5c_1(x) + 1)(5c_2(x)+ 2) = (5*7*c_1(x) + 7)(5c_2(x)+ 2)

and read math people claiming you cannot! They are trivially wrong.

I was published. Math people got the paper yanked and then, destroyed the journal.

Truth is stranger than fiction and because this scenario seems so impossible, I guess, to most people, it's big enough that it is being allowed to happen.

A small fraud might be more believable, but "top" academics around the world trying to hide over one hundred years of error is difficult to believe no matter how easy the math. But that is what is happening.

It's like, none of you can accept that an entire discipline worldwide is corrupted, especially the field of mathematics, but the consequences can be dire even for physicists, as ANYTHING using the flawed mathematics is not going to actually work.

People may think it works by rationalizations, but if you dig deep you will find it does not.

If you come to comprehend that this widescale academic fraud is real, then you probably will soon find it is difficult to get anyone to pay attention to it.

Big names in mathematics are, of course, entrenched in defending themselves, and with six years behind them, they're screwed on a massive scale if the truth comes out, so I'm seeing evidence that there is nothing that will move them.

They are no longer actually academics in the pursuit of knowledge: they are desperate men, holding on to a fraudulent scheme worldwide.

Wednesday, January 07, 2009


JSH: Then they turn to insults

It's easy for me to beat the math people with the mathematics, as it's mathematics. So I can prove what I say easily enough, then they argue around the details desperately trying to find some way to obscure the result, often simply lying about what I say, deleting out the argument to claim something that totally doesn't follow, or babbling incessantly about non-issues.

But ultimately they turn to insults.

So I got a paper published on this subject in a formally peer reviewed mathematical journal.

They insulted the journal. Insulted the editors, and insulted me. (And figured out how to break the entire freaking journal process. Seems editors are a weak link!!!)

I can easily show you the result with the distributive property for God's sake, and they argue around that result you used to think was so trivial, and eventually get to, yup, insulting me.

They refuse to follow any rules of academia. Throw out all rules of ethics, so journals don't even bother claiming an error with my papers, they just say they don't think the paper is "appropriate" for their journal.

I still get a kick out of the Bulletin of the American Mathematical Society, where the chief editor—yeah the big cheese—informed me it was a "survey journal" as she also begged that I not send any more papers! Said they weren't suitable for that journal.

They have broken all rules to make one rule: no matter what they will not acknowledge this error no matter how easily it is proven, or how clearly the proof is presented, or how much effort is made in getting them to acknowledge it.

For the theoretical physicists especially "string theorists" maybe it's not a big deal. Their stuff isn't working worth a damn anymore anyway, so going to "pure physics" probably makes a lot of sense to them.

But for you experimentalists, you may as well pack up and go home. None of the theory will work, and if you check knowing this error, nothing using Galois Theory really worked before! Oh yeah, I know group theory and ya da ya da ya da, blah blah blah, you, you true believers!!! Why don't you just start tithing with your religion as well as you're not doing physics.

You're praying.

You're so naive.

My research it seems opens the door wider on the nucleus. Probably helps explain quark behavior.

Could be key to bringing forward quantum chromodynamics to the level of QED.

In contrast, their crappy math just will not work (no matter how much you pray or just believe, believe, believe!).

Believe me now or believe me in a decade.

For you the biggest difference will be your wasted life and research, or not wasted life and research.

Tuesday, January 06, 2009


JSH: General distributive property result, helps?

There is a general result on the complex plane that with n, a natural number, given

n*P(x) = (n*f_1(x) + n)(f_2(x) + 2)

where P(x) is a polynomial and

f_1(0) = f_2(0) = 0,

you have a demonstration of the distributive property.

Notice it also follows then that with

n*P(x) = (f_1(x) + n)(f_2(x) + 2)


f_1(0) = f_2(0) = 0

you do as well.

It may seem trivial but it's a fairly powerful result if you pick a ring where the f_1(x) and f_2(x) exist, as there you have a factor result!!!

That is, given

n*P(x) = (f_1(x) + n)(f_2(x) + 2)

and f_1(0) = f_2(0) = 0

in, say, the ring of integers, you have that f_1(x) must have n as a factor.

That is true for the ring of integers, and for gaussian integers, but it is not true for the ring of algebraic integers, which is why math people fight that result, which is the generalization of the construction I've been using with n=7.

Here I'm not going to make it as easy for posters disagreeing with me.

If they dispute the result, then give a counterexample in some ring other than the ring of algebraic integers.

Like pick n=32 and show f_1(x) coprime to 2 or something, or do something with gaussian integers.

But when you fail to refute this result, act like you care at all about mathematics and just admit you are wrong.


JSH: Mathematical proof denied

Ok so yeah, it is of interest to study mathematicians and others in math society deny a mathematical proof, especially a trivial one.

Evidence so far is that with something with HUGE social consequences they are capable of denying anything, doing so with fervor, and doing so for years.

THAT is weird, as imagine a physicist finds out that some theory he has trusted in for decades is bogus, which casts doubt on ALL of his experimental findings, which he realizes were wrong by interpretation, where he simply convinced himself the theory worked but in retrospect he now realizes it ALWAYS failed.

Now then, imagine this physicist, shrugs, goes BACK to using the bogus theory, writing papers, and teaching it to his students!!!

Can you imagine a physicist doing such a thing?

That is the equivalent of what the mathematicians are doing though, and I think the difference is that they can get away with it!!!

Physicists are trying to explain the real world. Mathematicians can just do "pure math" which only requires the agreement of their colleagues.

For the sake of argument—play Devil's Advocate—imagine I have found a bizarre and devastating error in number theory that invalidates over a hundred years of mathematical works, and mathematicians simply can't accept the emotional pain so they just ignore it!

What happens then? Nothing. If they all ignore it, they can just keep doing what they've been doing: writing papers, teaching, getting grants and accepting and giving out prizes, with completely bogus stuff.

But what about physicists?

How long till people notice the physics doesn't work? Would it even take to not being able to build computers, cars, planes, or predict the motion of the planets or stars?

The physicists can't just turn to complete denial and continue as before, unlike mathematicians.

And what I'm proving to you—as I have trivially shown over and over again with a proof relying on the distributive property—is a remarkable error that entered into number theory over a hundred years ago--obscured at least partly by "pure math".

And I discovered it over 6 years ago!!!


Google: SWJPAM

Unlike physicists, mathematicians can remain in error as long as they willfully do so and all go along with the error!

And that is what they are doing now.


JSH: Demonstrating the distributive property

The problem with proof is, people can try to deny it, so here's yet another explanation of a remarkable error in number theory, where the proof only requires you accept the distributive property.

In the complex plane consider

7(x+1) = 7x + 7

which is a very simple demonstration of the distributive property. Now multiply both sides by x+2:

7(x+1)(x+2) = (7x + 7)(x + 2)

and on the left hand side simplify (x+1)(x+2) to get x^2 + 3x + 2, and you have

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

and the stepped out demonstration that yes, indeed, it is the distributive property, where the only addition to the distributive property is multiplying both sides by x+2 and simplifying the left hand side, slightly. (Yes, my proof really is this simple.)

Moving on to the more complicated argument consider

7(f_1(x) + 1) = 7f_1(x) + 7

where f_1(0) = 0, so it's a normalized function. Now multiply both sides by f_2(x) + 2, where f_2(0)=0, and you have

7(f_1(x) + 1)(f_2(x) + 2) = (7f_1(x) + 7)(f_2(x) + 2)

and now let (f_1(x) + 1)(f_2(x) + 2) be a polynomial P(x), and you have have:

7P(x) = (7f_1(x) + 7)(f_2(x) + 2)

as a demonstration yet again of the distributive property, which you'll notice encompasses the prior result with

P(x) = x^2 + 3x + 2, and f_1(x) = x, and f_2(x) = x.

Notice it is a proof on the complex plane that given

7P(x) = (7f_1(x) + 7)(f_2(x) + 2), where P(x) is a polynomial and f_1(0) = f_2(0) = 0,

you have a demonstration of the distributive property.

So that is true for ALL such cases. ALL of them, over infinity.

THAT is the result being fought by the posters arguing with me, as now let:

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

7f_1(x) + 7 = 5a_1(x) + 7, and f_2(x) + 2 = 5a_2(x)+ 7

where the a's are roots of

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

And you only need prove that f_1(0) = f_2(0) = 0, and you have the same distribution result on the complex plane, as before.

So that proves a general result encompasses both:

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


7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

where the a's are roots of

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

in a simple and elegant proof by direct demonstration of the distributive property.

Mathematical proof is irrefutable. What you have is an absolute argument which is not refutable mathematically.

Monday, January 05, 2009


JSH: Top of the intellectual foodchain

Ok, so I admitted in another post that I'm at the top of the intellectual foodchain. I don't like talking about it too much, but of course, I am very well aware that with my discoveries, history will probably record me as one of the major discoverers in human history, and resistance to the results I've found is probably a short footnote to the story from the perspective of history.

Like textbooks on the subject will note something like, for a time mathematicians disagreed on the key result. Or something like that. They may get more than a sentence or two, maybe.

So I really don't need anything from anybody in that sense, as if that doesn't happen, it probably just means the human species goes extinct, and soon. So no worries!

It is kind of interesting to me to read posts where people seem to think they can convince me that my results are wrong, as, um, I did find them. It was kind of hard, you know? And being someone who can find, say, an over one hundred year old error in modern number theory, I'm not exactly the kind of person who can be convinced I'm wrong by some posters on a newsgroup.


Seeing as I am probably right now the world's kind of, um, trying to find some humble way to present it...can't find it, so I'll glide past that, where the entire mathematical field worldwide is needed to block acceptance of my discoveries, the stress impacts the entire world, which is kind of weird, but normal if you understand why, and I like to point that out, with, yup, Google searches.

Like I defined mathematical proof. Don't believe me? Google: define mathematical proof.

And then go to Yahoo!, and search there on: define mathematical proof.

I'm lower down in the top 10, but moving up (I think).

I got a kick recently out of searching in Google on: devastating error

Seems maybe I have gotten the attention of some of you.

My place in human history is assured as one of the major discoverers.

Yours is not.

Posters can reply to this post with whatever they want. Just like crackpots can rail against Einstein, or insult Newton.

Doesn't matter.

In the big scheme of things, the war has long been over.

So why do I bother posting?

Ah! I think that is the one thing that befuddles many of you: Why, if I'm right, with my place in human history assured, do I bother posting on newsgroups?

I'm curious as to what answers that question might bring.


JSH: Bizarre mystery, mathematicians non-response

As a researcher now with multiple results, I've been long fascinated by the resistance to the mathematical proofs I've discovered from members of the mathematical community, but now I feel like a big clue as to why they have denied fairly easy algebraic proofs in order to maintain a remarkable error is about "pure math".

There is almost a feel as if modern mathematicians in "pure math" areas no longer even care if it is correct, but are more focused on the style of argument.

Back in 2003, for instance, I was at my alma mater Vanderbilt University, explaining a slightly more complicated form of this result—for a while I was using cubics instead of quadratics—to Professor Ralph McKenzie there, on his blackboard.

And, nothing happened. He went home after I'd explained and later emailed me thanking me for the discussion and he offered some books on number theory I might read.

I was shocked. Overcome with emotion I emailed back a blistering reply. Increasingly angry, I ended up emailing the head of the math department, several other mathematicians in the department and a dean at Vanderbilt, as I could not believe it. Um, of course that didn't really work out. That was years ago.

I had stepped through the proof of this problem with a top mathematician who hadn't disagreed with it, and had agreed on the key points, and he just acted like it didn't really matter.

Understand then, from the major sources I have agreement with this result. It WAS published in a formally peer reviewed journal. To date no established mathematician to my knowledge has shown an error in the argument, and no one has even disagreed with the result outside of math newsgroups.

By all the rules, I shouldn't be elaborating on how bizarrely the mathematical community is acting.

The error should be known, making headlines, as a remarkable find by an amateur of an over one hundred year old error lurking at the heart of number theory.

So how do they do it? How do academics consistently ignore an easy proof of a remarkable problem at the very heart of their field?

Google: SWJPAM

That journal was one of the few cases where mathematicians did the right thing here, albeit briefly, and those editors caved, pulled my paper after publication, and then SHUT THE JOURNAL DOWN in one of the more dramatic instances in this story.

Posters on math newsgroups will rationalize endlessly on how crappy the journal process can be, how crappy the editors must have been, and how it must have just been a mistake that they published a paper from an admitted amateur.

I've kept piling on mathematical results as well with my latest being a general solution to binary quadratic Diophantine equations.

Confronted with it, I've seen posters early on agree with the result, and later now, disagree with it, or simply keep going as if I didn't say anything when I mention it.

With this latest result using the quadratic construction I wrote up a paper, and sent it to the Bulletin of the American Mathematical Society, and the chief editor replied telling me it was a "survey journal" not appropriate for the paper and to please not send them any more papers!

I've sent the paper on to the Annals of Mathematics, a publication of Princeton University with the cooperation of the Institute for Advanced Study. Which I guess is kind of a lark (even if they were going to publish it can take years for them to accept).

It just seemed like the right next place. They've acknowledged receipt.

I wonder if maybe I should just let the process play out? Even if a couple of years go by, is it really all that bad?

Those of you who are theoretical physicists or experimental physicists trying to use number theory tools I've proven are useless, what are your careers worth?

I mean, in the big scheme of things?

So what? A few years go by, and everything you have simply gets tossed as useless, why should any of us at the peak of the intellectual food chain care?

You are all expendable, as, hey, more of you will be born. There will be more eager students to become professors who will have a chance, you will not.

And the world will keep on turning.

YOU are expendable, right?


JSH: Simple proof, but social response

I have found a remarkable error in established number theory which has physics implications, but while it's VERY easy to prove the error, mathematicians are invested fully in it, and have resisted efforts to get them to acknowledge mathematical proof.

The error is easily shown with a simple construction on the complex plane:

7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2)

where b_1(0) = 0, b_2(0) = 0, a_1(x) = b_1(x), and

a_2(x) = b_2(x) - 1, and the a's are roots of

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

Now you have NON-LINEAR functions.

Explicitly you have

a_1(x) = ((7x - 1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x - 1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

The result on the complex plane shows that 7 distributed through only one of the a's, but which one?

That's not visible because the square roots are not in general resolvable. Consider:

x^2 + 4x + 3 = 0, and with x=(-4 +/- sqrt(16 - 12))/2 = (-4 +/- sqrt(4))/2, which root has 3 as a factor? You can't say until you take the square root.

Despite all of the above, the reality of the proof does not seem to crystallize for most people without another example so consider:

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

I can let that be

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

where f_1(x) = 7x, and f_2(x) = x.

And those are LINEAR functions.

There is not a mathematician in the world who will argue with you about whether or not 7 distributed through the factor

(f_1(x) + 7)


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

with linear functions.

Notice though that with 7(x+1) = 7x + 7, you simply have the distributive property:

a*(b + c) = a*b + a*c

With b a linear function—no argument.

e.g. a*(f(x) + b) = a*f(x) + a*b

when f(x) is a linear function with

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

no argument from the mathematical community.

But with f(x) a NON-LINEAR function i.e.

7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2)

where b_1(0) = 0, b_2(0) = 0, a_1(x) = b_1(x), and

a_2(x) = b_2(x) - 1, and the a's are roots of

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

a_1(x) = ((7x - 1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x - 1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

now there is an uproar from the mathematical community as the result now follows that only one of the roots of

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

is a product of 7 and some other number, which leads to a conclusion in other rings.


Well consider again the simple example: f(x) = 7x

That is TRUE on the complex plane, but the equation is ALSO valid in the ring of integers, where it gives a factorization result.

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

a_1(x) = ((7x - 1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x - 1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

is true in the ring of algebraic integers, but it can be proven that NEITHER root can have 7 as a factor in that ring if the a's are not integers for an integer x.

(If the a's are integers, like with x=0, then exactly one root has 7 as a factor in that ring as expected from the result on the complex plane.)

The key point of all the discussion above is that the distributive property is key here, where with

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

the type of function shouldn't matter.

With a linear function f(x)—no debate.

With a non-linear function, furious debate.

But how does this relate to physics?

Well physicists were never "pure" in that their results are usually expected to work in the real world (ignore "string theory" for the moment), but mathematicians around the turn of the 20th century strongly endorsed "pure mathematics" known for not giving practical results, but they promised that in time the mathematics might be of value in practical areas.

But what if it was wrong?

This result indicates that over a hundred years of results in modern number theory are in fact, wrong, because it shows a contradiction between a key element in such theories—the ring of algebraic integers—and the field of complex numbers.

Quite simply, the field of complex numbers with the distribution argument, disagrees with the ring of algebraic integers, where 7 cannot be a factor, when the a's are non-rational with integer x.

There is a fight between the dominant field for physicists—the complex plane—and the dominant ring for number theorists—the ring of algebraic integers.

It is an unfortunate duel to the death for theories dependent on one position or the other.

Either the complex plane is correct and theories based on its correctness are correct, or the mathematicians are correct, and their research using algebraic integers wins.

BUT physicists work in the real world. Number theories more often work in the "pure math" arena.

So there is no real debate. They are wrong, the physicists are right. The complex plane wins and number theorists have a hundred years of error.

But being proud they resist the truth like so many people before them.

Wishing they were right, they betray their own field, hide from the truth, and continue to teach false ideas to young minds, despite one of the easiest proofs of error possible—for a problem that has lurked for over one hundred years.

Truth IS stranger than fiction. As mathematicians betray their field to hold on to error, history records one of the greatest intellectual challenges in the history of the human species.

It is a battle for the very heart of mathematics itself.

Thursday, January 01, 2009


JSH: Chaos theory and social situation

It's still something of a mystery to me why mathematicians would behave so blatantly against their own field with results so easily demonstrated to be major research finds, as, don't they expect to get caught?

If not, why not?

I think chaos theory may have the answer where you see acceptance of any given result as a matter of analyzing a complex social system.

At this time, Google and Yahoo! appear to be leading indicators that the clock is ticking against mathematicians who may not only be in actuality bad at math, but also bad with understanding how the world actually works.

I have long suspected for instance that they rely on the arguments on newsgroups to tell them whether or not my research is making headway in gaining acceptance and have long believed that it is making none BASED on the arguments that happen on newsgroups like those here.

But for the chaos theorists other data may be more telling.

As a research point consider the following search strings with Google:

define mathematical proof

devastating error

solving binary quadratic Diophantine equations

All those bring my research up highly in Google, and I think "definition of mathematical proof" is now bringing it up highly in Yahoo! as well.

The end of the resistance of naive mathematical professors—naive about chaos theory and its implications—may be a sudden reversal of public opinion and almost overnight worldwide acceptance of my research, followed by major investigations into their behavior, bringing almost overnight ruin for them.

So why would they put themselves in this position?

Best answer is they don't know how acceptance of research results actually occurs!

And that they overrate disagreement which inflates the apparent influence of minority opinions.

Actual results will of course tell the correct answer as that's what's great about physics!!!

It works in the real world.

Any chaos theorists out there able to make predictions about the moment of criticality?


JSH: Issue of fake claims of proof

Now some of you may wonder why it's so important to out mathematicians using this error I've found in number theory as it may be hard to understand why it matters if the complex plane disagrees with the ring of algebraic integers about roots of

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

The short answer is that mathematics is highly sensitive to error. An error in mathematics can mean you can appear—if the error is not acknowledged—to prove just about anything.

That is a BIG DEAL as consider there is now a yearly Abel Prize.

According to that Wikipedia article in 2008 the prize was $1.2 million U.S., but what if you found out that a researcher had that prize for work that this error invalidates?

What if the researcher found out about the error, should he give the money back?

The result brings into question a lot of "pure mathematics". It hits in any area where the ring of algebraic integers is relied upon at at all, which is a huge swath of modern number theory. It can mean that a tremendous amount of funding is being spent on research that is full of error.

Now when I first discovered it over 6 years ago I still believed that maybe today's mathematicians were just victims of a mistake that entered the field over a hundred years ago, long before any of them were even born.

But through the years as I've watched a math journal die and faced endless arguments on math newsgroups where posters clearly ignore the actual proof of the problem to make things up or insult me, and mathematicians I contact in other ways like through email simply choose to not acknowledge, it is increasingly clear that they know what the error is, and are exploiting it for their own selfish gain.

People calling themselves mathematicians choosing to live in error for the money or prestige or other selfish reasons.

Deliberate error in the field.

So yes, it is quite possible I believe that a recipient of the Abel prize KNEW his research was wrong, and KNEW that he was getting over a million dollars for a fraudulent result, but hey that's a lot of money!

And clearly they seem to think that my case isn't being believed and that none of you will choose a clear and direct mathematical proof over authority figures refusing to acknowledge it.

I suggest to you that the fraud is an expression of cynicism about you.

These people don't believe in mathematical proof. That is clear.

I do. After all I even defined it. Google: define mathematical proof

I am someone who is supposed to be listened to, one of the latest of a long-line of major discoverers and they must know by now who I am!

So their behavior is cynical on a scale hard to contemplate.

But clearly they also don't believe YOU believe in mathematical proof as math journals do not just die. Simple proofs of an extraordinary error do not just sit there without someone noticing in this day and age.

I suggest to you: these people believe you are with them in their fraud.

I suggest to you that they think you are one of them.

But are you?

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