### Friday, July 31, 2009

## JSH: The object ring

Possibly a more impressive search in Google is on my object ring.

Google search: object ring

The object ring was a discovery driven by my find of a problem with the ring of algebraic integers.

For some reason I only had #2 in Google when I searched, out of over 22 million.

The refusal of the current mathematical community to follow mathematical proof is irrelevant to me in one sense, as the likely result is simply the replacement of that community.

For a while there may be two dominant "number theories" on the planet.

And then there will be only one.

Google search: object ring

The object ring was a discovery driven by my find of a problem with the ring of algebraic integers.

For some reason I only had #2 in Google when I searched, out of over 22 million.

The refusal of the current mathematical community to follow mathematical proof is irrelevant to me in one sense, as the likely result is simply the replacement of that community.

For a while there may be two dominant "number theories" on the planet.

And then there will be only one.

## JSH: There is something about hatred

The poster "mensanator" decided to dredge up old history yet again (he's done it before) and again I answered with the story as I find it more interesting to consider the anger and the hatred that boils out of this newsgroup time and time again.

There is something about hatred.

You dehumanize your opponent and then nothing else matters. Their efforts are "trash". Their ideas are "junk".

You tell them to go away and feel satisfied that your hostility is justified.

Any defense they offer is an affront. Their protestations are a sign of their ignorance and low level.

But still the newsgroup is on Usenet. And still I can post despite the hostility of professors or regulars or anyone else.

And I can think a lot about what it means to be human, and to be dehumanized. And wonder what it means to be the people who engage in such behaviors.

Sure, yes, I've contacted Professor Ullirch's Oklahoma university when I thought he stepped way over the line talking about "racial slurs" in response to me.

He introduced race. Not I.

But silly things like facts matter not to some people who I think adore hatred. They love hate.

It is a thrill isn't it?

To think of destroying some person possibly distant, possibly far away?

To say some words and tear at their heart?

To rip at their minds? Drive them to distraction with comments?

To drive a stake through their soul if you can?

In such efforts we understand the world that demands our sacrifice. That needs our blood.

As yes, there will be blood.

We demand it.

There is something about hatred.

You dehumanize your opponent and then nothing else matters. Their efforts are "trash". Their ideas are "junk".

You tell them to go away and feel satisfied that your hostility is justified.

Any defense they offer is an affront. Their protestations are a sign of their ignorance and low level.

But still the newsgroup is on Usenet. And still I can post despite the hostility of professors or regulars or anyone else.

And I can think a lot about what it means to be human, and to be dehumanized. And wonder what it means to be the people who engage in such behaviors.

Sure, yes, I've contacted Professor Ullirch's Oklahoma university when I thought he stepped way over the line talking about "racial slurs" in response to me.

He introduced race. Not I.

But silly things like facts matter not to some people who I think adore hatred. They love hate.

It is a thrill isn't it?

To think of destroying some person possibly distant, possibly far away?

To say some words and tear at their heart?

To rip at their minds? Drive them to distraction with comments?

To drive a stake through their soul if you can?

In such efforts we understand the world that demands our sacrifice. That needs our blood.

As yes, there will be blood.

We demand it.

### Thursday, July 30, 2009

## JSH: Considering world class

What really intrigues me as I consider replies to my posts is the anger. And I think that is fascinating as it makes me wonder so much about what is going through the heads of certain people.

My research is world class if by that I mean the world considers it, which it seems it does!

NASA has a fascinating problem: there are people who do not believe that man ever landed on the moon.

Mathematics has at its base as a

But what if they simply choose to not?

As I've noted, my mathematical research is separate from me. Like any past discoverer, reality is that my research needs my promotion like the earth need humanity to keep spinning as in, NOT. Or as they might say in Zen, "mu".

But on these newsgroups I've noted a peculiar point of view which seems to think otherwise and sees my continued posting as an affront meant to promote that which if it is important—needs no such promotion!

Puzzling over human failings in intellectual thinking is more than just a minor exercise. It's a crucial way to consider what hope we might have as a species to defy the odds, and survive…at least for a little while longer.

The story here is grander because of Internet realities which is a statement meant to attract the anger. I have this year visitors from 108 countries/territories to my math blog, which is already more than all of last year when 107 countries/territories visited it—according to Google Analytics.

Whether that means anything or not is a matter for discussion. To me it is irrelevant in one sense, but interesting in another, as a stepping point for discussion—as a way to probe your minds.

At the end of the day, I suggest to you that you are all prisoners of your programming!!!

You are nothing more than genes and environment—unless you DARE to believe that on top of that you have a soul.

Questions intrigue me. Many of you are now questions. Here in this thread—you give your answers, if you can.

As best you can, as at the end of the day, you can do nothing more. You are limited by your reality.

You are maybe simply machines, programmed, determined, unable to do ANYTHING other than what is required by what you are.

You are, in a word, destined.

My research is world class if by that I mean the world considers it, which it seems it does!

NASA has a fascinating problem: there are people who do not believe that man ever landed on the moon.

Mathematics has at its base as a

**discipline**the notion that people can be expected to obey mathematical axioms and follow mathematical proof.But what if they simply choose to not?

As I've noted, my mathematical research is separate from me. Like any past discoverer, reality is that my research needs my promotion like the earth need humanity to keep spinning as in, NOT. Or as they might say in Zen, "mu".

But on these newsgroups I've noted a peculiar point of view which seems to think otherwise and sees my continued posting as an affront meant to promote that which if it is important—needs no such promotion!

Puzzling over human failings in intellectual thinking is more than just a minor exercise. It's a crucial way to consider what hope we might have as a species to defy the odds, and survive…at least for a little while longer.

The story here is grander because of Internet realities which is a statement meant to attract the anger. I have this year visitors from 108 countries/territories to my math blog, which is already more than all of last year when 107 countries/territories visited it—according to Google Analytics.

Whether that means anything or not is a matter for discussion. To me it is irrelevant in one sense, but interesting in another, as a stepping point for discussion—as a way to probe your minds.

At the end of the day, I suggest to you that you are all prisoners of your programming!!!

You are nothing more than genes and environment—unless you DARE to believe that on top of that you have a soul.

Questions intrigue me. Many of you are now questions. Here in this thread—you give your answers, if you can.

As best you can, as at the end of the day, you can do nothing more. You are limited by your reality.

You are maybe simply machines, programmed, determined, unable to do ANYTHING other than what is required by what you are.

You are, in a word, destined.

### Saturday, July 25, 2009

## JSH: Break in the wall

Sometimes people will ask why don't I just send papers to journals, and the answer is, I've done that. I even had one paper published and then retracted by the editors. I've contacted mathematicians directly mostly by email and even went back to my alma mater Vanderbilt University and talked to one directly.

Now then, the claim by those disagreeing with me is that I'm wrong, or my research isn't that important, and if you accept that mathematicians WOULD pick up on something important, then you'd accept that I'm simply refusing to accept reality.

For me that puts me in a fascinatingly difficult position, as mathematicians can just sit back, do nothing, and the standard opinion then is that my research is not important BECAUSE they do so, which gives me a conundrum.

There is a wall of silence. For most mathematicians, even if they find out I'm right, they may feel they can simply do nothing, and hope that nothing changes.

Recently though I've been pondering what may be a break in the wall.

Data from Google Analytics about hits to my math blog indicate various directions to world interest and show that the ideas are traveling. If you presume that there is active hostility in math circles against my research, then the pattern fits well.

People are going around mathematicians anyway, and that is proceeding slowly but it is worldwide.

I'm conducting an experiment on these two newsgroups around the top draw right now, to get some sense of what it may mean.

There are a lot of reasons I think why solving quadratic residues would be a big draw around the world, some of them kind of scary.

You see that research is a product of my factoring research which posters have gone out of their way to dismiss.

If the world is pulling it though, then that research may burst forward against very active mathematical resistance, breaking through it with a great deal of force.

Or not. The question then—but does it work?—is a critical one.

As if it DOES work well enough to pull world attention against active resistance from the mathematical community, then the likely outcome could be dramatic as a great deal of force, you could say, is involved.

So that is the open question. Posters can claim it's not open, but I'm looking at data that says it is.

People can say all kinds of things, so it's a big deal for me if there is evidence indicating that the wall is being breached.

And a big deal for people invested in that not happening, so it's kind of interesting to watch that play out as the back story to some recent threads.

Now then, the claim by those disagreeing with me is that I'm wrong, or my research isn't that important, and if you accept that mathematicians WOULD pick up on something important, then you'd accept that I'm simply refusing to accept reality.

For me that puts me in a fascinatingly difficult position, as mathematicians can just sit back, do nothing, and the standard opinion then is that my research is not important BECAUSE they do so, which gives me a conundrum.

There is a wall of silence. For most mathematicians, even if they find out I'm right, they may feel they can simply do nothing, and hope that nothing changes.

Recently though I've been pondering what may be a break in the wall.

Data from Google Analytics about hits to my math blog indicate various directions to world interest and show that the ideas are traveling. If you presume that there is active hostility in math circles against my research, then the pattern fits well.

People are going around mathematicians anyway, and that is proceeding slowly but it is worldwide.

I'm conducting an experiment on these two newsgroups around the top draw right now, to get some sense of what it may mean.

There are a lot of reasons I think why solving quadratic residues would be a big draw around the world, some of them kind of scary.

You see that research is a product of my factoring research which posters have gone out of their way to dismiss.

If the world is pulling it though, then that research may burst forward against very active mathematical resistance, breaking through it with a great deal of force.

Or not. The question then—but does it work?—is a critical one.

As if it DOES work well enough to pull world attention against active resistance from the mathematical community, then the likely outcome could be dramatic as a great deal of force, you could say, is involved.

So that is the open question. Posters can claim it's not open, but I'm looking at data that says it is.

People can say all kinds of things, so it's a big deal for me if there is evidence indicating that the wall is being breached.

And a big deal for people invested in that not happening, so it's kind of interesting to watch that play out as the back story to some recent threads.

### Friday, July 24, 2009

## JSH: Issue is real

I live in a reality where people are dedicated in claiming none of my research is of value, so of course, I'm intrigued by Google search results that appear to contradict them. Yes, I can understand how people can attack Google search results as well.

However, rational criticism of Google search results does not settle the issue or give an answer.

So I asked a question in a previous thread: but does it work?

Seemed simple enough to me.

Consider, imagine you do amateur mathematical research, and math people go out of their way to say your research is useless.

Ok.

But then you do Google searches and that SAME research dominates search results.

Math people THEN say that Google search results are meaningless.

Anyone else see a problem here?

However, rational criticism of Google search results does not settle the issue or give an answer.

So I asked a question in a previous thread: but does it work?

Seemed simple enough to me.

Consider, imagine you do amateur mathematical research, and math people go out of their way to say your research is useless.

Ok.

But then you do Google searches and that SAME research dominates search results.

Math people THEN say that Google search results are meaningless.

Anyone else see a problem here?

### Thursday, July 23, 2009

## JSH: But does it work?

I've noticed that when I do searches in Google—it has to be Google—on solving quadratic residues I get a page of mine at my math blog on a new way to do it I invented.

But does it work?

Question then is, are Google search order results for my research about things that work, or stuff that comes up for other reasons?

Search in Google is: solving quadratic residues

No quotes.

But does it work?

Question then is, are Google search order results for my research about things that work, or stuff that comes up for other reasons?

Search in Google is: solving quadratic residues

No quotes.

### Friday, July 10, 2009

## JSH: Simple enough refutation

Ok, so it seems a simple enough refutation of standard teaching on Galois Theory to consider in the ring of algebraic integers:

x^2 - 6x + 35 = 0

and let x = y + 35, to get:

(y+35)^2 - 6(y+35) + 35 = 0

which is:

y^2 + 64y + 1050 = 0

if I did my math right, as then each solution for y must have one of the solutions for x as a factor, but if you use the quadratic formula to solve for x and y, and check to see if y/x can be an algebraic integer, you find it cannot as you will get a non-monic primitive quadratic irreducible over Q.

So none of its solutions can be algebraic integers!

That directly contradicts with x = y + 35 and x being a factor of 35, showing that use of the ring of algebraic integers leads to a direct contradiction.

So there is a "core error" that is over one hundred years old.

x^2 - 6x + 35 = 0

and let x = y + 35, to get:

(y+35)^2 - 6(y+35) + 35 = 0

which is:

y^2 + 64y + 1050 = 0

if I did my math right, as then each solution for y must have one of the solutions for x as a factor, but if you use the quadratic formula to solve for x and y, and check to see if y/x can be an algebraic integer, you find it cannot as you will get a non-monic primitive quadratic irreducible over Q.

So none of its solutions can be algebraic integers!

That directly contradicts with x = y + 35 and x being a factor of 35, showing that use of the ring of algebraic integers leads to a direct contradiction.

So there is a "core error" that is over one hundred years old.

## JSH: Question about Galois Theory

I have some questions about Galois Theory.

In the ring of algebraic integers consider the simple quadratic:

x^2 - 6x + 35 = 0

The solutions are non-rational, and traditional view by Galois Theory is that the factors of 35, for instance, 7, split in ways I think that have to do with the class number, but let y = x - 35, then x = y + 35, so:

(y+35)^2 - 6(y+35) + 35 = 0

which is:

y^2 + 64y + 1050 = 0.

Now exactly one solution for x should be a factor of exactly one solution for y, so do the class numbers match? (Am I using the phase "class number" correctly? Sorry if not.)

Can you show solutions where y/x is an algebraic integer?

In the ring of algebraic integers consider the simple quadratic:

x^2 - 6x + 35 = 0

The solutions are non-rational, and traditional view by Galois Theory is that the factors of 35, for instance, 7, split in ways I think that have to do with the class number, but let y = x - 35, then x = y + 35, so:

(y+35)^2 - 6(y+35) + 35 = 0

which is:

y^2 + 64y + 1050 = 0.

Now exactly one solution for x should be a factor of exactly one solution for y, so do the class numbers match? (Am I using the phase "class number" correctly? Sorry if not.)

Can you show solutions where y/x is an algebraic integer?

### Tuesday, July 07, 2009

## JSH: Two remarkable things!

The issue of the "perfect tweet" can help some of you see two remarkable things which I see a lot on the sci.math newsgroup:

I've actually taken lessons from the sci.math newsgroup to consideration of lying behavior in other areas, like politics and economics, as well as business in general and even relationships, and concluded that on a far larger scale than most admit: people lie a LOT and routinely lie about lying, but most importantly, they lie in areas where it's easy to check them and determine they are lying!

Lying may be the most dominant form of human communication.

- People lie a lot, or maybe I should say, give falsehoods in even the most obvious areas.
- Objective tests are often ignored, and people still lie a lot!

I've actually taken lessons from the sci.math newsgroup to consideration of lying behavior in other areas, like politics and economics, as well as business in general and even relationships, and concluded that on a far larger scale than most admit: people lie a LOT and routinely lie about lying, but most importantly, they lie in areas where it's easy to check them and determine they are lying!

Lying may be the most dominant form of human communication.

### Sunday, July 05, 2009

## JSH: Perfect tweet, can it be re-done?

I posted a perfect tweet today, defined to be 140 characters that is grammatical, which defines itself. Can that accomplishment be matched?

Here it is, the perfect tweet:

In my opinion, crucial criteria for a perfect tweet is that it be EXACTLY 140 characters, have few if any abbreviations, and is grammatical.

Question is, and I leave off quotes as that adds to the tweet so that sentence is itself the perfect tweet, can that accomplishment be matched? Can you write another tweet that defines itself as a perfect tweet by those rules which is itself a perfect tweet?

Seems to me one option for attack would be to find synonyms of the same length for any of the words. Otherwise it's a matter of finding combinations with the same meaning where it MAY be impossible in the English language.

So it's a challenge. Go for it.

Here it is, the perfect tweet:

In my opinion, crucial criteria for a perfect tweet is that it be EXACTLY 140 characters, have few if any abbreviations, and is grammatical.

Question is, and I leave off quotes as that adds to the tweet so that sentence is itself the perfect tweet, can that accomplishment be matched? Can you write another tweet that defines itself as a perfect tweet by those rules which is itself a perfect tweet?

Seems to me one option for attack would be to find synonyms of the same length for any of the words. Otherwise it's a matter of finding combinations with the same meaning where it MAY be impossible in the English language.

So it's a challenge. Go for it.

## JSH: Thank you!!!

Thank you to posters arguing with me about the "perfect tweet" as I locked down another variant of the perfect tweet which defines itself in the tweet and then came up with another perfect tweet talking about it:

Ok, I've locked down both variants of the "perfect tweet" which defines itself as perfect within the tweet while also being a perfect tweet.

The above is a perfect tweet. I'm on a roll!!!

Hey, seriously though, what are the odds of a perfect tweet that is grammatical? And what are the odds of a perfect tweet which defines itself as perfect?

I'm guessing at least one in 100 trillion, which I think is rather conservative.

I may have locked down the only 2 perfect tweets that define themselves as perfect possible in the English language.

The game is on!!! There can only be so many and this is a once in human history kind of thing.

How much more fun can it get?

Ok, I've locked down both variants of the "perfect tweet" which defines itself as perfect within the tweet while also being a perfect tweet.

The above is a perfect tweet. I'm on a roll!!!

Hey, seriously though, what are the odds of a perfect tweet that is grammatical? And what are the odds of a perfect tweet which defines itself as perfect?

I'm guessing at least one in 100 trillion, which I think is rather conservative.

I may have locked down the only 2 perfect tweets that define themselves as perfect possible in the English language.

The game is on!!! There can only be so many and this is a once in human history kind of thing.

How much more fun can it get?

### Saturday, July 04, 2009

## JSH: Psychology of the denial of "core error"

Turns out that if you follow rigorous mathematics it is trivial to show a problem with the use of the ring of algebraic integers with a quadratic factorization that I've often given 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.

The primary problem here is that if you let the ring be the ring of algebraic integers, you get something never before seen in human mathematics which is a constraint on a constant factor revealed on the right hand side of

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

which is obviously not there on the left hand side.

To understand what I mean—I'm sure some posters would reply that they find it incomprehensible—consider a simpler example in the ring of integers:

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

where the 7 is still unconstrained, and you can in fact move it around at will, so

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

Now the mathematically astute of you may notice that the first example IS fixable so that you have equivalence on both sides of the equals and the 7 is unconstrained if you use normalized functions:

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

where

7b_1(x) = a_1(x) and b_2(x) = a_2(x) + 1

and the a's are still roots of

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

and the b's are normalized because they both equal 0 when x=0.

NOW notice that algebra still works and the 7 is unconstrained again and you can move it around, like before:

7(175x^2 - 15x + 2) = (5b_1(x) + 1)(5(7)b_2(x)+ 14)

or even divide it off completely:

(175x^2 - 15x + 2) = (5b_1(x) + 1)(5b_2(x)+ 2)

But now the ring of algebraic integers itself becomes the problem as unlike any other known ring in mathematics it will not allow you to do the above!!! The b's are not in general in that ring!

That is because

7b_1(x) = a_1(x)

means that a_1(x) has 7 as a factor which requires that one of the roots of

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

have 7 as a factor, and you can get a contradiction with x=1, which gives

a^2 - 6a + 35 = 0

as provably in the ring of algebraic integers NEITHER of the roots of that quadratic can have 7 as a factor.

That's the easy algebra and it reveals that the ring of algebraic integers is mathematically distinct from ALL other known rings, including fields of course, as the problem does not emerge in anything else!

The ring of algebraic integers actually blocks algebra itself, by not allowing certain algebraic operations to occur, and that is only revealed by this remarkable construct:

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.

What happens there is that you get the unconstrained 7 on the left—math students are normally taught to divide such factors off—which cannot in general be divided off from the right hand side ONLY in the ring of algebraic integers, which I call entanglement.

The 7 is somehow trapped in place, entangled, on the right hand side!

I've proven that this problem leads to the

With errors in mathematics you can get this odd ability to appear to prove just about anything.

Ok, the math is easy. I've simplified explaining over the years and explained over and over and over again in many different ways and even had a paper published in this area with slightly more complicated cubics before I simplified to quadratics.

Now that story is interesting as it reveals an amazing psychology here: publication supposedly means something in the mathematical community but when I got published sci.math'er said it meant nothing, some of them conspired on the newsgroup to attack the paper by emails and did so, panicked the editors who pulled my paper and the journal died after managing one more edition!

Now rational people knowing the math, knowing the error allows people to do fake math, would get very suspicious with that story, but this particular error appears to run so deep that the bulk of the mathematical community might rather hide it than deal with it.

That's because the ring of algebraic integers was introduced over one hundred years ago. The sheer volume of erroneous publications revolving around the error can defy imagination and may represent the BULK of number theory work over the last century plus encompassing the ENTIRE 20th century!!!

One dead math journal under remarkable circumstance is minor with such a situation.

Notice here also that practitioners in number theory with decades in the field and lots of awards or prizes have the most to lose from the revelation of the error.

Graduate students as well can have serious investment in the error as imagine being one who considers his graduate thesis and realizes it's junk!!!

So for years I've included math undergrads who are really in an awkward position here, as the people they rely on the most may be the most compromised but unlike professors and grads they don't have as many years or as many "successes" invested in wrong mathematics!

And remember, for someone who believed they were brilliant or had great accomplishments this error really can cut deep emotionally as well as in many other ways. It's like it can rip their entire world apart.

I have been careful in trying to figure out the best way to hopefully get some resolution here as I understand just how high emotions can run, and some of the replies I get can give readers some perspective, but make no mistake, the fight is to keep doing wrong math, and to teach it to others.

So from one perspective, the professors who didn't have a chance and the grad students who didn't have a chance are victims, yes, but they are also abusers when they are teaching to new students who DO have a chance to not be taught this error, and to do number theory and start advancing it again after, oh, about a hundred years of stagnation.

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.

The primary problem here is that if you let the ring be the ring of algebraic integers, you get something never before seen in human mathematics which is a constraint on a constant factor revealed on the right hand side of

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

which is obviously not there on the left hand side.

To understand what I mean—I'm sure some posters would reply that they find it incomprehensible—consider a simpler example in the ring of integers:

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

where the 7 is still unconstrained, and you can in fact move it around at will, so

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

Now the mathematically astute of you may notice that the first example IS fixable so that you have equivalence on both sides of the equals and the 7 is unconstrained if you use normalized functions:

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

where

7b_1(x) = a_1(x) and b_2(x) = a_2(x) + 1

and the a's are still roots of

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

and the b's are normalized because they both equal 0 when x=0.

NOW notice that algebra still works and the 7 is unconstrained again and you can move it around, like before:

7(175x^2 - 15x + 2) = (5b_1(x) + 1)(5(7)b_2(x)+ 14)

or even divide it off completely:

(175x^2 - 15x + 2) = (5b_1(x) + 1)(5b_2(x)+ 2)

But now the ring of algebraic integers itself becomes the problem as unlike any other known ring in mathematics it will not allow you to do the above!!! The b's are not in general in that ring!

That is because

7b_1(x) = a_1(x)

means that a_1(x) has 7 as a factor which requires that one of the roots of

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

have 7 as a factor, and you can get a contradiction with x=1, which gives

a^2 - 6a + 35 = 0

as provably in the ring of algebraic integers NEITHER of the roots of that quadratic can have 7 as a factor.

That's the easy algebra and it reveals that the ring of algebraic integers is mathematically distinct from ALL other known rings, including fields of course, as the problem does not emerge in anything else!

The ring of algebraic integers actually blocks algebra itself, by not allowing certain algebraic operations to occur, and that is only revealed by this remarkable construct:

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.

What happens there is that you get the unconstrained 7 on the left—math students are normally taught to divide such factors off—which cannot in general be divided off from the right hand side ONLY in the ring of algebraic integers, which I call entanglement.

The 7 is somehow trapped in place, entangled, on the right hand side!

I've proven that this problem leads to the

**appearance**of contradiction and allows a math practitioner to appear to prove things that are not true, or are not actually proven.With errors in mathematics you can get this odd ability to appear to prove just about anything.

Ok, the math is easy. I've simplified explaining over the years and explained over and over and over again in many different ways and even had a paper published in this area with slightly more complicated cubics before I simplified to quadratics.

Now that story is interesting as it reveals an amazing psychology here: publication supposedly means something in the mathematical community but when I got published sci.math'er said it meant nothing, some of them conspired on the newsgroup to attack the paper by emails and did so, panicked the editors who pulled my paper and the journal died after managing one more edition!

Now rational people knowing the math, knowing the error allows people to do fake math, would get very suspicious with that story, but this particular error appears to run so deep that the bulk of the mathematical community might rather hide it than deal with it.

That's because the ring of algebraic integers was introduced over one hundred years ago. The sheer volume of erroneous publications revolving around the error can defy imagination and may represent the BULK of number theory work over the last century plus encompassing the ENTIRE 20th century!!!

One dead math journal under remarkable circumstance is minor with such a situation.

Notice here also that practitioners in number theory with decades in the field and lots of awards or prizes have the most to lose from the revelation of the error.

Graduate students as well can have serious investment in the error as imagine being one who considers his graduate thesis and realizes it's junk!!!

So for years I've included math undergrads who are really in an awkward position here, as the people they rely on the most may be the most compromised but unlike professors and grads they don't have as many years or as many "successes" invested in wrong mathematics!

And remember, for someone who believed they were brilliant or had great accomplishments this error really can cut deep emotionally as well as in many other ways. It's like it can rip their entire world apart.

I have been careful in trying to figure out the best way to hopefully get some resolution here as I understand just how high emotions can run, and some of the replies I get can give readers some perspective, but make no mistake, the fight is to keep doing wrong math, and to teach it to others.

So from one perspective, the professors who didn't have a chance and the grad students who didn't have a chance are victims, yes, but they are also abusers when they are teaching to new students who DO have a chance to not be taught this error, and to do number theory and start advancing it again after, oh, about a hundred years of stagnation.

### Thursday, July 02, 2009

## JSH: Why choice rules

On my math blog Penny Hassett had a comment that got me to thinking and after pondering at her suggestion what others believe to be true in their disagreements with me about a certain serious mathematical issue, I concluded that infinity could help to clear the air.

So I have simple examples yet again to start (believe me it gets really hard later so please pay close attention to the easy part):

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

and let the ring be the ring of algebraic integers. And notice that reflects CHOICE. My choice for a simple example as the 7 can be factored an INFINITY of ways!!!

After all,

7(x^2 + 3x + 2) = (x + 1)(7x + 14)

is ALSO just as valid mathematically, and in fact there are an infinity of possible variations all equally valid.

Key here also to notice is that there are an infinity of choices FOR EVERY x, and remember the ring is the ring of algebraic integers.

So far easy and there should be no disagreement with the above!

Now to the hard mathematics.

Now we'll TRY to assume the ring is still the ring of algebraic integers and now a harder example:

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

where

7b_1(x) = a_1(x) and b_2(x) = a_2(x) + 1

and the a's are roots of

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

Notice the b's are chosen such that when x=0, both the b's equal 0, which is to say, they are normalized.

Now just like before you have human choice, and you can imagine moving that 7 around easily enough:

7(175x^2 - 15x + 2) = (5b_1(x) + 1)(5(7)b_2(x) + 14)

which is just one more human choice out of infinity.

And there are an INFINITY of possibilities for EVERY x. Every x has an infinite number of choices.

So how does the mathematics choose?

It doesn't. I did. I chose.

But that's where the fights start and the arguing and the dead math journal comes into the picture as what follows from mathematical logic and the rigidity of infinity is a conclusion that is devastating emotionally to thousands of mathematicians around the world which is that there is a core error: the ring of algebraic integers gives a false result.

Because now you have that one of the a's has 7 as a factor by the choice argument.

But the a's are roots of

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

and at x=1, that gives

a^2 - 6a + 35 = 0

and provably in the ring of algebraic integers NEITHER of the roots can have 7 as a factor and there is shown the direct contradiction.

Notice the ring of algebraic integers does not fail gracefully and its error is not fixable as your alternative is to introduce some mystical force to choose out of infinity, like, maybe God?

(Posters have been remarkably vague in past arguments about how a particular factor of 7 arises, often simply repeating what is taught in math classes about ways to find the factors at any particular x, but remember there are an infinity of choices for EVERY x, so the choice problem gives you infinity at every point. So who chooses? God?)

But not even God can choose for you here if you go with mathematical logic. Of course, you can like so many human beings before you simply choose to believe in what you've been taught, even though it is very, very wrong.

The mystical nature of the disagreement with me is remarkable but I think it's natural for human beings to turn to spirits or mystical forces in the face of something difficult to understand. That's what your ancestors did.

But then you're no longer really mathematicians.

So I have simple examples yet again to start (believe me it gets really hard later so please pay close attention to the easy part):

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

and let the ring be the ring of algebraic integers. And notice that reflects CHOICE. My choice for a simple example as the 7 can be factored an INFINITY of ways!!!

After all,

7(x^2 + 3x + 2) = (x + 1)(7x + 14)

is ALSO just as valid mathematically, and in fact there are an infinity of possible variations all equally valid.

Key here also to notice is that there are an infinity of choices FOR EVERY x, and remember the ring is the ring of algebraic integers.

So far easy and there should be no disagreement with the above!

Now to the hard mathematics.

Now we'll TRY to assume the ring is still the ring of algebraic integers and now a harder example:

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

where

7b_1(x) = a_1(x) and b_2(x) = a_2(x) + 1

and the a's are roots of

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

Notice the b's are chosen such that when x=0, both the b's equal 0, which is to say, they are normalized.

Now just like before you have human choice, and you can imagine moving that 7 around easily enough:

7(175x^2 - 15x + 2) = (5b_1(x) + 1)(5(7)b_2(x) + 14)

which is just one more human choice out of infinity.

And there are an INFINITY of possibilities for EVERY x. Every x has an infinite number of choices.

So how does the mathematics choose?

It doesn't. I did. I chose.

But that's where the fights start and the arguing and the dead math journal comes into the picture as what follows from mathematical logic and the rigidity of infinity is a conclusion that is devastating emotionally to thousands of mathematicians around the world which is that there is a core error: the ring of algebraic integers gives a false result.

Because now you have that one of the a's has 7 as a factor by the choice argument.

But the a's are roots of

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

and at x=1, that gives

a^2 - 6a + 35 = 0

and provably in the ring of algebraic integers NEITHER of the roots can have 7 as a factor and there is shown the direct contradiction.

Notice the ring of algebraic integers does not fail gracefully and its error is not fixable as your alternative is to introduce some mystical force to choose out of infinity, like, maybe God?

(Posters have been remarkably vague in past arguments about how a particular factor of 7 arises, often simply repeating what is taught in math classes about ways to find the factors at any particular x, but remember there are an infinity of choices for EVERY x, so the choice problem gives you infinity at every point. So who chooses? God?)

But not even God can choose for you here if you go with mathematical logic. Of course, you can like so many human beings before you simply choose to believe in what you've been taught, even though it is very, very wrong.

The mystical nature of the disagreement with me is remarkable but I think it's natural for human beings to turn to spirits or mystical forces in the face of something difficult to understand. That's what your ancestors did.

But then you're no longer really mathematicians.