## JSH: Understanding asymptotic convergence in exclusionary rings

Some recent threads simply explaining yet another way to see the coverage problem of the ring of algebraic integers have floundered on the issue of convergence and what it means with rings like the ring of algebraic integers, where I introduce the concept of an exclusionary ring.

The concepts are simple, luckily, as I can mostly use something basic:

S = 1 + x + x^2 + x^3 +…

where the issue is convergence, so that if S converges in whatever ring you're in—notice none given yet—you can go to

S = 1 + x*S

and solve to get

S = 1/(1-x)

where to get convergence in the ring of complex numbers, which of course is also the field of complex numbers, let's choose x = sqrt(5) - 2, considering only the positive solution, then in complex numbers, I have

S = 1/(1 - sqrt(5))

which is a complex number, and it's also an algebraic number, but it's NOT an algebraic integer.

But if I go back to

S = 1 + x + x^2 + x^3 +…

and plug in x=sqrt(5) - 2, I'll have

S = 1 + (sqrt(5) - 2) + (sqrt(5) - 2)^2 + (sqrt(5) - 2)^3 +…

and if I stop at some point like just look at

1 + (sqrt(5) - 2) + (sqrt(5) - 2)^2

I have an algebraic integer, and in fact, I can have an arbitrarily large number of terms added together, and stop—ever closer to 1/(1-sqrt(5)) and STILL have an algebraic integer.

BUT in the ring of algebraic integers, you cannot reach 1/(1-sqrt(5)) because it is NOT an algebraic integer!

Yet you can approach it out to infinity, so you have an asymptotic approach to that value.

A corollary to this is 1/x where you can let x go out to infinity and that approaches 0 but never reaches it, or you can let x approach 0, but never reach it, but the difference with the infinite sum example is that the asymptotic nature is created by the exclusionary nature of the ring of algebraici integers!!!

That is, the reason you can never reach 1/(1-sqrt(5)) with the series is that 1/(1-sqrt(5)) is NOT the root of any monic polynomial with integer coefficients, which is the rule that defines algebraic integers and excludes that value!

So the mathematics holds on a definition, as logically, if 1/(1-sqrt(5)) is not an algebraic integer, then there is no way to reach that value in the ring, so

S = 1 + (sqrt(5) - 2) + (sqrt(5) - 2)^2 + (sqrt(5) - 2)^3 +…

approaches it asymptotically, in the ring of algebraic integers.

But now you have a problem with

S = 1 + x*S

as it's just not necessarily true in the ring of algebraic integers.

Why not necessarily true?

Because the exclusion does NOT apply if 1-x is a unit in the ring of algebraic integers.

So if it is a unit then the series can converge within the ring, but otherwise the exclusionary rule—the definition of algebraic integers as roots of monic polynomials—prevents.

Now I feel confident there are posters who will want to dispute me on the ability to use convergent infinite series in the ring of algebraic integers, but I want more than namecalling, or other childish rants, and I want more than someone saying they read it different in some number theory text.

The problem here again is that people who don't know mathematics that well, can write books.

And other people can believe things that are wrong for quite some time, and LOTS of people can believe things that are wrong, so simply saying a lot of people have believed other things for a long time is not the way to reply back to me here.

If any of you actually know mathematics versus being people who can repeat what you're told as if that's all that matters, then you can explain the status quo view.

You can explain the mainstream beliefs of the mathematical community in this area without relying on insults, without simply claiming that I'm wrong, and without doing anything other than objectively replying.

I know that can't be done, so I say that upfront to remind readers that a lot of people around the world are fighting the truth with my research to preserve their delusion of expertise, which is all about a lot of people getting some mathematics wrong.

[A repy to someone who noticed that James had previously asserted that if all the terms in a series were in a ring the sum must also be in the ring.]

Well that's wrong, so yes, I had to, following what is mathematically correct.

I make mistakes, but I care about what is correct.

So I can admit when I'm wrong, as mathematics is beautiful in that what's right is absolutely right.

And people fighting for their own delusions of worth do not matter to that correctness.

After all, these arguments will one day be gone, all of you will be dead and your children, and even your children's children will be dead, but the correct mathematical arguments will still be correct.

The mathematician looks beyond the moment to any point further when all the arguing is meaningless, which is why to some mathematicians are unearthly or almost mystical.

Only true mathematicians care nothing for social crap or the accolades of the moment as they quest for absolute knowledge knowing that even God cannot change it.

And in that way, the mathematician stands in the presence of the divine.

While lesser beings stoop at the feet of social needs, begging for approval, priding themselves on social acceptance, and when they are gone—there is nothing left.

## JSH: They are crackpots

These number theorists are not going to admit the truth.

They are pretenders to the title of mathematician.

I have shown that easily and in multiple ways.

Now I yet again answer all objections and even this latest infinite series objection that I think started with W. Dale Hall, and you know what?

If he convinced you, then you just are not what you think you are.

The equations x^2 - (3+t)x + 2 = (x - r1(t))*(x - r2(t))

with

r1(t) = 1 - t + 2t^2 - 6t^3 + 22t^4 - 90t^5 + …

r2(t) = 2 + 2t - 2t^2 + 6t^3 - 22t^4 + 90t^5 - …

when t is an algebraic integer where the two series converge are in the ring of algebraic integers because the functions are roots of a monic polynomial with integer coefficients.

If someone puts up a different infinite series that does not converge in the ring—guess what?

It does not converge in the ring.

Posters routinely lie.

They have to, they're fighting for their egos, which is why when I got published, they immediately trashed the journal, which is why when I answered all their objections, they just kept saying I didn't, which is why when the journal died they acted like that was the most natural thing, hardly worth mentioning at all.

They are fighting to be something they are not—mathematicians, really mathematicians.

They will not give up simply because they have been repeatedly proven wrong.

If they did, then they wouldn't be crackpots.

[A reply to someone who wanted to know who would think a physics degree makes him or her a capable mathematician.]

A degree does not make a mathematician at all.

That's why you can't get it.

You think it's about social crap.

Some person tells you you're a mathematician and you think that's it.

A degree is a social label—information saying that some people believe you are something.

But mathematical proof is beyond society and beyond humanity, standing as an intellectual achievement appreciated by any being who can follow the logic of it.

I present mathematical proof and in retaliation people like you talk about degrees.

You are a crackpot. You know that society supports you in your delusion because most people don't know any better.

Since I have mathematical proof, all you have is—your ability to convince.

These kind of fights have happened throughout human history and before your kind has always lost.

If I lose here, and well I might, then that first may be the last.

## JSH: Stop being dense

So why would people known as top number theorists around the world NOT admit the coverage problem with algebraic integers?

Because in admitting it they are taking away that very position.

This thing is huge. It stepped in over a hundred years ago—before anyone currently alive was even born—and in mathematics a small mistake at a key spot can be devastating.

This error re-writes the math history books over a hundred plus years, yanking expert status from so many people over that time, and to admit it now, people called top, would suddenly find they are nowhere near there.

What do they have that this error doesn't touch?

Dedekind in saying that he proved a result that was key in convincing others that the algebraic integers did give full coverage, said he used ideal theory, which at that time was a relatively new idea.

He couldn't figure out any other way to do it, and now we know that was because there was no way to prove full coverage—not correctly.

So one of the first things ideal theory was used for was to convince people of this faux result about algebraic integers and a lot of history passed from that point until today, till we have people who are currently thought brilliant, or even "beautiful minds" who if they admit the truth, are just, well, I guess maybe still highly intelligent, but otherwise, ordinary people.

You have to be fairly dense to look at all the evidence, understand what is at stake, and not go with mathematical proof.

Our civilization can be affected by the decision you people are making.

Mathematics is crucial for our continued development and "pure math" has been dubiously distinguished by being mostly useless.

Now the explanation is that much of it is wrong.

But what about the correct mathematics? What great advances might humanity make with it that it cannot make without it?

Sacrifice the future for the egos of some people who got some math wrong?

Would you be satisfied today if Newton were sidelined for some people who found his ideas troublesome, or Galileo couldn't get past people like these that I'm dealing with, or Gauss himself never got the recognition he deserved?

Maybe there wouldn't be computers, or cars, or planes just for some people doing what many of you are doing today.

But so what, right? We got ours, so screw the future, right? Forget the unborn, who may never be born (maybe better off?) because people like you failed where past generations did not.

If humanity never manages to really get off this planet out to distant stars, and ultimately finally dies sometime in the future having done some things sure, but ultimately falling short, what if it is here and now that the die was cast?

What if here and now you decided to end humanity—take away the future—and put limits on what was possible for all generations to come by failing like no generation before you failed, to protect some people who just can't handle the mathematical truth?

What if here you betrayed humanity and condemned it?

## JSH: Gene Ward Smith approach, coverage error in "core"

I am going to rather easily explain how you can show a remarkable but devastating problem with the coverage of the ring of algebraic integers, but what makes the interesting twist here is that I didn't think of the base equations for this approach, or find a crucial infinite series as a poster who goes by Gene Ward Smith did:

"Consider the polynomial x^2 - (3+t)x + 2. The two roots of this can be expanded in series as

r1(t) = 1 - t + 2t^2 - 6t^3 + 22t^4 - 90t^5 + …

r2(t) = 2 + 2t - 2t^2 + 6t^3 - 22t^4 + 90t^5 - …

If t is 2-adically a nonunit, so that |t|_2 < 1, then these converge in the 2-adic integers. One root, r2, is even, the other, r1, is odd. The 2 does not split up. If t is a rational integer, this means that when t is even, the roots split into an even root and an odd root 2-adically."

He goes on about adics, but in another post he said he just got the series out of Maple.

Now there's the issue of convergence of the series, but before getting into lots of technical stuff, notice that with his r2(t) in EACH TERM you can clearly see 2 is a factor.

I'll assume that goes out to infinity, which I'm sure is easily verifiable.

Now then, if t is an algebraic integer, and the series converges, then necessarily it converges to the roots of x^2 - (3+t)x + 2. Some of you may be confused on this convergence thing, but the gist of it is that yes, you can expand out a solution with an infinite series and it only apply in certain ranges, as otherwise the series blows up, as in goes infinite.

Here you have

x^2 - (3+t)x + 2 = (x - r1(t))*(x - r2(t))

with

r1(t) = 1 - t + 2t^2 - 6t^3 + 22t^4 - 90t^5 + …

r2(t) = 2 + 2t - 2t^2 + 6t^3 - 22t^4 + 90t^5 - …

when t is in a range where the infinite series do not blow up.

So, like, you know that r1(t)*r2(t) = 2.

And it's easy to see that r2(t) has 2 as a factor in EACH TERM so it makes sense that r1(t) will give a unit, when you have t an algebraic integer for which the series is valid.

And in fact, you can see that at the trivial t=0.

So now if you wish to disbelieve the result you have to believe that with every term multiplied by 2, the sum does not have 2 as a factor—when the actuality is that it may not in the ring of algebraic integers.

What's interesting here is that you have

r1(t) = 3+ t - r2(t)

so the infinite series is the same for each function, BUT r1(t) and r2(t) must be algebraic integers if t is an algebraic integer because they are roots of a monic polynomial with algebraic integer coefficients.

So the infinite series if it converges will converge to algebraic integers.

Oh yeah, you can get answers easily enough since

x^2 - (3+t)x + 2 = (x - r1(t))*(x - r2(t))

with

r1(t) = 1 - t + 2t^2 - 6t^3 + 22t^4 - 90t^5 + …

r2(t) = 2 + 2t - 2t^2 + 6t^3 - 22t^4 + 90t^5 - …

when t is in a range where the infinite series do not blow up.

as you can just plug in some t, like t=sqrt(5) - 2, and get to a monic polynomial with integer coefficients to verify for yourself that you have algebraic integers. Proving the series then converges is a different matter, but that can be done as well.

To ignore this result and continue with what you think you knew about mathematics you have to accept that 2 can be a factor of EVERY TERM but somehow, mysteriously and against any logical position, just fade away as a factor of the full sum just because it might not be in the ring of algebraic integers.

Note: Standard teaching is that BOTH r1(t) and r2(t) would have non-unit factors of 2 because they would in the ring of algebraic integers if 2 is not a factor of r2(t) in that ring.

But despite each term having 2 as a factor in the ring of algebraic integers, the sum may not have 2 as a factor—because that sum may not be able to be the root of a monic polynomial with integer coefficients!!!

The rule limits whether or not you get an algebraic integer, but notice algebraically it has no impact, so with each term having 2 as a factor, the sum must have 2 as a factor, in a ring that algebraically makes sense and does not have the coverage problem.

Now I have proven a problem with the ring of algebraic integers multiple ways over the years since I found it, and even had a paper published, but mathematicians have decided to go against proof in this area because of the fallout from the coverage problem.

Part of my point though to undergrads is that the graduate students and professors who grew up with the coverage problem and built their careers on flawed foundations are screwed.

You are not.

But they are not going to help you as they ARE screwed.

So math undergrads can go along if they like to protect people who thought they were brilliant mathematicians but missed an error that destroys their "proofs" and maybe their careers, but why?

Why keep working hard at your studies to learn crap? Why fight for knowledge that doesn't work? Why take tests to show your mastery of wrong ideas?

And make no mistake, I've proven this problem multiple ways and even got a paper published—before some sci.math'ers with Ph.D's who are screwed by this problem emailed the editors attacking the paper and managed to get it yanked.

They are SCREWED. This error shatters their illusions of mastery in mathematics, so they fight it as if they admit the truth then they are ordinary people, and not brilliant mathematicians.

So they will sell out the undergrads who have a chance for a fresh start, to bring you in, so that when you have spent years invested in bad ideas, you will fight for them too, and humanity loses out as mathematicians turn against mathematical proof, while hiding the truth from everyone.

## JSH: The "pure math" debacle

The real world is great for figuring out when you just screw-up, as there have been plenty of ideas people thought were great—until they tried them out in the real world and failed.

But over a hundred years ago mathematician moved away from the real world and began promoting "pure math" which had no practical value, so there was no way to tell when massive screw-ups happened, unless people got lucky, AND were willing to admit the truth.

In the arguments I'm having on these newsgroups you can see why people cannot be trusted to go with logic, which is why "pure math" fails, as some person gets some harebrained mathematical idea that he thinks is brilliant, and if he's considered tops in his field, the other mathematicians go along with him.

You challenge him, and you're pushed out of the group.

You push and you're labeled a crackpot.

Without the real world to show he's full of it, the ideas become part of the established teachings and get ever harder to unseat.

So "pure math" fails because human beings cannot be trusted to tell the truth, even when it hurts, especially when it hurts their egos!!!

So I have multiple ways to prove the coverage problem with the ring of algebrai integers, and you can watch how math people dodge every proof, every obvious point, and even play stupid with basic algebra, to keepthe wrong ideas in place, showing why "pure math" is a complete failure.

In contrast, in the physics field, someone claims they have a device that generates lots of free energy, well, people want to plug it up, right?

But in the math field, some professor claims he has a great proof, and two or three other professors chime in and say he does, you have to fight, and fight, and fight to argue that he does not, no matter what you find, unless other mathematicians decide they wish to agree with you!

Every example where mathematicians have gone along with saying someone is wrong, shows you how easy it is for someone that is wrong to get credit for being right.

Students pay the price.

You learn wrong mathematical ideas, can appear to prove ANYTHING with wrong ideas when you actually prove nothing, and eventually history will correct, and you will mostly be the invisible victims.

Make no mistake, if you come up with what you think is a brilliant mathematical argument using flawed ideas that argument is not brilliant. And neither are you.

[A reply to someone who thanked James for for saying some of what he had been thinking.]

I wish I could trust you on that but I get so many people playing what they think are funny games that I'll have to doubt your sincerity, but make several crucial points in reply:
1. I got published. My paper went to a peer reviewed mathematical journal, where I even told them that I was not a professional mathematician, and got quite a bit of support from the editors—up until some sci.math people starting an email campaign against it.

They had the paper for nine months. Sci.math'ers got them to yank it in a DAY.

Nine months of waiting down the drain in 24 hours.

2. I can explain points down to basic axioms, even getting into debates where I easily proved that my conclusions follow trivially from the distributive property, only to get posters claiming, yes, they believe the distributive property, but still saying I had to be wrong.
That and more shows you what's wrong with "pure math" as a real world activity as people can just lie.

Acceptance of mathematical proof depends on people willing to follow rules, and over and over again people here refuse to follow rules, so the impasse continues.

I've wryly noted that if my ideas could build a better atomic bomb there wouldn't be a problem.

But that goes to the heart of the issue—if there is nothing but the word of some people then those people can lie.

And a lot of them can lie and they can lie for long periods of time especially when their interests are at stake.

So "pure math" fails because people can do "pure" junk, deny that's what they're doing, and fight like hell when you prove that is what they're doing, as they try to make damn sure no one believes you!

## Coverage problem with algebraic integers

I came across yet another way to show the coverage problem of the ring of algebraic integer, again using mostly simple algebra.

Use the polynomial

x^2 - (3+t)x + 2 = 0

as when t is an algebraic integer both roots must be algebraic integers, and also when t = 0, the roots are quite simply 1 and 2.

So express the roots r_1 and r_2 as

r_1 = 1 + t^k*f_1(t)

r_2 = 2 + t^k*f_2(t)

where k is some non-zero rational number, and f_1(t) and f_2(t) are algebraic integer functions of t.

I know I have t^k as a factor because those terms zero out when t=0, as then you just have the trivial

x^2 - 3x + 2 = 0.

Notice then that

r_1 + r_2 = 3+t = 3 + t^k*(f_1(t) + f_2(t)

so

t^(1-k) = f_1(t) + f_2(t)

so abs(k) is less than or equal to 1.

Multiplying the root together you have

r_1 * r_2 = 2 = 2 + t^k*f_2(t) + 2*t^k*f_1(t) + t^{2k}*f_1(t)*f_2(t)

so

t^k*f_2(t) + 2*t^k*f_1(t) + t^{2k}*f_1(t)*f_2(t) = 0

so

f_2(t) + 2*f_1(t) + t^k*f_1(t)*f_2(t) = 0

which tells you that f_1(t) must give unit results.

Now it's trivial to show the coverage problem by using t=2 as then

I have

x^2 - 5x + 2 = 0

and the roots are

r_1 = 1 + 2^k*f_1(2)

r_2 = 2 + 2^k*f_2(2)

so one root is coprime to 2, which forces the other to have 2 as a factor, meaning k=1.

But let x = 2y, and I have

4y^2 - 10y + 2 = 0

and dividing both sides by 2 gives

2y^2 - 5y + 1 = 0

but that is a non-monic polynomial with integer coefficients irreducible over Q—that is, it does not have rational roots—so it cannot have algebraic integer roots proving that neither root of x^2 - 5x + 2 = 0 can have 2 as a factor!!!

The seeming contradiction is resolved by realizing that the ring of algebraic integers excludes some numbers, which isn't so remarkable if you consider evens as a ring, and note that 2 is coprime to 6 in that ring because 3 is not even.

But mathematicians haven't noticed this problem for over a hundred years while using the ring of algebraic integers during that time, so there are mathematical arguments believed to be proofs which are not.

As an analogy continuing with the evens example, imagine someone had what they thought was a proof that settled on 2 being coprime to 6, in evens. But it's not a proof if they realize that 3 is a factor of 6, though not even, but they just don't know it somehow.

That's kind of what happened over the last hundred plus years in number theory.

It's easy to prove the problem, but quite a few people around the world have careers built upon it, like if many people had arguments depending on 2 being coprime to 6 in evens, and someone finally pointed out that 3, while not even, is a factor of 6.

The sad thing is that modern mathematicians have chose to ignore the error and keep teaching the wrong mathematics!!! Which they are doing to this day.

Like, college students around the world—even at premiere schools like Harvard or Yale—tomorrow will get homework on or be tested on bogus mathematics that doesn't work because of this weird error, when the information is out there, but mathematicians are avoiding the truth maybe because it's easier!

## JSH: Actually, easy answer anyway

I noted that a poster going by the name Gene Ward Smith claimed that

r1(t) = 1 - t + 2t^2 - 6t^3 + 22t^4 - 90t^5 + …

r2(t) = 2 + 2t - 2t^2 + 6t^3 - 22t^4 + 90t^5 - …

are the two roots of x^2 - (3+t)x + 2, and I've explained how you can easily use that to show the coverage problem I talk about—if he's right.

But you can make a fairly easy argument regardless of whether or not that series is correct, as consider the solution when t=0, as then the roots are definitely 1 and 2.

So you should have something like those two series no matter what, as everything else has to go away when t=0.

Given that the roots of

x^2 - (3+t)x + 2 = 0

must be algebraic integers if t is an algebraic integer, then you can use the fact that

r_1 + r_2 = 3+t

and use something like t = sqrt(6) - 2, which forces r_1 and r_2 to be coprime to each other.

Since t has sqrt(2) itself as a factor then, and given the earlier situation where I noted that every term must have t itself as a factor in any expansion because when t=0, you just have 1 and 2, it's then trivial that one root will then not have 2 as a factor.

But now you just substitute

x^2 - (3 + sqrt(6) - 2)x + 2 = 0

and get to a polynomial with rational coefficients, and see if it's reducible over Q.

I think the problem here is partly that many number theorists lack the proper intuition in this area to realize that there was something wacky about how Galois Theory supposedly worked, as why would there be this weirdness only for non-rationals?

But worse than lacking the proper intuition modern number theorists clearly have a lot of political know-how, as I've been blocked repeatedly, where even publication hasn't mattered, while they've kept teaching the wrong information, protecting their careers against the interests of humanity.

After all, mathematics is an important subject. The lack of linking between "pure math" and physics in most areas is probably a result of the "pure math" being wrong.

So there was never ever going to be a time where the ideas using ideal theory would be practical as they were crap, but with non-experts being able to masquerade as experts and block people with mathematical know-how, there has been no way to get the truth out.

So the real story is bizarrely the opposite as I'm not the crackpot here—modern number theorists are.

I am the actual number theory expert trying to handle a situation where the crackpots took over a field.

[A reply to someone who told James to prove that modern number theorists are crackpots.]

Easily done. The polynomial again is

x2 - (3+t)x + 2 = 0

where when t is an algebraic integer both roots must be algebraic integers, and also when t = 0, the roots are 1 and 2.

So express the roots r_1 and r_2 as

r_1 = 1 + t^k*f_1(t)

r_2 = 2 + t^k*f_2(t)

where k is some non-zero rational number, and f_1(t) and f_2(t) are algebraic integer functions of t.

Notice then that

r_1 + r_2 = 3+t = 3 + t^k*(f_1(t) + f_2(t)

so

t^(1-k) = f_1(t) + f_2(t)

so abs(k) is less than or equal to1.

Multiplying the root together you have

r_1 * r_2 = 2 = 2 + t^k*f_2(t) + 2*t^k*f_1(t) + t^{2k}*f_1(t)*f_2(t)

so

t^k*f_2(t) + 2*t^k*f_1(t) + t^{2k}*f_1(t)*f_2(t) = 0

so

f_2(t) + 2*f_1(t) + t^k*f_1(t)*f_2(t) = 0

which as a sidenote tells you that f_1(t) must give unit results.

But, of course, it can't in the ring of algebraic integers.

Notice then the coverage problem is trivially shown by using a t that has 2 as a factor:

r_1 = 1 + t^k*f_1(t)

r_2 = 2 + t^k*f_2(t)

because then one root must have 2 as a factor while the other then is a unit.

But now I can let t just equal 2, and note that

x2 - 5x + 2 = 0

and you have this interesting result because then

r_1 = 1 + 2^k*f_1(t)

r_2 = 2 + 2^k*f_2(t)

as one root has 2 as a factor, but provably does NOT have 2 as a factor in the ring of algebraic integers.

Readers wishing to attack the result run into the problem of what happens when t=0, as how else can you get the zeroing out of all the extra terms unless you have t^k as a factor with k a non-rational number?

I like this way of explaining as then I don't have to worry about whether a particular infinite series converges or not, as the complexity is wrapped up in the function f_1(t) and f_2(t).

Easy proof. Trouble is, it kills ideal theory.

Mathematicians though who refuse mathematical proof are not actually mathematicians but must then be crackpots!

This result in and of itself does not give crackpot status to posters arguing with me, but refusing to acknowledge mathematical proof does.

### Saturday, September 23, 2006

Hey! A Gene Ward Smith made a post in reply to me where he put the following:

"Consider the polynomial x^2 - (3+t)x + 2. The two roots of this can be expanded in series as

r1(t) = 1 - t + 2t^2 - 6t^3 + 22t^4 - 90t^5 + …

r2(t) = 2 + 2t - 2t^2 + 6t^3 - 22t^4 + 90t^5 - …

If t is 2-adically a nonunit, so that |t|_2 < 1, then these converge in the 2-adic integers. One root, r2, is even, the other, r1, is odd. The 2 does not split up. If t is a rational integer, this means that when t is even, the roots split into an even root and an odd root 2-adically."

So I say let t equal a non-rational algebraic integer unit.

Chew on that one for a while, and let's see where this discussion goes.

Needed, (1)verification of Smith's claim. (2)Verification that a non-rational algebraic integer unit will be within the size range.

(3)Some goddamn acceptance from math people when precious ideas that DO NOT FREAKING WORK are proven to be wrong.

Show some damn balls—SOME courage.

Any? Any of you have an ounce of courage in your miserable selves? An ounce of courage for the truth?

## JSH: Nothing but ridicule

If you look over the thread where I talk about the Catch-22 of catching members of a major discipline with a massive and very old error, which invalidates too many of them for them to acknowledge it on their own, you will see post after post in reply that is just about ridicule.

Math people have this down to a science.

They do not answer objectively. They do not concern themselves with the facts.

They just ridicule and it works!!!

Yup, many of you will expect—no matter what I remind about when careers are challenged—that if there were anything to what I'm saying SOMEONE would reply agreeing with me.

But they just ridicule, and some of you may have an inkling that maybe there is something to what I'm saying, but despair working it all out, and why bother? So you just wander off, maybe a little more skeptical of what people in math areas might claim, but then again, is it really your problem?

Well, consider just how broad the strokes of these people are by doing a web search on Britney Gallivan, a young woman who gained distinction for figuring out something that may seem trivial—equations on paper folding when she was a teenager.

Now do a web search on Karl Gauss, the great German mathematician who among many, many other things found the gaussian distribution, otherwise known as the bell curve. He also did mathematical research in astronomy, figuring out planetary orbits, worked out a lot of calculus and so many other areas that it is hard to imagine his full impact on our world.

But what did he gain a lot of early fame for?

He figured out how to draw a 17-sided figure using only a straight-edge and compass, as, yup, as a teenager, like Britney Gallivan.

Yes, Ms. Gallivan has had recognition, but the fluffly celebrity-ish kind, and some mention here and there on mainstream math sites but NOTHING like the official recognition she should get, and what impact might that have?

Are we losing a potential Gauss because math society is dominated by people who challenge real discovery, partly maybe to hide serious flaws in their own research?

Allowing mathematicians to behave this way with a young woman should be criminal, and should sicken those of you who care about our intellectual future, and may even wonder why women are less represented in mathematics and the sciences.

If despite all of that modern mathematicians can just mostly ignore a dramatic accomplishment of one young woman, and not have to answer to anyone, then maybe you need to wonder about your role in allowing this sort of thing to go on.

YOU fund more mathematical research than you probably know, if you are in a country like the United States or in the European Union or other areas where there is a lot of public funding for mathematical research.

These people can do what they do because of your tax dollars in those cases, and they can escape accountability by snowing people, using ridicule when challenged—and ignoring important research from outsiders—like Britney Gallivan.

It is a corrupt society that you help create, every day, every moment that you sit back while your tax dollars work—to keep people satisfied and secure in academic jobs when those same people spend their time fighting the discoverers.

You may disdain sports, but ask yourself: Would Mario Andretti, a race car driver, spend much time attacking amateur drivers? Would Michael Jordan go out of his way to lament "crackpot" amateur basketball players? Can you see Jack Nickolson, ridiculing golfers who dare to try his sport?

Why then do you think it normal and ok for mathematicians, supposedly well-trained professionals, to use playground tactics, like namecalling and ridicule against people who dabble in mathematics?

And if you don't think it matters—what if Britney Gallivan were your daughter? Or your sister? Or your friend? Or just someone you knew in school?

These people have a contempt for knowledge and they are worse than just being wrong, as my research has shown they are wrong in some key areas of number theory, but they also go out of their way to ignore or attack people who are right.

And they do it with your tax dollars.

## JSH: Simple explanation, coverage problem

Having noted that it is unlikely that any number theorist will acknowledge the coverage problem of the ring of algebraic integers on their own—as it invalidates their degrees—I might as well explain, yet again, just how easy it all is, and why there is this problem.

Remember, algebraic integers are defined to be roots of monic polynomials with integer coefficients.

That definition can be proven to require that NONE of the roots of a monic polynomial with integer coefficients that is irreducible over Q can be coprime to any prime factor of the last coefficient.

It is a quirk of the ring, and mathematicians wrongly concluded over a hundred years ago that meant that non-rationals were somehow special in comparison to rationals, so that you had this odd and special result.

How special?

Well, look at some simple quadratic examples:

x^2 - 3x + 2 = (x-2)(x-1)

is a pairing of a rational unit with a rational non-unit in the ring of algebraic integers.

THAT CANNOT HAPPEN with non-rationals in the same ring.

So mathematicians assume that with the roots of something like

x^2 - 5x + 2 = 0

where you do not have rationals, that 2 is somehow split up, and factors between the two roots.

But what my research proves is that nope, it doesn't have to split up (though it still might) as the ring of algebraic integers will simply NOT allow either root to have 2 as a factor IN THAT RING.

That does not mean, however, that 2 is not a factor of only one root while the other is a unit.

Confused? Consider evens with 2 and 6, and note that in evens 2 does not have a factor in common with 6 because 3 IS NOT EVEN.

Because 2 can't be a factor in the ring of algebraic integer with my simple irreducible quadratic example of one root, the ring of algebraic integers just does not allow the other root to be a unit. Easy. But it still can actually be a unit.

So how do you know? Turns out that you can prove it with some careful mathematical analysis and a special tool I call non-polynomial factorization. Factor a polynomial into non-polynomial factors and you can see all kinds of neat and weird things, as well as RIGOROUSLY prove the problem.

I have a published paper that says I accomplished the proof, and a retraction of that paper, and the later death of the journal to let you understand how big the politics here are.

The math is easy. The consequences are what's hard.

Proving that the ring of algebraic integers has this coverage problem requires that you factor polynomials into non-polynomial factors, which is the new thing that I brought to the table, which showed the problem.

The problem hasn't been proving that the techniques I use work, as the proof is easy relying on mostly trivial algebra, but in climbing over the waves of denial that have swept the mathematical world, or maybe I should say the Usenet math world, though I have had contacts off of Usenet.

It's easy to prove the coverage problem, but if number theorists accept it, they lose any results that depend on the flawed ideas, and that is a huge thing, as it can be shown that it takes away the theory of ideals, and changes how Galois Theory is seen to work.

So how does it change the view on how Galois Theory works?

Simplest answer is that taking away the supposed difference between rational and non-rational numbers means that Galois Theory tells you only as much with non-rationals, as it does with rationals.

Number theorists have shown that they will not accept the truth on their own.

And it's not a big surprise why they will not, and why others when they realize the scope of the problem turn tail and run, because, for instance, Wiles did not prove Taniyama-Shimura, which is then still a conjecture, as the theory of ideals and the old way of using Galois Theory just went away.

With this result, people like Wiles may have no major proofs discovered at all in their entire careers and with the truth, may not be able to justify their positions and status.

So, if Wiles acknowledged this result, next thing he'd need to do would be to step down from any positions that would come from thinking the flawed results were correct, as would mathematicians around the world.

Understand the problem?

It's not about mathematical proof but about human denial of mathematical proof, where the people with the power would lose that power by telling the truth, by invalidating their own expertise. It's a freaky thing that I guess has never happened in the world's intellectual history, where the very people who are tasked with acknowledging a result are the ones who would lose their expert status by doing so.

[A reply to someone who said that any mathematician that would find an inconsistence in the definition of algebraic integers would become famous and therefore would not hide it.]

The problem is the definition leads to a ring with a coverage problem.

And no, they wouldn't be famous—they'd be infamous among all the mathematicians invalidated by the result!

Like, let's say some relatively unknown mathematician at some university takes this result and runs with it, down the road, it topples Wiles, Ribet and Taylor, who end up with NO MAJOR RESULTS for their entire careers.

That mathematician would have to run the gauntlet with just about every other number theorist in the world gunning for them.

Before it were over that person would be called every name in the book, have people attacking personally, and even maybe trying to probe into personal information—all in a growing campaign to shoot that person down, no matter what.

Nope. Math people don't have those kind of balls. The reaction to this result that I've seen is terror or denial, like with that math grad student at Cornell University.

Rather than being my poster boy for total fear in the graduate community, he could have been out there pushing the result, but by running and hiding, he can still try for a career as a mathematician.

You know, he does read sci.math, as he's responded to me a few times about ripping on him.

But abject terror defines the math community in this area, or total denial.

None of you have the courage to push the truth here.

### Thursday, September 21, 2006

I am stuck in a situation I HATE. What school is there for this? Where can you learn to deal with something so unimaginable?

Well I don't know what the hell I'm doing. I'm scared as hell, and often I wish I could just wake up from this stupid and annoying dream.

So I'm saying, hey, this is a dream, and at this point I'm supposed to say, you, make a choice, check the facts, use some goddamn common-sense and act like an adult, and just consider the possibility, please.

I hate this, from going from being a science nerd, who devoured any news from that world with interest, to being someone who is skeptical of just about everything I read from academics because of what I am facing now.

So some star blew up, and now astronomers say they're not sure about their models because it was bigger than the Chandrasekhar limit?

You know what I think when I read that? I think, who gives a damn what these losers think.

I don't.

Not any more.

I can't stand this, and what makes it that much more horrible is all of you.

You, the same people who sit by, and let George W. Bush invade Iraq, with barely a peep out of most of you. You, the same people who sit by with global warming. You, the same people who sit by while torture becomes the law of the world.

You, the same people who wonder when you scream and die, finally caught in the wrong place at the wrong time, and no one listens.

While you, sit back, comfortable, certain that you're safe in a world that is becoming ever more dramatic in its extremes, and less and less forgiving for people like you, who refuse to think.

Is your son in Iraq? Does that matter? Or do you blindly follow President Bush, even though that means he may be torturing someone, broken in spirit, betrayed in mind, because YOU are not what you claim to be either.

Or maybe you're in Iraq, or in some other country, like Pakistan, or Yugoslavia, or Venezuela, maybe getting a tummy-tuck, or some other plastic surgery.

Humanity today is defined by its decisions NOT to think through its choices.

Believe in God to save you? Or maybe in your atheism and disbelief in God to save you?

Who cares? This world today shows the value of your beliefs—their impotence.

What you allow defines you. What you say is meaningless in the face of what you do.

So you choose. Every day you choose. And with your choices you make this new world, whether you admit it or not.

## JSH: What if? Academic nightmare supposition

Just for the sake of argument, suppose that someone found a massive error in some major discipline that went back over a hundred years and was big enough that it invalidated the degrees of most of the people in this area, as their coursework heavily relied on the flawed area.

Is there any reason to believe that anyone in that discipline would acknowledge such a find?

Even if there were some support given for the person who found the error, what if people who figured they'd lose everything they'd worked their entire lives for, simply chose to try to hide the error?

Would there be any way that it could be found out regardless?

How about a journal publishing a key paper, retracting it under pressure and dying soon after? Might that have an impact?

It didn't.

That scenario is a current reality as I'm the poor slob who found an over one hundred year old error in some abstract algebraic number theory.

Mathematicians in number theory have taken the run away road, and I have lots of stories that are dramatic, and true, where they've gone out of their way to avoid the result, even in one case, after proving it themselves.

But it's a Catch-22. The world trusts these people and counts them as experts in the area. My result takes away their expertise, so they deny it.

Hey, I even got a paper published. Some sci.math people emailed the editors of the math journal claiming the paper was wrong. The editors yanked it immediately, then put up that it was withdrawn—they withdrew it, not me. They managed one more edition of their journal, and then quietly shut down.

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

and

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

Of course the sci.math'ers rant about how horrible the journal was, and rationalize the destruction of a math journal as if it were an every day, minor occurrence.

All within the bounds of the scenario I painted.

So who can you believe?

I have multiple math results, some of them comprehensible to regular people, as one of them counts prime numbers. But beyond that, hey, I've been a computer programmer. Some of my software coding may have helped save your lives, but keeping with the traditions of my field I will not talk specifically about what company I mean. And also, the math people can get vicious so I am wary about how much personal information I give.

Nonetheless, some of my work may have helped save your life or that of a loved one of yours.

So you can't check that but you can do a web search on "Class Viewer" and see a little tool I wrote for Java programmers. Yup, Java is currently the most popular programming language on the planet. I have one of the top apps in its domain, which is a tool for people who help build your technological world.

And if none of that grabs you, read over some of my research in logic:

http://mymath.blogspot.com/2005/06/three-valued-logic.html

I explain why logical systems need three values: true, false, and unresolvable.

Oh, hey, or you can read the first prime counting function article on the Wikipedia that I wrote, where my final version is still available in the history:

http://en.wikipedia.org/w/index.php?title=Prime_counting_function&oldid=9142249

Mathematicians will not acknowledge the truth because it invalidates them.

This scenario has never occurred where an over one hundred year old error has been found in a mature modern discipline, with enough impact to invalidate every top researcher in the field, as well as the middle ones and the bottom ones.

That means that any supporter I get will have to fight the top mathematicians in this area, who are fighting for their prestige, their positions, and their very sense of self.

It's harder to imagine a greater challenge in our highly political world.

But I am hoping that the some of you do the math, or check the facts, and slowly the idea goes out that some very powerful people are hiding a very embarrassing truth, and eventually enough momentum is gained that they are forced to defend themselves, and when it is obvious they cannot go against the mathematical proofs I have, people will quit giving them the benefit of the doubt.

## JSH: Proving is the easy part

It's not at all difficult for me to prove my result showing the coverage problem of the ring of algebraic integers. Most of the mathematics after all is just basic algebra.

I can give examples to explain it simply enough for most of you to understand as that 2 coprime to 6 in evens tells exactly what a coverage problem looks like.

And the result was published, but then some political stuff got it yanked and the journal died.

If you were reading about all of this as ancient history, you would be amazed that any sensible people could ever fight against the result, and be shocked that denial could last for years.

But this isn't ancient history—it's right now—and for some of you the truth means you did not get a valid doctorate, or that argument you thought was a brilliant proof was not, not brilliant, and not a proof.

If you are a professor, you may have the sinking feeling in your gut of having taught students wrongly for years, even decades.

But what if you just deny, right?

That is the choice that has been made up to this point, so more students can be taught wrongly, and more people can go through years with the wrong mathematical ideas.

Proving the mathematics is the easy part. Getting past basic human denial—people who can't accept not being brilliant, who can't accept being the wrong ones, and now being closer to crackpots than actual mathematicians as they refuse to accept, that is what's so hard.

## JSH: How stupid do you have to be?

I not only have that HUGE algebraic number theory result that got published in a peer reviewed math journal which retracted under social pressure and then committed seppuku, I have lots of other mathematical and logic results as well.

LIKE I have my prime counting function that hey, you can get to a partial differential equation from, and hey, that has never been seen before in history, but wow, supposedly brilliant people ignore the result.

But, um, they're not really brilliant.

Didn't get that part?

My results prove they are NOT really brilliant!

But it proves that many of you are not as well, right?

You study hard to learn Galois Theory or the theory of ideals and that gives you this ownership, right?

So why admit that energy was wasted?

Because if you have what it takes to be a real mathematician versus a pretend one, you admit it.

The real story here is how few of you have even an ounce of what it takes to be a real mathematician.

But why would you?

Many of you will never amount to much in this life, and will never compare to those brilliant people you want to be.

Telling the truth here just takes away an area where you could pretend to be something—and have the world recognize you—that you could never actually be.

And pretend can be good enough, can't it?

When that's all you have, why not?

Who of you could be anything like someone like me?

Someone with real results, fought for, battled out, where the mathematics says I am right versus some people holding on to their wishes and hoped for's.

Who among you has greatness in you?

And that's the problem. So few do that the many can choose to lie, feel comfortable in the lie, and keep going knowing that without it, they are just ordinary people.

## JSH: Thoroughly tested result

I have a peer reviewed and published result, which has been argued out in extreme detail for years, and there is no room for error.

That result shows a coverage problem with the ring of algebraic integers, so for instance,

x^2 - 12x + 65 = 0

has one root that can be shown to be coprime to 5 but it can also be proven that none of the roots are coprime to 5 in the ring of algebraic integers.

An analogy to understand what happens can easily be seen by considering evens, and noting that if you have evens as a ring, then 6 is coprime to 2. Wow!!! How can that be??!!!!

It's true because 3 is not even, that's how.

So, with x^2 - 12x + 65 = 0, two of the roots have sqrt(5) as a factor—but not in the ring of algebraic integers—just like 6 has 2 as a factor, but not in evens as a ring.

It's not even hard to understand, so why all the arguing, and how is it possible?

Well, it turns out that it's the definition of algebraic integers as roots of monic polynomials with integer coefficients that is the culprit.

That highly specific definition blocks certain numbers, which still behave more like integers than like fractions, that is, more like 2 than 1/2, which is why I abstracted out the key properties of the ring of integers, to get the object ring:

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:
1. 1 and -1 are the only rationals that are units in the ring.
2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.
And that's it. Turns out that you only need two key properties and you have numbers that are integer-like, or are in fact, integers. I call such numbers objects.

Even that concept isn't terribly hard as consider 1/2.

Do you ever get 1/2 or do you get 1/2 of something?

You always get 1/2 of something.

So 1/2 is like an instruction—a set instruction—telling you to take one of two equal objects from a set of two.

Hey folks, this isn't rocket science. But it is overturning, as if you do the mathematics right, wow, out the window goes—<gasp>—ideal theory!!!

And there you have it. Why mathematicians would ignore this result—they're not real mathematicians!

Now Wiles has nothing, neither does Ribet, nor Taylor.

Suddenly, they're just ordinary people who join the ranks of people who thought they had something proven mathematically, but did not because they missed something small, that nevertheless overturns their results.

Would you tell the truth if you were them? Knowing that your ENTIRE CAREER could go up in smoke?

So there, people with Ph.D's in mathematics whose thesis would go away, have lots of motivation to keep arguing over this until they keel over and freaking die, because the alternative is to admit being an ordinary person—not a brilliant mathematician after all.

Just another person who tried and failed.

But make no mistake, Andrew Wiles, Ribet, Taylor, Granville and any number of other supposedly top mathematicians are not.

They are failures, who today are simply succeeding at hiding the reality.

Well, at least they have that.

## JSH: When anger hurts

So you can prove something mathematically, cover all the dots, handle all the objections and then have people just ignore you, or worse, say you didn't do what you did, and you wait, waiting for validation from SOMEWHERE and it doesn't come.

So what might you do then?

Well, you may GET REALLY ANGRY and it's not hard to post some angry stuff, and next thing you know, guess what?

You're labeled a crackpot or a crank.

It's a perfect system in a way, as while people trust mathematicians to tell them the truth as a group, they will just dismiss small groups of people or one person telling them that mathematicians routinely lie—for their careers.

I mean, most people aren't stupid, they know that lying for one's career can be a common thing, but they don't connect with large scale lying by people they are also told are brilliant, even when you can prove that lying is the best way for those people to protect themselves from truth, versus biting the bullet and just handling the consequences of what is mathematically true.

Before you learn that though, if you make that discovery that mathematicians routinely lie, you can get angry, and express that anger, and then they happily label you a crank or a crackpot, and guess what?

They keep doing what they are doing.

The world needs to look at the reality and comprehend that yes, academics do lie.

How do you know with any academic speciality whether or not you are getting lies fed to you by some person or persons trying to help or maintain their career?

People know politicians lie, and it's time for them to understand that academics can lie as well, and they can lie as a group.

So here it is directly for people who do not get it:

ACADEMICS CAN LIE AS A GROUP TO HELP OR MAINTAIN THEIR CAREERS.

And that can anger people who find out, and their anger can be used against them by people who know what it takes to keep the lies going.

[A reply to someone who wanted to know when did James “prove something mathematically, cover all the dots” and so on.]

And that's part of the politics of all of this, no matter what, people like you come in and act like there are no proofs, despite the reality, and people trust you.

But the point I'm making is, people must check academics and not just trust that if a lot of professors claim something is true that it is.

Reality is a LOT of people in a discipline—like mathematics—can decide to hold on to something that is just not true, and they know that people trust them.

Problem is, someone like me comes along, and people like you do your thing, and I keep at it. You sit back, satisfied that you did your jobs.

Then I talk some more, and you do your thing, and nothing happens.

You sit back, satisfied.

Then I talk some more and people listen, and you go into complete freaking shock as your world is totally destroyed because you believed that because you never saw anything happening, nothing ever would happen.

And history repeats.

## JSH: So what's the result?

I think I keep stumbling a bit in trying to explain this thing, so I'm going to keep replying in the other threads, but try to bottomline it in this one so that it makes sense.

Bottomline is that you can try to stay in the ring of algebraic integers with a polynomial you can't factor into polynomials, like

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

with a factorization that CAN be done in the ring of algebraic integers:

P(x) = (f(x) + 1)*(g(x) + 2)

where you have f(x) and g(x) that will give you algebraic integers, if x is an algebraic integer.

There are an infinity of such functions.

There are an infinity of them with the requirement that f(0) = g(0) = 0.

Despite that infinity though, a few more steps and no functions out of that infinity can still remain.

And that's as simple as going to

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

and it's almost simple enough that most of what I did in my previous threads was unnecessary.

Now the a's can't be algebraic integer if f(x) and g(x) are algebraic integer functions.

Or if the a's are algebraic integers, then f(x) and g(x) can't be.

It's that simple.

One way to see what can happen that algebraic integers can't handle is to consider

2x^2 - 3x + 1 = (2x - 1)*(x - 1)

and you may say, trivial. Yeah, well, the ideas here aren't that terribly complicated people as notice you have one root that is an integer, while the other is a fraction as it's 1/2.

That's what you can't get with algebraic integers.

So

2x^2 - 5x + 1 = 0

can't have an algebraic integer root, so mathematicians decided you can't have a pairing like I just showed you, but they were wrong.

It's that simple. They made a mistake that can be explained with some quadratics.

The reality is that non-rationals aren't so different, and a number that is more like 1 than it's like 1/2 can be paired with a number that's more like 1/2 than it's like 1, but not in the ring of algebraic integers.

So if you follow the math and go with what is proven, ideal theory doesn't work, as it's not doing anything. There is none of this class number anything. No ideals that mean anything.

Nothing to any of that, it's just crap.

Human nature being what it is, people can deny that, and deny all of my research to deny it, deny publication or a dying journal, just like people before you could deny that the world wasn't on the back of elephants, or believe that the sun could be made to stand still by the will of God for some stupid battle of some small group of minor people in a story about something that never actually happened.

You people in denying mathematical proof are no different from people who denied that man landed on the moon. You are no different from Creationists who will argue day and night that the earth was created in six days, or will tell you that soon the end of the world is coming and angels will fly around killing people in God's name.

You people in fighting mathematical proof are no different from any other group of people in the past or present who when faced with a truth they do not like, just decide to ignore the truth, against all facts, against even dramatic happenings, and may even fight to the death for their beliefs.

You are no different.

And for that reason, you are not really mathematicians.

## JSH: Showing what you can't do

So now I've gone to showing what you can't do with just algebraic integers. I want to make another post to just these two newsgroups though to explain a little more about how politically things have worked out.

My original paper steps through quite nicely in a very rigorous way a path to understanding there is something wrong with the old ideas on algebraic integers, but it's abstruse and complicated.

It is a much more difficult argument in many ways as I use a degree three polynomial, and other stuff, while using a much more stylized language—and sci.math'ers went to town confusing people about the mathematics.

So I've had to simplify, moving to quadratics, and trying to explain in simple ways, but posters on sci.math repeatedly found ways to confuse, so I finally—years have passed now—came up with the notion of showing what you CANNOT do with the old ways of thinking.

So why was all that necessary?

After all, I've done much, much more than just argue about this on Usenet. Barry Mazur himself saw an early draft of my paper in this area and commented on it, so I know he read it. I sent it past Andrew Granville who passed on telling me the New York Journal of Mathematics wouldn't publish my paper, sending me to the chief editor who did pass on it.

And the paper, did, of course, get published in the now defunct Southwest Journal of Pure and Applied Mathematics, and some sci.math'ers sent some emails and convinced the chief editor to withdraw it.

What could be so huge? What is the big deal?

Well I found a small little problem with the idea that the ring of algebraic integers is big enough to cover all the numbers you need it to cover, when mathematicians thought it was.

So they have mathematical arguments that just get to the point of saying something like, and this cannot be this or that in the ring ofalgebraic integers, so this or that is true.

Well, I prove that is not sufficient, so that means that arguments thought to be proofs, are not.

How far back does this go?

Over a hundred years, all the way back to Dedekind.

How much impact?

Well, you can possibly toss textbooks in number theory into the trash going back, oh, about a hundred years back to Dedekind.

Whole swaths of the mathematical world are just wiped away, like flipping a switch.

Reams of papers just totally yanked away. Books, supposed accomplishments, the hopes and dreams of thousands of people who thought they had it right, but were wrong.

The mathematics here is in a way unimaginably cruel. It is about as cruel as you can get in a way because mathematics is so absolute, so hard.

So people fight the truth. Mathematicians turn away from mathematical proof—maybe even disgusted by it—and cling to just believing what they believe, and can look at social support as validation.

So as I've made my discoveries the mathematical world worldwide has shifted to being more of a social organization. Mathematicians are more about convincing each other than proving anything, as they shfit away from mathematical proof.

They have to, or face what for many of them could be considered, the end of the world.

By showing what you can't do with the old thinking I'm helping some of you understand that I am right, but you also need to understand that I've proven this result many ways over the years, and been heard by people at the top of math society.

It's not about mathematical proof at this point.

It's about a deep human need to just cry no!!!—at a reality deemed too cruel.

It is just so damn cruel and harsh to accept the truth here that a lot of people…just don't.

## Problem with algebraic integers

The results on prime numbers are telling but nothing compared to the sheer reach of my most controversial result, which shows a problem with what are called algebraic integers, as in, they can't do the full job.

Algebraic integers are kind of simple in that they are roots of monic polynomials with integer coefficients.

But now I'm going to step through a very short bit of algebra:

I will use a polynomial P(x), functions f(x), g(x), a_1(x) and a_2(x), where x is an algebraic integer.

Now let

P(x) = 175x^2 - 15x + 2 and consider the factorization

P(x) = (f(x) + 1)*(g(x) + 2)

where f(0) = g(0) = 0.

Now multiply by 7, so I have

7*P(x) = 7*(f(x) + 1)*(g(x) + 2)

and there are an infinite number of ways that you can distribute 7 through the factors on the right side, but we don't have time to check them all, so let me just pick one, and have

7*P(x) = (7*f(x) + 7)*(g(x) + 2).

But now let

a_1(x) = g(x) - 5 or a_1(x) = 7*f(x)

and

a_2(x) = 7*f(x) or a_2(x) = g(x) - 5

which gives

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

which is necessary for the a's to be algebraic integers if f(x) and g(x) are not rational.

And, get this, if a_1(x) and a_2(x) are non-rational algebraic integers—that is, given an algebraic integer x you get an algebraic integer result—then f(x) and g(x) cannot be algebraic integers!!!

That is the crucial point.

Remember f(0) = g(0) = 0.

Seem minor? Well, that result is big enough for a math journal to die in a fight over a paper in this area because if you pull the thread, so to speak, that is, consider the necessary mathematical conclusions that follow, then ideal theory can't work.

Run yourselves around on that one for a while, and hopefully some of you will understand the flaw with current views on the ring of algebraic integers and why my object ring is necessary:

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:
1. 1 and -1 are the only rationals that are units in the ring.
2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.
Faced with the inability of the ring of algebraic integers to cover all the ground necessary, I figured out the key properties needed for a
ring that does.

## Problem with algebraic integers

The results on prime numbers are telling but nothing compared to the sheer reach of my most controversial result, which shows a problem with what are called algebraic integers, as in, they can't do the full job.

Algebraic integers are kind of simple in that they are roots of monic polynomials with integer coefficients.

But now I'm going to step through a very short bit of algebra:

I will use a polynomial P(x), functions f(x), g(x), a_1(x) and a_2(x), where x is an algebraic integer.

Now let

P(x) = (f(x) + 1)*(g(x) + 1)

where f(0) = g(0) = 0.

Now multiply by 7, so I have

7*P(x) = 7*(f(x) + 1)*(g(x) + 1)

and there are an infinite number of ways that you can distribute 7 through the factors on the right side, but we don't have time to check them all, so let me just pick one, and have

7*P(x) = (7*f(x) + 7)*(g(x) + 1).

But now let

a_1(x) = g(x) - 6 or a_1(x) = 7*f(x)

and

a_2(x) = 7*f(x) or a_2(x) = g(x) - 6

which gives

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

which is necessary for the a's to be algebraic integers if f(x) and g(x) are not rational.

And, get this, if a_1(x) and a_2(x) are non-rational algebraic integers—that is, given an algebraic integer x you get an algebraic integer result—then f(x) and g(x) cannot be algebraic integers!!!

That is the crucial point.

Remember f(0) = g(0) = 0.

Seem minor? Well, that result is big enough for a math journal to die in a fight over a paper in this area because if you pull the thread, so to speak, that is, consider the necessary mathematical conclusions that follow, then ideal theory can't work.

Run yourselves around on that one for a while, and hopefully some of you will understand the flaw with current views on the ring of algebraic integers and why my object ring is necessary:

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:
1. 1 and -1 are the only rationals that are units in the ring.
2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.
Faced with the inability of the ring of algebraic integers to cover all the ground necessary, I figured out the key properties needed for a ring that does.

## JSH: Why wasn't proof enough?

So I started with

P(x) = (f(x) + 1)*(g(x) + 1)

in a post where I stepped through an example to show how the ring of algebraic integers is NOT sufficient.

With P(x) a polynomial and x an algebraic integer, follow that argument out and you get to easily defined functions, which if algebraic integer functions reach back to block f(x) and g(x) from being algebraic integers, only if f(x) and g(x) are non-rational.

I built up versus working backwards which is how I usually prove the result.

And remember, I have mathematical proof, have had it for years, yet a few people—names many of you know who post regularly on sci.math—repeating over and over again that they refuted a mathematical proof convinced the majority of you, or you just sat despite knowing the truth.

How many people? Not really many. You people got beaten by a few names, Dik Winter, Arturo Magidin among others, saying a mathematical proof was wrong.

Your delusion is in believing you could not be so easily controlled.

You were.

Even publication in a peer reviewed mathematical journal did not matter.

How is that? How can that kind of power exist? How did those few people control so many of you so easily?

Some of you might have put your lives on the line versus accepting the mathematical truth, just because you had a few people—just a few is all it took—telling you something was wrong when it was right.

P(x) = (f(x) + 1)*(g(x) + 1)

to show the rather remarkable flaw deep in "core" of number theory.

That flaw takes away ideal theory, changes the view and usage of Galois Theory, and revolutionizes mathematics.

Who knows what massive intellectual accomplishments not possible with the wrong mathematical ideas are now possible.

This saga is one of the biggest events in the intellectual history of humanity.

And most of you picked the wrong side.

But why? Why was mathematical proof not enough?

Going forward if the truth sinks in, I want some attempts at honest answers about why so many of you were convinced when most of the mathematics here is basic algebra.

Why wasn't mathematical proof enough?

## JSH: Understanding some crucial math

I want to step quickly through some very basic algebra to help people undestand an important result.

I will try to keep the operation within the ring of algebraic integers.

I will use a polynomial P(x), functions f(x), g(x), a_1(x) and a_2(x), where x is an algebraic integer. The point of this exercise is to show how you can find that the ring of algebraic integers is NOT sufficient, even with very basic algebra.

Now let

P(x) = (f(x) + 1)*(g(x) + 1)

where f(0) = g(0) = 0.

Now multiply by 7, so I have

7*P(x) = 7*(f(x) + 1)*(g(x) + 1)

and there are an infinite number of ways that you can distribute 7 through the factors on the right side, but we don't have time to check them all, so let me just pick one, and have

7*P(x) = (7*f(x) + 7)*(g(x) + 1).

But now let

a_1(x) = g(x) + 6 or a_1(x) = 7*f(x)

and

a_2(x) = 7*f(x) or a_2(x) = g(x) + 6

which gives

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

which is necessary for the a's to be algebraic integers if f(x) and g(x) are not rational

And, get this, if a_1(x) and a_2(x) are non-rational algebraic integers—that is, given an algebraic integer x you get an algebraic integer result—then f(x) and g(x) cannot be algebraic integers!!!

That is the crucial point.

Notice at this point, you have P(x) is a polynomial, and I didn't even need the condition that it have integer coefficients.

Mathematicians who think they can show me wrong here need only demonstrate a counter-example.

So what is the dramatic conclusion?

Remember, I started with

P(x) = (f(x) + 1)(g(x) + 1)

so the conclusion from everything above is that if f(x) and g(x) are NOT rational, then they cannot be algebraic integers if a_1(x) and a_2(x) are, so you CANNOT STAY IN THE RING OF ALGEBRAIC INTEGERS.

Remember f(0) = g(0) = 0.

Freaky, eh? I'm wondering if I need P(x) to have integer coefficients but I don't think I do.

Run yourselves around on that one for a while, and hopefully some of you will understand the flaw with current views on the ring of algebraic integers and why my object ring is necessary:

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:
1. 1 and -1 are the only rationals that are units in the ring.
2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.

## JSH: Aftermath

Maybe the biggest problem is that many of you don't understand what I'm trying to convince you of, so you overrate your importance and think I'm working hard to make you believe I'm correct.

Nope.

If I am correct, on my blog is a rather simple approach that could solve the factoring problem and lead to a breakdown in current Internet security.

I don't need you to believe me for that to happen.

So what is the point? Why do I go on and on, trying to convince when proof has failed, and even publication has failed?

Because you need to remember later, this time.

I'm not worried about what you believe now.

I am concerned with what you believe later.

If enough of you become convinced this is somehow my fault, then hey, that could be a problem for me.

I want you to remember that you made a choice, bet on the wrong horse and your choice was what gave you your consequences.

I am not out there making your choices for you.

Anger is a nasty emotion. People who find it hard to think in the first place can find it impossible when gripped by that emotion.

Just remember, you may have deluded yourself into thinking that most of the important stuff was figured out, but later, ask yourself, how could so many people have been so wrong?

Ever hear about how a frog can sit in water that is slowly brought to a boil versus jumping out, as if you do it slowly enough, it can't sense what is happening?

Yet now our entire planet is heating up and the amphibians are dying, going extinct.

There is a kind of weirdness to that, like some weird story, now isn't there?

Even in the subject line of this post, clever? Think so?

What if you just do not quite understand reality the way you think you do?

What if the word "aftermath" came into existence for just this very moment?

History pivots on the next few decades. Everything that has come before for humanity is about what will happen in the next thirty or so years.

How many of you are like frogs? I don't know, and no longer care.

After all, time will tell, quite soon, as every book gets balanced, and every question for humanity gets answered, so that the next step can be reached.

And I think and hope that the next step is the real leap in science and technology that will take us to distant stars, or what was the freaking point?

But I fear many of you will be left behind because this is a world you can't comprehend—a reality you can't fathom because of its logic.

It is so logically precise. And that precision is what can hurt so deeply.

The rules are never broken.

It will be too logical if you live in a human-focused world centered around yourself and your community, where frogs don't just go extinct, with polar bears, quite a few bird species, horse species and probably most primate species as well along with so many others, don't forget plants…so much death.

For many of you, your mind will tell you that the death will not occur because you think you know the limits.

Man is not the measure of all things.

Our time on this planet is running short, and soon we will have to move, or go extinct ourselves.

It doesn't matter if I don't convince you now, as the fix is coming, and the answer is that the people who don't get it, are the people who don't survive, so hey, get it all out now, as soon there will be a lot of quiet.

No more arguing. No more debate. Reality will be doing all the talking.

## JSH: We may have a problem

Almost as a lark years ago I thought I'd play with mostly simple algebra and use brainstorming to see if I couldn't figure out some neat math stuff that got missed. Low-hanging fruit. I was going for something simple but slightly subtle that might not have been noticed.

But something weird happened when I talked about math ideas on Usenet—math people went after me.

The insults flew almost from the start and kept going until this day, over ten years later.

Well, what would you do? I doggedly kept at it, and then found reasons for the insults—the math people were full of hot air in key areas.

So I traced out several areas and if you look over my research you'll find I answer things from supposed logical contradictions like the so-called Liar's Paradox, to finding a super ring like my ring of objects, which leads into abstruse areas.

BUT I found math people kept up the insults and lied about all of my research, so I pondered the problem: what could I figure out that they couldn't get away with lying about?

Easy answer—the factoring problem because it underlies Internet security.

And I may have done it, or come close enough that others can build on my research, but I'm not sure, and the same math people of course just keep giving insults, and the usual mantra claiming I'm wrong.

So why is it our problem?

Well, these people don't just lie about dinky little crap like randomness with prime numbers, where I'm sure many of you can see right through the statistical arguments—like why use statistical arguments against physics people?—to see why p mod 3 with p a prime greater than 3 is probably random like I say.

I fear that they are capable of lying about yet another clever idea that can be used for factoring.

But hey, what can I do? I say they lie, they say they don't, and years go by, so I put the information on my blog.

If they are right, no problem!!!

If not, you may go home one day to find your are broke, no matter who you are.

If they are wrong on this one Bill Gates may wake up broke.

So that's the problem. I had this idea a bit over two and a half months ago, and put it on my blog about two months ago, so hey, I'm thinking, maybe no worries?

I may as well toss it in here as it's kind of short, or maybe not. Does it really matter what the idea is?

The real point is it's not just an esoteric exercise if these math people are actually lying about this prime stuff, as they may be lying about much bigger stuff as well.

And the result may be that you are soon going to be really poor without having had a chance to do a damn thing about it. Or maybe this is your chance, but not really, as what can any of you really do?

I don't even know what I can do. The situation is bizarre beyond belief. Oh yeah, I did send the idea to some ex-CIA guy and someone with the U.S. Navy and they passed on it, so, no worries?

## JSH: Forget the lies, my paper

The real frustration for me when dealing with sci.math regulars is the insistence on lying about the details, so let me set the record straight on my paper on non-polynomial factorization, which was the one published and then withdrawn by the editors after some convincing emails by some sci.math regulars.

In the original paper, I declare the ring to the ring of algebraic integers.

That can be proven to not be true by using other arguments.

So it is the declaration at the beginning of the paper which does not remain true over the length of the paper which is the source of objections.

Two ways to handle that, of course, I could simply declare something different.

Or, I could do what I did in my revision, which was to note that the ring declaration COULD NOT HOLD which is the point of the paper.

That is no one can declare anything less than a field and have the operations hold through the paper, which is the over-turning result.

Now I came back, after I got over the anger enough to even bother with the sci.math newsgroup again, and argued with at least one of the people who falsely attacked my paper to the SWJPAM editors, and conceded all of his main points, and still showed that my conclusion held.

What did he do?

The fucker just kept on claiming he refuted me.

I conceded all of his main points—the mathematical ones—and STILL my conclusion holds and away he goes claiming I'm still wrong, when I am FUCKING CONCEDING HIS MAIN GODDAMN POINTS!!!

You people just lie.

You lie, and then lie some more, and then you lie about lying.

## JSH: We argue

I may only have an undergraduate degree in physics but that is enough to have experienced the joys of arguing about physics on so many things and so many ideas.

That's what I hate about how this forum is corrupted, how arguing is not useful, as people turn to personal attacks and anything to appear to win versus caring about what is true.

How do you know that everything you have ever been told is not a lie?

How do you know that this thing you call existence is not just some game of a superior intellect running you through crap like rats in a maze?

I believe that physicists can know. And I believe that if this were just some stupid crap game where I were a pawn to some cruel higher intellect I could prove it, and with that proof, make my choice to leave it.

How do you know that everything you have ever been told is not a lie?

And not giving up until you're satisfied.

Yes, you may settle when you should keep going, but at the end of the day, all you can do is your best? Right?

And if you do your best, you never need apologize to anyone.

You may anyway, but it's not a necessity.

I think that in asking questions I turn to the best that history has to offer, to great thinkers like Socrates—known as a gadfly—who annoyed so many people till they pushed him to suicide.

The group can hate the truth. But if you wish your life to mean something other than being a follower behind the herd, sometimes, you have to…do more than wish.

You have to dig, get dirty, fight, argue, wish, find it futile, and fight some more, and maybe get nothing, as who in the hell promised you anything?

But I just say, I'll do my best, and then I need apologize to no one.

Arguing is in the best interest of intellectual discovery.

Physics almost died when people decided that consensus meant too much.

I say, fuck consensus. Give me disagreement.

Make people fight to prove what they believe with NOTHING GIVEN no consideration, no benefit of the doubt.

GIVE ME NOTHING. MAKE ME PROVE EVERYTHING.

## JSH: Grooming behavior

Human beings are primates, so it's not surprising to see primate behavior in the way things go with postings and I find it fascinating!!!

Like look over arguments I have with posters, and you'll find an interesting pattern: I make an argument, some poster replies supposedly refuting it, and some other poster comes in, thanks them in some way or another, praising them for their supposed superior knowledge.

It's just primate grooming behavior.

Of course, at times I come back in, destroy the reply and may even show it reflects a very poor knowledge of basic logic or mathematics, but it doesn't matter as the purpose of the behavior on Usenet is the same as in the wilds with other primates: to reinforce an accepted dominance hierarchy.

Neat!!!

In many ways, human beings are more primate—that is they show more in common with other primates like gorillas and chimpanzees—than they are human.

## Wikipedia and me

Maybe to shift somewhat to somewhat lighter subjects I think it interesting to discuss my experiences with the Wikipedia and some of math people's attempts at slanting that encyclopedia with their own rather hostile antics.

First off, like most I came upon the Wikipedia as a useful resource, where often it's true you need to check things in other sources, but I've found that usually the Wikipedia is right, and you get more information that is useful.

But one day while doing the routine searches that I HAVE to do as I monitor the math wars, I found a search result showing up on the Wikipedia to me!!!

Some nincompoop had put me on the Wikipedia as a sci.math crank.

I thought about it a bit, and decided to edit the article to make it more accurate, clearing up some factual errors, like how long I'd posted, and removing more libelous and hostile remarks.

Someone came back, reverted my edits.

I changed again, correcting the article.

My edits were reverted again, and remember, the math community is HOSTILE, so we're talking about me finding some person had decided to put up some nasty negative about me on the Wikipedia—trying to set hostile opinions more firmly—and I was now in a fight against someone trying to block me from making it into an objective encyclopedia article—versus a rant against one individual.

Tiring of the stupidity, I put the page up for permanent deletion.

And it was so deleted, after a vote, and that was that.

The winning argument I gathered was that having a history of making wrong claims on newsgroups did not justify getting into the Wikipedia!!!

Oh well, all I cared about was preventing math people in the continual fight of the math wars from getting that big of a leg-up by having a hostile article against me in the Wikipedia, and remember, there are a lot of them so it just wan't worth it to fight against a bunch of nasty people by continually trying to edit the article to make it objective.

There may have been other paths to blocking them, but I like the one I took.

Still that wasn't all with me and the Wikipedia as I've talked about writing the first prime counting function article—which happened after what I mentioned above—but there is more.

Some other math community knuckle-head decided to STILL try and get me on the Wikipedia under the Cranks article, which I, of course, edited. Luckily that article is in dispute already so this time it's harder for my edits to be reverted. If you're curious go to the article and check the history to see my changes.

My point here is that I prize knowledge. But math people are capable of using means that are far beneath what you'd think, so I have to be vigilant in the math wars, as I track them across cyberspace. And I think the Wikipedia is valuable as an un-corrupted tool, while math people are quite capable of trying to use it in personal vendettas and attacks against individuals because they do not give a damn about the truth.

The math wars are a political war, so yes, this is a political posting, but hey, with people like I'm facing, if you do not step up to play politics, you may find yourself just another victim.

Math people try to exploit the Wikipedia. You might say I try to exploit the Wikipedia.

And our battles change your world.

THAT is how history is about INDIVIDUALS and GROUPS FIGHTING individuals, versus what you may think history is about, as I can talk about other battles across Blogger, where one is currently on-going as some math twerp took over a blog page after I deleted it.

## JSH: Harris integral?

http://sciphysicsopenmanuscript.blogspot.com/

But hey, I read over it, and what I saw when I did so made enough sense that I thought I'd elevate the attention given to it a bit and take the opportunity to talk about discrete mathematics and physics.

I don't know if the "Harris" in things like the Harris integral on the website is supposed to be about me, but I like the topic of discussion on the first post. I don't know anything about the person doing the blog so I'm not really endorsing it, as I hedge a bit in case it is some weird flame thing from the math community.

But hey, on to the idea that I see which is that Planck length is the smallest length in our universe, with the idea that we live in a discrete reality.

So you can say there is no way to consider knowing an absolute length as if our meter has meaning outside of this reality, as we can just go down to the Planck length and just find out how many of those lengths fit into a meter or a foot.

And, wasn't the meter started off of the average length of a man's arm? And the foot just came from, well, feet length?

So, we don't know what in some absolute sense any length is, just what it is relative to the smallest length in out reality.

That has huge consequences for physics because it implies that ultimately mathematics in physic will be discrete mathematics, which I know is terrifying to many as discrete mathematics is harder.

But hey, that's reality.

I am curious about current efforts in using discrete ideas in physics, who is tops in this area and what is the current research?

Oddly enough to me, as when I was a physics student years ago, I didn't care about discrete mathematics, a lot of what I've been doing the last few years is discrete mathematics, figuring out number theory stuff, which totally bored me as a teenager, and the more I consider things with primes, the more I'm impressed with the possibility of remarkable answers from a shift from a continuous view.

Sorry Cauchy, but it's time to move on.

And, oh yeah, what is the point?

Well, if the proper direction now for physics is into the discrete, then the next generation of science has not been done yet, and technology follows science, so the next generation of technology is not yet possible.

My own hope—a lot to do with how I've taken so much crap over the years from some very insulting people—is that the next generation of science will lead to technologies that put us deep into space, for real, like out to distant stars.

Technologies that will give us the ability to move out into the stars as explorers, where what's actually possible, and what the future brings is beyond what any science fiction ever dreamed, where science and technology take us beyond the simple concepts of space travel, and space ships of Star Trek, or Star Wars and other popular fictional approaches, in a way that is currently, hard to imagine.

## JSH: Math journals do not just die

My previous post was meant to scare you, as the story here is so far beyond what I thought was possible, like I DID get a paper published in a mathematical journal.

If that is news to you considering how much I talk about it, then think about how easily information is controlled by focusing people's attention.

I had a paper published in a peer reviewed mathematical journal, which also happened to be an electronic journal.

When word of publication reached the sci.math newsgroup, the group erupted in fury and some of the people there sent emails to the editors of the math journal that published my paper claiming it was wrong.

The NEXT DAY the chief editor pulled my paper. He sent me an email claiming it was a mistake, and didn't allow me to defend.

The editors first left a blank spot, but to get it all to look right, like with pages, they finally settled on putting that the paper was withdrawn. I didn't withdraw it. They withdrew it.

Now then, if you believe that the protection against "crackpots" not actually being wrong, but being wronged legitimate researchers is that benchmark of publication, think again.

AT this point mathematicians have a system where there is no way that someone like me can break it, when even publication can just be dismissed.

THERE IS NO WAY with the current system to break through, if the group decides against you.

They break their own rules in areas that should be dramatic and it doesn't matter.

In the last few years, there have been major scandals at journals in other areas, but in mathematics, who cares?

A supposed crackpot gets a paper published, the journal yanks it, and nothing.

Oh yeah, and a few months later the journal died. It just quietly shut down.

If you are naive and believe that it died because my paper got through, then think again.

To try and get that same paper published I enlisted the help of a Ph.D in mathematics with a slew of his own published papers, who signed on as co-author and went to journal after journal to be quietly informed through back-channels that it wouldn't get through.

It was politics.

That is control. And if you don't think that kind of control doesn't teach lessons, then think again.

And if you wake up in a world where what you say doesn't matter, and governments have more power than you ever thought possible—like I NEVER thought mathematicians had this kind of power—then you sat like a frog as the water came to a boil, and lost your freedoms.

And you deserve it.