## JSH: Test of cognitive function

One of the weirder things I face is the possibility that rejection of my work is about an inability of people to hold information about it in their working memory so here's a test of that assertion with what many have maintained on math newsgroups is a meaningless prime counting function that is just one of many others, but what you are about to see here is not a trick.

For reasons that escape me, that function cannot be held in human memory, so to give it you have to SEE it first.

To see the fully mathematicized function you can go to the Wikipedia Talk pages of the prime counting function:

http://en.wikipedia.org/wiki/Talk:Prime-counting_function

You may have to read a ways down, but I give the full function below "Questioning Controversy".

Look at it and see if you can comprehend it.

Look away and see if you can remember it.

No one in human history has derived THAT particular function before, or since. And I'm hypothesizing that maybe 1 in a thousand of you can even remember it—no matter how hard you try—but I doubt any of you can fully re-derive it completely from memory.

## JSH: Simple proof, but difficult response

I have found an easy demonstration of a very damaging error that has hold of number theory.

Mathematical argument is about as easy as it gets:

In the complex plane:

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, to emphasize those are LINEAR functions.

Now compare and contrast with

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.

So in one case 7 multiplies times a linear function, while in the second it multiplies times a non-linear function.

There is no way the type of function changes the distributive property as the distributive property with functions is as follows:

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

So all I did was figure out a creative way to put a non-linear function in a position where it had to be multiplied by 7 in a highly specific way, where it's trivial with linear functions, and the answer is also clear with non-linear ones…but I blow up about a hundred years of "pure mathematics" in number theory with the result.

"Pure mathematics" actually arose AFTER this error came into number theory in the late 1800's so it may be an artifact of the error as of course, wrong mathematics can't be useful for situations where you need correct answers.

By claiming "pure" math practitioners escaped having to have mathematics that could be checked against the real world, and the field became corrupted, and now it's VERY difficult to get past that corruption even with a very basic proof.

With the error people can appear to "prove" just about anything, and so Andrew Wiles has no proof of FLT. Guys winning Abel prizes in "pure math" errors related are getting money for failure.

The system broke here, but there is so much social inertia that it is difficult to get the word out.

Go over the argument above. Ask yourself: is there any way the TYPE of the function can change the distributive property?

Ask yourself again and again and again until it sticks.

Your heroes are nonesuch, and have proven they will hold on as long as society lets them, then I expect they will run for cover, and use every excuse in the book. Expect it, steel yourselves for it.

Those you respected so much, will come crashing down to the ground very hard, and reveal themselves to be, craven human beings, who held on to failure as long as they could, because they were never great, or even all that good, but just opportunists.

## JSH: Simplest explanation, Occam's Razor unleashed

In the complex plane with

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

you normalize:

a_1(x) = b_1(x), and a_2(x) = b_2(x) - 1, as then you have

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

where b_1(0) = 0, b_2(0) = 0.

Now compare with

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

and what is the primary difference?

The type of function: in one case you have non-linear functions b_1(x) and b_2(x), and in the other you have linear functions 7x and x.

I suggest to you that the distributive property does not care.

If some yahoo told you that the 7 bounced around like a kangaroo with

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

based on the value of x, you'd say they were bonkers.

But if some Ph.D's scared of losing some math beliefs and their precious notion that the mathematical field is immune to upheavals bitch and moan for years against it, you believe them, and screw the complex plane.

Simplest explanation.

None of you have the spines for the situation. You all lack the balls to confront a massive failure by your freaking colleagues.

They failed. Get over it. For a hundred years plus a nasty little bug lurked in modern number theory in "pure mathematics" and no one knew because none of their math applied to anything in the real world.

And I figured it out and these turds proceeded to do their best to punish me for it.

Because I yanked away their baby blanket. Took away their fantasy.

Destroyed their little bubble.

So they're bitching, whining and running away from the damn problem versus getting to the truth, confronting their loss, and moving forward with real research versus the crap they are still doing and teaching now.

The only extra is you are part of the duplicity now, if you do nothing.

Doing nothing is letting them get funding, get prizes for failure, and teach students junk.

Do you know how much the Abel Prize is?

Over a million dollars U.S. given every year to someone who probably did not succeed at anything at all.

Lying for a million bucks isn't noble, or academic survival in the real world: it's just being a con artist.

## JSH: Over a hundred years, wrong stuff

Academics in the mathematical field are refusing to acknowledge being informed of a massive error in their field which entered it over a hundred years ago despite the simplicity of the proof of the error.

It has been six years since I discovered the error while trying to prove Fermat's Last Theorem.

At one point I DID get a paper published in a journal called the Southwest Journal of Pure and Applied Mathematics or SWJPAM for short. The editors caved to social pressure from emails sent against the paper, and pulled it after publication:

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

The listing for my paper has "Withdrawn" under it, but I didn't withdraw it; the editors did.

EMIS, a European agency, is hosting the archives of the journal as not only did the journal die a little while after that withdrawal, but all mention of it was wiped by its hosting university, so all the papers over 10 years would have been lost if not for the action of the Europeans, despite the journal being an American journal previously hosted by an American university.

The error that is being held on to so desperately allows mathematicians to appear to prove things that are not true.

So it allows faux research results.

Academic mathematicians who have had their entire careers under the error may be afraid that without it, they cannot appear to do mathematics at all, so for them, it may seem to be a survival issue.

As an intellectual issue it may be one of the biggest in human history.

Academics aware of it, have a duty to act for the betterment of the human species by outing the academic fraud, and forcing the mathematical community to acknowledge the error.

That is an ethical requirement.

## JSH: Mathematicians cheating is kind of weird

Ok, so I have a simple proof of a massive problem in number theory and even got a more complicated cubic version published in a formally peer reviewed mathematical journal which is now dead. It died about a month after it pulled my paper after publication after some math people sent a bunch of emails claiming it was wrong.

No other journal would touch that paper and a physicist I had shopping it around told me he got the answer back that they just weren't going to touch it, not that it was wrong.

I personally sent the paper finally to the Annals of Mathematics at Princeton, and was never told it was wrong, or anything else for months. After six months I contacted them asking what was its status and was told the database had been noted that a rejection had been sent by email. I never received a rejection. I asked if I could be told why, and was told that was all the info in the database, so no reviewers report.

I have recently greatly simplified my presentation of the problem, and written another paper. It can be found at a math group I recently created:

(Hope that works if it doesn't, well, um, hope it works.)

That paper is now at the Bulletin of the AMS.

If they reject it will go to the Annals.

If they reject I will work my way down and work journals around the world as this time I am very serious.

If necessary later I will pursue prosecutions for academic fraud and have no problems with pushing government agencies to issue subpoenas for people at Princeton or wherever.

I have no respect for any institution in the world. I have no problem with having any academic at any level brought before a jury.

So you grad students know you have no protection here. I may need any number of you to testify about what your professors were saying.

But what's more interesting to me is this weird behavior from academics who should be terrified about not only losing their careers by ignoring this result, but dealing with massive humiliation along the way, so it occurs to me that they do not believe that proof will matter!

Like I can talk about Barry Mazur at Harvard as a little twerp who got a look at my earlier paper, but so far to my knowledge has done nothing.

Of course there is no way if the result gets accepted I would just sit still about him remaining at his position, so he never believed that it would be accepted. And he will not be at Harvard much longer. I guarantee it.

I can go on and on.

There is Andrew Granville who infamously told me his grad students often found something like my prime counting function, who also looked over the paper.

Kenneth Ribet is safe as he never commented on my work and actually helped me a while back with a stupid bet I made (and lost).

I can rip through a Who's Who of academics at institutions around the world.

So, then, they never expected anyone to accept mathematical proof.

Now isn't that odd?

What do they think they know that I don't.

Some of you have been remarkably arrogant as well, especially considering the massive hits you are going to take in the public eye as it will be very public and none of you will be anonymous any longer.

Your faces will be of interest as will be your living circumstances, especially certain ones of you like "Uncle Al" and "mensanator" who seems to have the delusion that he can hide when the world is actually looking for him.

This issue is of great importance to me as I've been very curious about why people would act against their own self-interests on such a scale when the end was not really in doubt.

It has always been just a matter of time.
So there finally is the last mystery.

Over the last six years I have worked through possibilities and closed doors to other answers so that the final mystery is: why would academics with everything to lose behave as if mathematical proof wouldn't matter?

Simplest answer is, they don't believe in it.

That would indicate a notion that academics actually only work to convince people of what is true, not to find what is actually true, so if the belief is that the weight of thousands of established mathematicians around the world was all that mattered then it would make sense that one man couldn't emerge victorious in that situation—if you thought it was just about one man's word against thousands of others.

To me that is remarkable and well worth the time to make certain that that judgment is the correct one.

As make no mistake, at the end of this situation, probably early in 2009, quite a few established people will I'm increasingly certain no longer be in their current positions with not only a loss of status but also of tenure, pensions, etc., and may face governments requesting their money back.

Who would risk that over some math if they thought it was possible?

Ergo, they don't believe in their own system. Don't believe in formal peer review or any other kind of review. Don't believe in grad students or even undergrads as people who cared primarily about the truth.

It follows that these people cynically do not believe in the world's academic system at all.

I consider them through their actions to be the most damning witnesses against the modern academic structure—worldwide.

After all, these are MATHEMATICIANS. If they could hold their own for over six years with the current setup and act as if they'd never get caught, then why should people trust, say, English Literature professors?

I consider the case closed. The only thing left in getting the mystery solidly ended is the testimony of these people and of the people around them especially their grad students:

What have they been doing these last six years as they dove into a world of fraud and academic pretend?

## JSH: Simple math but bad math habits

It is hard to hear that you have been taught wrong, and harder still to confront dogma, because "mathematical proof" is just a phrase for most of you as you've never been put in a situation where you really did not wish to accept a result, so you avoid the mathematical proof but by rationalizing continue to believe yourselves to be mathematicians or real students of mathematics, when you no longer
are.

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

I come up #2 now in most venues.

Physics students should do better. Physicists know about resistance to results and hard to understand results which challenge accepted views, but the field has softened because of the dominance of mathematical techniques, so people who are really mathematicians get to pretend to be physicists, when they are not, and there is a worship as well of authority, so that when the math people say false physics students follow along because, what else can they do? Resist authority? But, but, but…how can they?

All of that is a preamble for one of the simplest most powerful mathematical arguments in the history of human civilization which mathematicians have resisted for over 6 years now despite how easy it is to prove.

It only requires you accept the distributive property and believe that proof is, well, proof.

The distributive property is: a*(b + c) = a*b + a*c

In the complex plane with

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

you normalize:

a_1(x) = b_1(x), and a_2(x) = b_2(x) - 1, as then you have

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

where b_1(0) = 0, b_2(0) = 0.

And you have one factorization out of infinity and in THAT factorization the 7 has distributed in one way, which is easily verifiable at x=0, as then you it distributed through

5b_1(0) + 7,

as that equals 7(0 + 1), so with the distributive property you have a=7, b=0, c=1.

EASY. But remember 6 years of mathematicians arguing against this result!!!

Now if we consider that a*(b+c) = a*b + a*c, is true if one of the elements is a function then I have

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

and the TYPE of the function does NOT MATTER; however, that challenges current mathematical intuition, so while with something like

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

math people found it hard to dispute the distributive property, hide the 7 away with:

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

and they disputed the result as they don't want to believe that with

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

only one root should have 7 as a factor as that's what the distribution shows.

(Remember the factorization is normalized and

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

Now in case you forgot your algebra classes, it is NOT taught in them that only one of the roots of

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

in general should have 7 as a factor. You can verify one case easily enough, x=0:

a^2 + a = 0

as one root is 0, and 0, of course has 7 as a factor. But use x=1, and you get

a^2 - 6a + 35 = 0

and only one of its roots actually has 7 as a factor, but you can prove in the ring of algebraic integers that NEITHER root has 7 as a factor as neither root does—IN THAT RING.

The ring gives bad results. Techniques based on it give wrong results.

So you have an advancement of human knowledge: it's now possible in this case to determine 7 is a factor of a root without being able to see it directly, where it's also not determined which root.

There is an ambiguity which cannot be removed which means the solutions are paired or entangled in a way that cannot be beaten. You must take them by two's. With quarks you must take them by three's.

Which may indicate some cubic function which controls some aspect of quark behavior, which cannot be probed without advanced analytical tools meaning some aspect of quark behavior may be beyond the vision of humanity with currently accepted mathematical tools, but possible to analytically study with the ones I've introduced.

For perspective I used some of those tools to also generally solve binary quadratic Diophantine equations.

Don't know what those are? Google it. I think I come up #1.

So there may be a ceiling on what humanity can do in physics at this point because it is not using the full mathematical knowledge available, where mathematicians have been resisting this result as it overturns past beliefs, and they wrongly believe their field is immune from revolutionary upheavals.

Their belief is holding back the scientific progress of our species.

And THAT is how you go from a very simple quadratic argument which requires that you only believe the distributive property to understanding how math people could fight for belief for over 6 years against mathematical proof, and hurt the physics community and the progress of the entire human species in the process.

## JSH: So why all the arguing?

The situation is critical so I need to give you a synopsis so you understand why, and understand the weird arguments.

You may have noticed me talking a lot about a trivial example:

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

where it's clear how I multiplied by 7. That's trivial, so why would anyone argue about it?

Well, I one-upped the trivial with something more complicated:

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.

So I switched from linear to non-linear functions. That picture is somewhat muddled so I made one more minor step, which is to normalize with a_1(x) = b_1(x), and a_2(x) = b_2(x) - 1, so that the b's equal 0 when x=0, and I have

7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5a_2(x)+ 2)

and if

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

is a guide--after all I only switched from linear functions to non-linear ones—then one would suppose that I picked 7 to multiply the same as before but that implies that with

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

one of the roots is a product of 7 and some other unknown numbers, which is a HUGE shift in mathematical thinking, as if you plug in numbers like x=1, then that means that with

a^2 - 6a + 35 = 0

only one of the roots one would suppose actually has 7 as a factor, but by standard mathematical teaching NEITHER of the roots can possibly have 7 as a factor in what is called by mathematicians the ring of algebraic integers, as the belief is that the 7 splits up in some way, and mathematicians can make calculations to GIVE you that split, so it's an odd situation.

To hold on to their current mathematical beliefs, posters are arguing that the 7 can shift around depending on the value of x, while I'm arguing that how the 7 distributes is independent of what is being multiplied and the argument is mostly the same as it has been for about six years or so, except at times (recently is the second time) I've pointed out that my distribution argument holds true on the complex plane, as it's not really a factor argument, so I've tried to hold math people to the complex plane.

Now all that may make you wonder, what's the big deal?

Well, I can explain ALL of their current mathematical teaching and include my distribution argument, and explain why they think the 7 splits up and think they can give you evidence that it splits up, but they can't answer my distribution argument, which is why I emphasize

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

as it's simple enough that you can know intuitively that I'm right, but then of course, the idea of authority figures kicks in, and I'm sure most of you suppose that I just can't be completely right because, well, for instance, if a math error entered over a hundred years ago, wouldn't somebody notice?

And what about the massive success of our current science and technology using mathematics?

Well I have a degree in physics so I know that most of the mathematical underpinnings of modern science and technology actually existed long before the error was introduced. Also the error is in number theory where mathematicians often say their research is "pure" as in, it has no practical application.

That is not a relief though as in, none of it matters, as some of this research actually seems related to quantum physics, so we may not be able to do certain things or explain certain things, say, in quantum chromodynamics without the correct number theory, as number theory not being a bigger part of modern physics may simply be because of the error.

This error while it can impact the explanations of some current theory doesn't impact the predictive value from what I've gathered of any working modern theory, while it may completely shoot down "string theory" which is kind of infamous for lacking predictive value.

Given the stature of the modern math field and many of its practitioners and their refusal to follow their own rules regarding acceptance of mathematical proof, it's not clear if the error will be acknowledged any time soon, but that is not really a win for them.

Given our history as a species eventually the tide will turn, the error will be acknowledged and over one hundred years of mathematical work will be re-thought. A lot of big names in the math field may turn into much lesser names, or even simply be considered to be examples of people who never discovered anything of value but were only thought to have done so.

I strongly suspect that it will be the largest upheaval in the intellectual history of the human race. And it is inevitable, but it's not clear when it will occur.

In the meantime, the arguing continues. Math people with YEARS of their lives invested in false information are holding on to it, and society is rewarding them as usual, so they have validation in that way, but it's just a delay of the inevitable when society will turn. From what I've gathered though, they're hanging on for every day with the error as a precious gift. Sadly enough, but if you understand all the issues, I guess it's understandable for people who may, if they acknowledge the truth, wake up to a world where all their "great" "accomplishments" are gone.

For many of them, their faith in society in terms of mathematical knowledge was simply not realized in what they were taught at even the best institutions.

It is clearly a case of a massive it's not fair situation.

The weight of history is against them, but for now they are considered geniuses, leading researchers, and "brilliant minds".

Losing that fantasy is the last thing they want to do.

## JSH: Difficult problem, as you can see

The problem with showing this huge error isn't that it's hard to prove it exists; it's that it's a huge error so it grabs just about everyone in the mathematical field along with people who today probably think they are top theoretical physicists, so a lot of egos.

So far there is one dead math journal to show you how far they can go.

Nothing is working so far. Math people are simply throwing out all process. Refusing to accept reason, and going about their business as usual—taking government grants, giving each other prizes, and teaching new students the error.

To them every day continuing with business as usual is a victory.

So far they've managed over 6 years.

Ok. I've given you lots of data. It's a hard problem but the future of the human species depends on resolving it.

Without the correct mathematics, there is no future. Right now, most of you do not know the correct mathematics because no one is teaching it, but me.

## JSH: A devastating error

It has taken me years to work through the full mathematical explanation of the bizarre error that took hold in number theory over a hundred years ago, but it's going to take years more for the mathematical and scientific communities to come to grips with the devastation it has wreaked—which can only start when they accept that
it exists.

I think, for instance, you can safely toss "string theory" entire. As in, forget about it.

But losing Galois Theory might bite a bit harder especially when you figure out how you lose it, as it's not exactly wrong.

To understand why considering some rational examples, don't focus on doing what comes natural:

x^2 + 4x + 3 = 0, solves with the quadratic formula as x = (-4 +/- sqrt(4))/2

and you're probably wondering why I don't simplify the square root, but bear with me, as consider

x^2 + 5x + 6 = 0, solves with the quadratic formula as x = (-5 +/- sqrt(1))/2

and if you do not resolve the square roots you can employ Galois Theory on those results and do the class number thing and everything else and convince yourself that you're doing something mathematically important, when you are not.

Notice that Galois Theory cannot tell you something as simple as: both cases have 3 itself as a factor!!!

A lot of mathematical tools are built around not being able to resolve the square roots where mathematicians were in error, as it's like needing to fly with your instruments.

Unable to physically SEE the roots like with rationals, you can still logically determine things about them using analytical tools, which I've done and demonstrated with a simple quadratic construction.

And notice the arguments in that thread!!!

The math community has a LOT invested with the flawed math, but the physics community in certain areas got dragged into the mess as well, which will be most devastating for theoretical physicists but may affect you all in terms of, entire experimental directions that have no worth.

Resolutions to long standing physics problems that may be extremely simple, but destroy the value of years of efforts in certain areas.

I don't want to hint as I'm not certain in every respect, but the very way we look at our physical world will probably shift as a result of correcting the flaw.

The good news is that a lot of mathematics in physics will actually get easier! My work simplifies HUGE areas of mathematics.

But there is more bad news.

Increasingly I'm concerned about a "lost generation" of physicists who are so thoroughly trained in a certain way of thinking and so along in their careers that they will find it difficult to impossible to start over, as yes, in many areas, the impact of the error will be, starting over.

For those of you with the juice left in you for it, it should be exciting though.

You have an opportunity to learn some of the greatest secrets of our physical world which were so close before in a way, but impossible to SEE with flawed mathematical tools.

In some ways it may be the beginning of modern physics as far as the future is concerned.

When humanity finally grew up in its understanding of mathematics.

## JSH: Was never a factor argument

Now isn't that amazing, I go to the complex plane and can give an example like

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

and posters who arrogantly argued with me for years, convincing untold numbers of people that I was wrong here fall apart just because I'm forcing the issue that the equations are in the field of complex numbers.

But factor arguments don't matter on the complex plane!!!

Yup. You're right, as I never had a factor argument. I've had a distribution argument.

So you can see how the 7 distributes through the factorization of x^2 + 3x + 2. That is about the distributive property.

Now going to a slightly more complex example STILL on the complex plane

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.

is actually in all key respects the same as the factorization above!!!

Normalizing using a_1(x) = b_1(x), and a_2(x) = b_2(x) - 1, you get

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

where b_1(0) = 0, b_2(0) = 0, which looks a lot more like

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

and it's still a DISTRIBUTION argument and I remind you are in the FIELD of complex numbers!!!

Without factor arguments available, the once derisive posters who arrogantly told you for years that I was wrong have no mathematical tools to distract, have no "counter examples", and in fact have nothing at all but their basic denial.

They were wrong.

They are wrong.

If mathematical society is tired of the field of complex numbers then fine, you keep acting like you have now for six years, but don't pretend that you are actually mathematicians or actually doing valid mathematics.

You are practicing your religion. And you don't give a damn about what is actually true.

## JSH: Questioning Galois Theory

One of the more remarkable things I have is a weirdly simple result on the complex plane which brings into question Galois Theory, which is just such a huge thing that it's hard to surmount disbelief despite the simplicity of the proof.

The way it works is I figured out this inventive way to factor a polynomial in the complex plane:

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.

It may seem simple enough but it's complexities have sparked arguments for YEARS as in literally years on math newsgroups, so I want to step back a bit and show you another more traditional factorization:

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

which is trivial, but is actually in all key respects the same as the factorization above!!!

That is where the issues start as, of course, people disagreeing with this result would attack that claim and besides, you can see above that there are two 7's on the right hand side, not one, with my weird, inventive factorization.

But, normalize and you get a different picture as using a_1(x) = b_1(x), and a_2(x) = b_2(x) - 1, you get

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

where b_1(0) = 0, b_2(0) = 0, which looks a lot more like

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

except, wait a minute, the 7 is obscured by the function b_1(x), while it is visible with the linear function, and that's it.

So yes, essentially they are the SAME in terms of you have 7 multiplied times one factor of a factorization, where in one case that 7 is obscured by a function, which is non-linear, while it is visible with the second case where you have a linear function.

So in a way, all the arguing over the years has been an issue about non-linear versus linear functions, as by the distributive property:

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

doesn't care about the TYPE of function f(x). But if you accept all of the above there is no argument, you now have that with

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

one of the roots should always have 7 as a factor as only one of the functions actually is multiplied by 7, and that's where things get dicey for Galois Theory.

Too big of a leap? Well let's step back a bit and stick in some numbers!

Let x=1, then you have a^2 - 6a + 35 = 0, which you can solve using the quadratic formula to get

a = (6 +/- sqrt(-104))/2 = 3 +/- sqrt(-26)

and I don't see 7 in there at all! It's obscured by the square root. Here's an example of that with easy numbers:

a^2 + 4a + 3 = 0, so

a = (-4 +/- sqrt(4))/2

and I don't see a factor of 3 in there—unless I resolve the square root.

But the problem with 3 +/- sqrt(-26), is that you CANNOT resolve the square root, so you CANNOT see the 7, which we just proved on the complex plane MUST be there.

So you have mathematical proof as an instrument, like for a pilot flying at night—you cannot see the 7 directly but you proved it's there on the complex plane so you know logically it HAS to be there—so in a way you're flying at night, like a pilot, needing to rely on his mathematical instruments.

But here's where things get really messy, as mathematicians well-trained and taught in over a hundred years of number theory will tell you, if you tell them there's a 7 in there, that you are wrong, and they will tell you they can prove you are wrong!!!

So what gives? I just stepped through a simple argument showing how it was just linear functions versus non-linear where just being non-linear doesn't change the distributive property so we know a 7 is there, but now I'm telling you that the experts in the field of mathematics will tell you it's not there, and say they mathematically prove it's not there, and oh yeah, what does this
have to do with Galois Theory anyway?

Well, if you understand how the experts in the mathematical field are wrong—like, hey, you can trivially prove on the complex plane they are wrong—and understand WHY they are wrong, you come across the problem that they built Galois Theory on those reasons why they are wrong!

Now they are fully invested in their beliefs!!!

You're talking about prizewinning theorists and top-ranked academics who work in highly prestigious institutions with all kinds of authority and social power to tell you that no matter how well you THINK you know the complex field, or how logical the argument above about linear functions are not that different from non-linear ones when you multiply them by 7 sounds to you, it's all crap because they're the experts and they have a hundred years of edifice built on it being crap.

So they will tell you you are wrong.

## JSH: So what are you made of?

Ok, I found a way to show an apparent direct contradiction with standard teaching of Galois Theory.

Some of you seem to think you are mathematicians.

Let's see what you're really made of by what you do in the next few minutes, hours and days.

If you're not really mathematicians then you will betray what you truly are.

Mathematics is a hard discipline.

So of course there are wannabe's who wish they were something they cannot be.

But pretend is over. Those of you who do not make the cut will be asked to leave the field.

## JSH: Breaking Galois Theory

In the ring of algebraic integers consider the special construction:

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

with x=1+sqrt(-6) which is chosen because it is a factor of 7, as (1+sqrt(-6))(1-sqrt(-6)) = 7.

Notice that

7*(175x^2 - 15x + 2) has 7(1-sqrt(-6)) as a factor

and

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

where a_1(x)*a_2(x) has 7(1+sqrt(-6)) as a factor, but a_1(x) is coprime to a_2(x), because

a_1(x) + a_2(x) = 7x-1

as I remind that the a's are roots of a^2 - (7x-1)a + (49x^2 - 14x) = 0.

There are multiple apparent contradictions with this example.

If you divide 7 from both sides, then 1-sqrt(-6) remains a factor of both sides, but then factors of 7 that remain for BOTH a's are coprime to it, while also being coprime to each other, but in each case with

(5a_1(x) + 7) and (5a_2(x)+ 7)

the 7 on the right in each the result after dividing 7 from both sides must be coprime to the 'a' opposite it, but then in at least one case what remains must have factors in common with 1-sqrt(-6).

Those attacking the validity of this example as a counter-example to standard Galois Theory should give the factors of the a's in common with 7.

Oh, the resolution? It's easy. There is no actual contradiction.

## TSP algorithm, using hidden variable solution

I came up with an approach to the Traveling Salesman Problem by imagining two travelers—one the traditional traveler moving forward in time long the optimum path, and a second backwards traveler moving backwards through time through the optimal path.

The two travelers meet and the solution collapses to a single traveler moving forward in time along the optimum path, while the algorithm works by minimizing the total weighted distance between the two travelers at each step.

I went ahead and generalized beyond the traditional TSP problem to a general solution for node and a traveler which allows moving to one node more than once, allowing what I call hubs, and nodes not reachable from all nodes.

The complete algorithm follows.

Given two travelers T_1 and T_2 and m nodes N, where m is a natural number, each traveler can be at a particular node, and each node has a distance from every other node in the space. However, it is not assumed that each node is reachable from every other node, but that at least one path exists between each node and one other node.

If no distance information is given then the travelers are assumed to be in a m-1 dimensional space with each node the same unit distance from every other node.

There is also a weight associated with the path from each node to another, which in general is considered to be cost. But it can be that or as many things as desired.

For instance, if it takes \$200 U.S. to go from N_1 to N_2, then that cost is what's used for a traveler at N_1 considering going to N_2.

Assume T_1 is at N_1 and T_2 is at N_m. For the first iteration T_1 considers moving to N_2 which is reachable, and then T_2 considers moving from each reachable node in turn EXCEPT N_1 or N_2 as it excludes the node that T_1 is already at, and also excludes the one being considered.

For each potential move T_2 calculates the cost from that node to N_m, as T_2 is moving backwards in time, which I'll call cost_2, and then it calculates the straight line distance (a unit for the distance normalized algorithm) to T_1 at N_2--not at N_1, as it is checking where T_1 will be—and it multiplies the cost for that move times the cost of the move T_1 is considering, which I'll call cost_1, times the distance to T_1, and stores that data.

After T_2 has gone through every possible move, it simply takes the one that is smallest cost_1*cost_2*distance, and stores that info. If there are more weights, like time, it'd multiply time_1*time_2*cost_1*cost_2*distance, and so forth with as many multiplications as there are weights.

Now T_1 considers moving to another reachable node, like N_3 if there is a path from it current node to that one, and T_2 does the same calculation again, and stores the smallest cost_1*cost_2*distance.

This process continues until all possible moves by T_1 are done, and now T_1 has a set of cost_1*cost_2*distance values from the calculations by T_2, and selects the smallest one. If there is more than one path with the least value then it doesn't matter which is taken, so the first can be taken.

Now both T_1 and T_2 actually move to their respective nodes given by that smallest value.

Now the paths from nodes N_1 and N_m to the chosen nodes in the direction moved is removed from consideration. However, the reverse path for each is still available.

And you have a complete iteration.

Note that you have a maximum (m-2)^2 checks, so the algorithm is polynomial time.

If the travelers start at different nodes and every node is reachable from every other node, but you still have nearly the same number after the first iteration as now only paths are being removed and not nodes, but the algorithm is still polynomial time.

T_1 and T_2 continue until there are no more paths available available or there is only one node available.

Note that in the latter case then EITHER can move to complete, so you can arbitrarily say that T_1 moves forward in that case.

And that is how every node is reached, even if some nodes are not reachable from certain other nodes.

To have a complete circle back to a starting point, you have both T_1 and T_2 start at the same node, like N_1.

The cost for a round trip path with a starting point from each node in turn is stored with the optimum path being the one with the minimum total cost.

Notice with this algorithm you may go to a node multiple times, and a well-visited node is called a hub.

If the optimum path from each node is the same then the graph is said to be perfectly correlated, and corresponds with a case where the costs correlate very well with the distance between nodes.

If only one path is optimal then the graph is said to be dis-correlated, and that corresponds to a case where costs have little or no relation to distances between nodes.

The percentage correlation can be found then by dividing the number of starting nodes that give an optimal path by the total number of nodes, where a graph for which there is a high percentage correlation is said to be a rational graph, where it seems to me that a 67% correlation or higher would qualify.

That could have real world relevance for determining, say, whether or not prices along paths correlate well between measures along the path, or it might reveal problem spots, where, for instance, there is a high cost associated with an otherwise desirable path, which, for instance, might help a city plan places for new roads.

## JSH: Probably about the degree

I think that the reason physicists side with mathematicians—even just quietly doing nothing if you've been reading is a defense of them as you know what that does, it gives them freedom to continue as they have—is sympathizing with the fear: what if, what if the very basis for your Ph.D is bogus? Is it still valid? What would you do if society said it weren't? Work flipping burgers?

And maybe physicists feel like it's the mathematician's problem, their field, their issue, their over one hundred year old quirky error, so why should physicists be bothered?

Because I am not actually going to lose here, and when it's done that academic wall of silence will come crashing down, and when it's your job and your funding that is being questioned your perspective will change.

What I'm doing here is trying to shift that perspective now, ahead of what you cannot see happening.

As you will not prepare for what you refuse to see.

Like many in the United States didn't prepare for the current down-turn. A down-turn some at least partly blame on physicists who reportedly build a lot of the mathematical systems that the finance people were using.

No matter what, academics are going to take a massive prestige hit. And it will be MASSIVE. But how much that hit affects any one of you individually is about the choices you make now.

I warned, vaguely, in multiple ways about the challenges that would be facing the world economy, though I was worried I'd precipitate it personally, which is a fear that has been allayed, but still I DID warn.

What you do not know about reality is what can change your life.

Many of you may have done research on major discoverers or read about them.

Maybe even imagined what it would be like to be around one, or even to confront one, to test yourself against one.

But you have no idea.

What I can tell you I do is, every day I sit and I try to imagine all people on planet earth, and what they are doing. All of them. And then I run models for what I think will happen next.

Futile effort that may be, but it keeps my mind occupied, somewhat. You see, I look at trendlines, and the larger the group of people the more they seem to trend, and I directly interfere as well.

I'm currently somewhat occupied with the idea of major radio stations with their audio libraries being warehouses accessed by at-home DJ's, which seems like a trivial idea to me.

I've projected the demise of satellite radio after that stupid merger of Sirius and XM radio.

I have defined Google itself and the new media market. And gave Yahoo! business advice which they seem to be only grudgingly taking because they're not smart enough to do otherwise.

And I project 50% to 75% decreases in public funding for research almost across the board so most of you will not be getting public funding as physicists anyway within the next five years regardless of your current status, even Nobel prize winners.

You will be out on the street.

I'm not really asking anything of you with these posts.

I'm working through my guilt before the fact. That is all.

I'm convincing my future self that I did everything that I could with people who at the end of it all, simply were not smart enough to see the train coming, as they lay on the tracks.

Who wouldn't move no matter how loudly I yelled to try and save them.

## JSH: Why algebraic integers do not really work

Before I came up with my special construction any one of you if presented with the following would call it easily:

7*P(x) = (f_1(x) + 7)(f_2(x) + 2), where f_1(0) = f_2(0) = 0

I'm sure you'd call it trivial as well that f_1(x) is the product of some unknown function times 7, that is

f_1(x) = 7*g_1(x)

so you must have

7*P(x) = (7g_1(x) + 7)(f_2(x) + 2), where g_1(0) = f_1(0) = f_2(0) = 0

by the distributive property and then you would probably angrily ask why you were being bothered with a triviality without even knowing exactly what P(x) is.

But I came up with a special construction and you're asked to throw all that out the window to believe that suddenly the 7 behaves bizarrely:

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.

And all because NOW if you think it's simple you have to believe that only one of the root of

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

can have 7 as a factor and mathematicians will tell you that you're wrong, and that NEITHER of the roots can have 7 as a factor, and you will be overwhelmed by the authority figures who are the world's presumed experts in this area, but why do they make that claim?

Turns out they are relying on mathematics associated with what they call the ring of algebraic integers and you can prove that neither of the roots have 7 as a factor in that ring.

But watch an interesting example with integers where I'll show you how their arguments work, except here you'll be able to see what is happening directly:

x^2 + 3x + 2 = 0

but let's say only ONE of the roots has 2 as a factor (of course only one does), and substitute with x=2y, then

4y^2 + 6x + 2 = 0, so 2y^2 + 3x + 1 = 0, and aha! You have a non-monic polynomial!!!!!

Of course you now have roots of -1 and -1/2 because you divided them BOTH by 2 with your substitution.

Well the ring of algebraic integers has a very special feature which is that NONE of its members can be the roots of a non-monic polynomial with integer coefficients that is not reducible over rationals. (That is VERY important, consider it carefully as everything revolves around this point!)

That feature it can be proven results from the definition of algebraic integers as roots of monic polynomials with integer coefficients.

You can say the ring of algebraic integers has a monic prejudice. It prefers polynomials that have a leading coefficient of 1 or -1, which is what monic means.

And THAT is the basis for why mathematicians will tell you in general you cannot say that only one of the roots of

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

has 7 as a factor. At x=0, you can see that's not true as one root is 0, and 0 has everything as a factor, but at x=1, you end up with a = 3+/-sqrt(-26) and you cannot resolve that square root, and if you can prove that in the ring of algebraic integers neither root can have 7 as a factor—because if one did then an algebraic integer would be the root of a non-monic polynomial with integer coefficients irreducible over rationals!!!

It's circular. The definition creates the requirement that reinforces the definition.

And it's wrong—one of the roots DOES have 7 as a factor, trivially. So the ring of algebraic integers is wrong. But the ring of algebraic integers was invented in the late 1800's.

Get it then? Incredibly the mistake in thinking occurred over a hundred years ago!

So there is a lot of inertia behind the wrong result, so what you thought of as obvious before my research as I demonstrated at the start—which is correct—is challenged by the people the world has designated as experts because they have over a hundred years of practitioners in their field believing something that turns out to be wrong.

While what they believe is based on something circular: where a definition creates a rule, which reinforces the definition that spawned it.

Now if mathematicians could be convinced to just follow mathematical proof then they'd realize their definition creates problems, but instead they've chosen to mostly ignore me while you can see some math people arguing with me, continually dodging the simple reality that with something like

7*P(x) = (f_1(x) + 7)(f_2(x) + 2), where f_1(0) = f_2(0) = 0

the thing being multiplied by 7 cannot tell the 7 where to go. The tail does not wag the dog!!!

Worse, once you know the problem you can figure out that a lot of what number theorists do must be wrong but enough of it is wrong that it can challenge the worth of Ph.D's themselves, as in, if a professor's research over decades is completely invalidated by a quirky math error, did he really do anything to deserve the Ph.D? Or the social standing based on the research?

Thorny question.

If a quirky math error invalidates the entire thesis of a professor with a Ph.D on which he got his Ph.D and it turns out that all of his research over his career is now known to be wrong, is he still really a professor?

Does he still have a Ph.D?

I'd guess that enough math people fear that question that the impasse continues and they challenge very basic algebra to hold on to a world where those questions aren't being seriously asked, at least, not by anyone not maligned globally as a "crackpot".

## JSH: Then consider this example

It seems that I haven't convinced with my talk of the distributive property so here is an example which should test your understanding of Galois Theory to the limit.

In the ring of algebraic integers, let x=1+sqrt(-6), with

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.

I picked x so that it is a factor of 7, as (1+sqrt(-6))(1-sqrt(-6)) = 7. Also notice that despite x having a factor in common with 7:

7*(175x^2 - 15x + 2)

still only has 7 as a factor.

Further note that a_1(x)*a_2(x) has 7(1+sqrt(-6)) as a factor, which is the point, as if the a's do not share factors in common with 7 carefully, then you contradict with only 7 being a factor of

(5a_1(x) + 7)(5a_2(x)+ 7).

Give the a's and solve for the factors in common with the a's for this value of x.

Have fun! Preserve your deluded view of Galois Theory—if you can.

The challenge is in front of you, are any of you good enough? Smart enough?

## JSH: And that is the cycle

I went ahead and demonstrated what I've faced for years from the math community where I say one thing, explain carefully, and notice they tag-team dodging what I actually say, making up things I don't say but claiming I do, or simply tossing up non-sequiturs until it's all a muddled mess, no matter how much I repeat, and carefully explain it.

Then when I give up the pointless arguing, they celebrate and claim victory.

But here I gave a very simple and direct construction, put it all in the complex plane, and explained a mathematical proof that relies on its key part on the distributive property. I carefully explained how it relies on the distributive property, and gave simple examples elucidating the proof.

And they would delete out the math, the explanations, or babble on incoherently and claim it was me doing so, and then at the end, claim victory, when I got tired of the exercise.

But the point was for you to see what I'm facing: the math community has gone rogue.

They cannot be convinced by mathematical proof alone. They are beyond reason.

## JSH: So of course I'm right

It's as simple and direct a demonstration of a major result that you can probably get mathematically, but the problem here is that something is happening that's not supposed to be possible: given various events I'm increasingly certain that leading academics around the world are trying to hide a major result in their own field.

They killed a math journal. Have kept accepting public funds for bogus research. And most tragically I think, have kept teaching new students the flawed techniques, assigning them homework, testing them.

So what is the flaw in how posters present how to look at the argument, well I have to show the special construction again to explain it, and remember, in the field of complex numbers, with

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

where b_1(x) = a_1(x) and b_2(x) = a_2(x) + 1,

which is the normalization, where the a's are roots of

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

What math people claim is that the 7 moves around dependent on the value of x. But if

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

then why does the function change what is multiplying times it? It cannot.

Trivially the equivalent can be seen with a classical factorization (the one above is what I call a non-polynomial factorization):

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

and if you think this math is basic, yes, you are right. It is basic math. There is no way the 7 can bounce around because what is being multiplied changed. You may say, too simple! Those are weird funky roots of some quadratic functions that can be weird bizarre mathematical beasties!

But the distributive property doesn't change no matter how weird the math beastie, right?

a*(weird_math_beastie + b) = a*weird_math_beastie + a*b

Notice that by shifting to factor arguments posters try to convince you to defy the distributive property with the notion that the function is changing things because the function can behave differently based on different values, but consider

7*f(x) + 7 = x + 7

here f(x) = x/7, which is hidden to some extent, but so what? Still doesn't change the distributive property!!!

If a*(f(x) + b) = a*f(x) + a*b then I'm right.

So why do they lie? Because they aren't decent people that's why.

You've met unsavory academics I'm sure. Here is just a critical mass of a LOT of them running things in mathematics and just lying, taking money, and teaching trusting students, crap.

Watching how they shift their behavior over the years I am increasingly certain they read the newsgroups trying to see if the lie can hold so you DO have a role to play. The insults and endless debates over trivial algebra give these people, if I'm correct, calculated comfort—they believe then they can keep lying indefinitely, keep teaching students lies indefinitely, keep getting public funds for bogus research indefinitely.

The newsgroups are their comfort, I fear. I wouldn't be surprised if "top mathematicians" are reading these threads: people many of you might see as heroes, coldly checking to see if they can keep getting money for nothing and lying to students.

Using you.

## JSH: Kind of weird, eh?

So I have an extremely easy proof of this extraordinary somewhat subtle error that entered the mathematical field in the late 1800's, which I can dramatically prove with a simple construction:

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

where b_1(x) = a_1(x) and b_2(x) = a_2(x) + 1,

which is the normalization, where the a's are roots of

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

You may think the problem is arguing it out on Usenet, but there are years of history here. I had a more complex argument using cubics, which was still understandable to a trained mathematician which was published in the now dead mathematical journal SWJPAM. I explained the key result in that paper in person to a Ralph McKenzie who is a leading mathematician at my alma mater Vanderbilt University as well as being on the faculty of the University of California at Berkeley.

He just did nothing. When I'd explained it to him, thinking that would be it, and he said it was time for him to go, I was just kind of shocked.

I say their minds snap. It's weird too. They just kind of fade out or something. Or go on sabbatical.

Maybe mathematics for a lot of mathematicians is like a religion. For me to prove a problem is one thing, but it's kind of like trying to prove Jesus was a fictional character to a devout fundamentalist Christian.

Evidence will not matter against belief. Here absolute rigorous mathematical proof does not matter to mathematicians because their belief system is built around the false ideas, like someone who's life experience and focus are built around Jesus as a real person.

For them there is no debate then. No choice. It's about faith.

But the consequences for mathematics are fairly huge. I pulled the thread so to speak and watch Galois Theory mostly go away, but found what I call the object ring, which vastly simplifies huge stretches of number theory.

Allowing me to prove Fermat's Last Theorem, advance mathematical analysis, and recently generally solve binary quadratic Diophantine equations, as just some of what the advance allows.

Quite simply, number theory entire had to be re-worked. As part of the task I also had to re-work some logic, and re-work set theory.

You do not know the mathematics you need to advance physics much further than it is today.

I know you don't, because I do.

## JSH: Really depressing

So yeah turns out I've explained my major mathematical find using the field of complex numbers before.

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.

Any person with even moderate mathematical ability can look at that system in the field of complex numbers and rigorously prove that 175x^2 - 15x + 2 is being factored into two elements only one of which is being multiplied by 7.

Normalizing makes it trivial to the extreme:

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

where b_1(x) = a_1(x) and b_2(x) = a_2(x) + 1, which forces b_1(0) = b_2(0) = 0, which is the normalization, where again the a's are roots of

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

Then one of those elements is 5b_2(x) + 2 while the other has been multiplied by 7, easily verified with absolute rigor at x=0.

I mention that I HAVE explained this result in the complex plane before so that those of you clinging on to that sappy love of the inner goodness of all human beings which explains why you still trust the mathematical establishment can accept that math people have DELIBERATELY clung to error for years now, despite explanations that are trivial to the extreme.

They are willfully wrong.

The problem is that I found a clever way to prove that it should be the case that only one of the roots of

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

should have 7 as a factor as provably IN THE FIELD OF COMPLEX NUMBERS only one of the roots was actually multiplied by 7. Since the other root was multiplied by nothing, why should it have factors in common with 7?

But provably it DOES in the ring of algebraic integers, so you have an apparent contradiction, easily shown, to even novices in mathematics.

It's really depressing that there is no one in the world to champion mathematical truth besides me. That with all the supposed brilliance of our modern era, such stupid lies can go on for years, when even a moderately bright mathematical student can verify that something MASSIVE must be wrong.

What's wrong with the world? Where are the smart people who care about such things?

## Congruence based factoring algorithm, revised

An initial presentation of a prior version of this algorithm had a fatal error. The underlying equations are the same but I was using them wrong. Here is the corrected algorithm.

It defies conventional wisdom by showing a congruence based factoring method that allows you to pick a large enough prime p, and solve for f_1 and f_2, where f_1*f_2 = T, where T is the target composite to be factored mod p.

With all positive integers, given a target composite T to be factored, first find a prime number p greater than sqrt(T). Then set a variable I call 'a', to a=2 and calculate a variable I call k, with the following congruence relationship:

k^2 = (1 + a^2)^{-1}(T) mod p.

where if it does not exist, you increment 'a' by 1 and try again. There is a 50% chance that any particular 'a' will work as there are (p-1)/2 quadratic residues modulo p.

If you get a k then you check for a solution for f_1 and f_2,where

f_1*f_2 = T

with

f_1 = ak mod p and f_2 = a^{-1}(1 + a^2)k mod p.

If ak is greater than p, and you've succeeded when you take ak modulo p, you will have a factor of T. If it is less than p, see if a(p-k) is greater than p, and check that mod p if it is. However, f_1 may be the larger factor so also check f_2, as then it will give the smaller factor exactly or p - f_2 will.

A key equation is

T - k^2 - f_1^2 = 0 mod p

so if T - f_1^2 is not a quadratic residue modulo p, no k will exist, so if you find an 'a' and k that fit the criteria above but do not factor T, then change p. (It can be shown that instead you've factored T mod p greater than T.)

Note there is an existence condition, where you can find a value I call k_0, which is determined by finding k such that

abs(T - (1 + a^2)k^2)

is a minimum, as a k that will factor T non-trivially must be within floor(k_0/2p) steps from k_0 where each step is in increments of 2ap, so if you find that you do not have a solution you can check to see how close you were, which is how the focus is on T. The k_0 result is a remarkable and surprising one where I will try to show some of the theory and the underlying equations at the end.

Example: Let T=119. Then p=11 is greater than sqrt(119), and trying k=10 gives

a^2 = (119)(10^2)^{-1} - 1 mod 11 = 8 mod 11,

but 8 is not a quadratic residue modulo 11, so no 'a' exists for this case. Trying now k=9, gives

a^2 = (119)(9^2)^{-1} - 1 mod 11 = 4 mod 11

so a=2 is a solution. And ak = 18, which is greater than 11, so

f_1 = ak = 18 mod 11 = 7 mod 11.

And you have a non-trivial factorization, as 7 is a factor of 119.

The equations result trivially from considering a simple system of equations:

z^2 = y^2 + nT

where n is some non-zero integer, you can introduce z = x+ak, and substitute out:

(x+ak)^2 = y^2 + nT

multiplying out and simplifying I then have

x^2 = y^2 + nT - 2axk - a^2*k^2

and you could could factor nT - 2axk - a^2*k^2 to get y, but the problem with that is that you don't know x, so I use a trick:

2ax = k + p*r.

Then the above becomes

x^2 = y^2 + nT - (1 + a^2)k^2 - k*p*r

which removes x, and that's the point, as now I can move modulo a prime p of my choice. If k = 2ax, then

x^2 = y^2 + nT - (1 + a^2)k^2.

And finally

nT - (1 + a^2)k^2 = 0 mod p.

Existence of k_0 is found by considering

x^2 = y^2 + nT - (1 + a^2)k^2 - k*p*r

as r = 0 at the correct k, as let k = k_0 + p*j, where j is an integer, then

x^2 = y^2 + nT - (1 + a^2)(k_0^2 + 2p*j*k_0 + p^2*j^2) - (k_0 + p*j)p*r

and as k_0 is constant as is x, y, a and nT, as you move k either forward or back from k_0 the p^2*j^2 term will tend to dominate so that r will tend to be negative, assuming k is positive, to counterbalance it.

It can be shown that the k_0 is the minimum value for k.

It is my hope that I've generated some interest in evaluating this congruence algorithm for factoring. According to the theory it scales well with the only additional time with increasing size of T coming from finding p and calculating 'a' which requires finding a quadratic residue modulo p.

I would appreciate comments or criticisms on this approach and apologize if it's not close to some standard form. I am cutting and pasting from several sources.

## JSH: Normalization issue?

Here's a case where normalization opens up a huge argument where a math journal has died over this thing.

I have the special mathematical construction:

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.

(If you doubt the correctness of the construct, multiply everything out.)

By inspection it appears that 7 may be a factor in some way of BOTH factors (5a_1(x) + 7) and (5a_2(x)+ 7), but I find a problem with x=0, as then, the a's are roots of

a^2 + a = 0

so a=0 or a=-1, so arbitrarily choosing a_1(0) = 0, and a_2(0) = -1, I have

7*(2) = (5(0) + 7)(5(-1)+ 7)

and the mystery is resolved but it's also clear (I think) that normalization is needed, so using

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

so that both functions equal 0 at x=0, I have

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

which by inspection implies that 7 is only a factor of (5b_1(x) + 7).

But that contradicts with what follows from what mathematicians call the ring of algebraic integers, when

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

does not have integer roots with integer x. That is, everything is ok, with x=0, as I just showed, but if you have x an integer and the roots are NOT integers, then there is a problem, as, for instance, with x=1:

a^2 - 6a + 35 = 0

So the result would indicate that only one of the roots has 7 as a factor while the other is coprime but solving gives

a = (6 +/- sqrt(-104))/2 = 3 +/- sqrt(-26)

and prior to this type of construction, conventional wisdom was that 'a' could not have 7 as a factor for either of its two values.

And that is the basis for the arguing. I say the tail does not wag the dog and that the 7 cannot be controlled by the factorization of 175x^2 - 15x + 2, but math people disagree.

And they have a lot invested.

If I'm right then Galois Theory is one of the casualties. And an error is realized from over a hundred years ago.

So it's not a minor thing. If I'm right then humanity can now peer into numbers like never before and know that somehow 3 +/- sqrt(-26) has one value for which 7 is a factor and one value which is coprime to 7.

The inability to say which does or does not is created by the inability to resolve the square root, as for instance, given 1 +/- sqrt(4), which one has 3 as a factor?

Can you tell without resolving the square root?

Acceptance of the result could lead to understanding of things like how quarks cannot be individually distinguished as the mechanism could be from number theory itself, where finally number theory becomes integrated with physics because it is correct.

The stakes then are very high.