Harristotelian Logic

One of the things I've liked about the Bulletin of the AMS is that they're rather quick with rejections, which is all I've gotten over the years, so I don't spend much time wondering about them, versus like the freaking Annals of Mathematics which is now on my crap list.

They keep you waiting for the inevitable rejection, where they never give a good reason either!

So like I said after I tested things a bit with some posts on newsgroups—which is one of the purposes for my posting as I look to see if people can point out errors—I sent my paper off to number theorists around the world from Australia to Washington state here in the US, and I submitted it for publication to the Bulletin of the AMS.

And I haven't heard back from them, which is a first! They're usually quick with a rejection, normally from the chief editor. Maybe that person is on vacation?

In any event, in the meantime the paper has changed! I've been fiddling with it after arguing with people as I realize I can add clarity in various ways.

The CURRENT paper is on a page at my Extreme Mathematics Google Group:

http://groups.google.com/group/extrememathematics/web/non-polynomial-factorization-paper

The direct link to the pdf is:

http://groups.google.com/group/extrememathematics/web/NonPolynomialFactorization.pdf

But now I'm at a bit of a loss, as I'm just waiting on rejection from the Bulletin, which has an older draft, but then again, maybe I should update them with the current paper? But how?

And why bother? Like it matters, as the AMS hates me.

One thing I know is that no matter what happens the American Mathematical Society will despise me, which is kind of funny.

A major amateur mathematician hated by the organization for mathematics in his own country.

But that is America—the land of contradictions. So brilliant in so many ways, and so completely lost in others.

But I like it. Of course it helps to see the humor in it.

[A reply to someone who wrote that James will never get a reply from the editors.]

I have always heard before. I do this rather regularly.

ALWAYS before, I'd get a rather quick email in reply, usually from the chief editor. It's just a perk of being me.

You on the other hand may not get a reply.

I always do.

Um, what errors?

[A reply to someone who wrote that James doesn't give a crap about the math and that all he wants is fame and fortune.]

And you care about math? You loser who'd rather fight for faux proofs than admit basic math?

You care about your job.

You are scared that if the truth comes out then all your posts come back to haunt you, and you'll be fired. Guess what? You will.

You coward.

I can be crazy. Most mathematical geniuses at my level are freaking bonkers.

But what's your excuse?

Everything I've said before fits in with who I am.

It's hard to stay sane with knowledge like I have. It's hard to control the truth fighting to be freed through you without your choice.

The genie will not be denied. But that is my burden and my pain that goes on and on and on.

What's your excuse?

[A reply to the same poster, who wrote that he has a job because he knows the math and he doesn't rely on someone to tell him whether or not something is right or not.]

You have a job because the people in charge of you keep you.

If it is true that I am this great discoverer, and your posts get back to your employers, then reasonably, you can expect that you no longer will have a job, as they will no longer want to keep you.

That is your motivation.

You ARE a coward.

You do not give a damn about mathematics and THAT is why you will be fired, not because of your previous posts, but because of these recent ones.

Mathematics is not just some thing, something you can use to feel important.

Without mathematics we would not have human civilization.

People like you are a threat to the future of humanity itself.

Make no mistake, I will feel no pity, and show no mercy.

None of my predecessors did, not Newton, not Archimedes.

And neither will I.

[A reply to someone who wrote that James cannot accept that he is a failure at Mathematics.]

They didn't.

Failing at math is not a bad thing. Most people do fail at math.

Pretending to be someone who succeeds, relying on other failures to keep up the lie, now that is something else entirely.

Mathematics has a heart and soul, and ultimately it is Mathematics that is coming for you people.

But because you're actors who think mathematics is just some game of pretend you have no comprehension that the discipline itself is what you should be afraid of, not me.

I'm starting to spread things out a bit, as I go to primes, while pushing everything else, while there is also action off the newsgroups.

More than likely it will be the copy protection idea that breaks you people, and then the rest will pile on, as money is what matters, so yes, you people could block some prime stuff, as if people REALLY care about prime numbers, and how many people care if you have some silly one hundred plus year old mistake?

But give the world a way to keep people from casually copying DVD's and see what happens.

You people need to understand how unimportant everything you do really is by considering that comparison.

The world didn't give a damn about prime numbers and couldn't care less about the screw-ups with the ring of algebraic integers.

But copying DVD's? Yeah, the world cares about that so you can just wait to see how much.

What I think might finally actually happen is that soon some of you will no longer be pretending to be mathematicians and actually getting paid, but you can keep posting and be cranks!

Which is what you actually are now anyway.

Don't be afraid of me. In many ways I'm dragged along with all of this myself, being pulled by higher powers who seem to have it in for you people in a big way so there is a big setup here with a lot of humiliation to be delivered…which is why it took so long.

The discipline wants revenge against the pretenders, and wants it served cold…

Posters in attacking my proofs showing inconsistency with the ring of algebraic integers routinely move outside the ring. Here is a post meant to show you how they do it.

In the ring of integers, consider x^2 + 3x + 2 = 0, which of course factors as

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

and now solve for it using the quadratic formula, but kind of weird by NOT resolving the square root then

x = (-3 +/- sqrt(1))/2

and now make the substitution, x=2y, so you get

4y^2 + 6y + 2 = 0, so you can divide by 2 to get

2y^2 + 3y + 1 = 0,

and your solution now becomes

y = (-3+/- sqrt(1))/4

which is two solutions where one is not an integer.

So you moved outside the ring of integers.

So what's the trick?

Well, with integer solutions you can resolve the square root and throw away one solution, which is what most people routinely do, so they say that sqrt(4) = 2.

When you do not resolve the square root—or cannot when it is non-rational—then you cannot throw away the other solution, so it gets dragged along, and if you do what posters typically do in replies against my research, and blanket divide a variable like x above so that you divide MORE THAN ONE SOLUTION you end up pushed out of the ring of algebraic integers.

When I've pressed them on the reality that the sqrt() returns more than one value, posters have replied with derision noting that mathematicians have DEFINED it to have one value, so that they can continue their trick unabated, as if it were a legitimate criticism against my research.

But as I've noted repeatedly, the ring of algebraic integers is inconsistent, and you cannot prove that is is from within the ring!

It is too weak as a ring, to allow you to prove that certain results are not within it, so posters are forced to go outside the ring to try and make their objections.

As a reminder, the updated paper—I had to clear out some errors noted by Rick Decker—is linked to at my Extreme Mathematics group:

http://groups.google.com/group/extrememathematics/web/non-polynomial-factorization-paper

One crucial addition to the paper besides error fixing is the noting that I use identities mostly, and one equation that is not an identity, so that equation MUST drive the conditions, and it can be placed easily enough in the ring of algebraic integers.

This result is one of the biggest in mathematical history demonstrating an actual inconsistency with a well-known mathematical object, which mathematicians have unknowingly used for over a hundred years without understanding how it can lead to false arguments that appear to be proofs when they are not.

Readers should note that I have multiple mathematical discoveries at this time where all have been vigorously attacked by posters who clearly have a need to deny any mathematical result if they feel it will give credence to my research.

They are dogmatic in their resistance, which is part of the reason I call these continuing arguments against mathematical proof—and even publication in a peer reviewed mathematical journal—the Math Wars.

I have rebutted the sci.math newsgroup which killed a mathematical journal with false claims, and bears a responsibility to accept accountability.

Now at least it is possible to carefully explain exactly what is wrong with the ring of algebraic integers as using it you can appear to prove two different and opposite things.

So I can start with an identity, the factorization:

175x^2 - 15 x + 2 = (f(x)+2)(g(x)+1)

which I do in my paper, and proceed to use only identities and a key monic expression in my derivation, to appear to prove that one and only one root of

y^2 - 6y + 35 = 0

has 7 as a factor, in the ring of algebraic integers.

That is not in doubt and has not been refuted, and in fact it stands as the beginning of the objections raised against my research, as you can then go, say, to the field of algebraic numbers and prove that 7 is NOT a factor of EITHER root in the ring of algebraic integers!!!

That is the basis of my rebuttal to posters who have long argued against my research, even attacking it in emails to the math journal that published a key paper of mine, and died after the editors trusted them.

I have rebutted these posters but they persist in ignoring basic proof, like that all their own claims of counterexamples depend on going outside the ring of algebraic integers, and that my work clearly shows using identities and expressions valid within the ring of algebraic integers, you can appear to prove that 7 is a factor of only one root.

So why is this a big deal?

Because mathematics is very particular about error. If people deny the error then they can "prove" things that are mathematically NOT true, and if you built a career on faux proofs, would you want that known?

My key paper proving the inconsistency problem starts with an identity. I use identities throughout much of the paper, only at one point finally introducing a single conditional expression:

r^2 + rs - (2 + 2xt + tQ(x))st + s^2 = (1+2xs+sQ(x))t^2

If you know anything about mathematics at all then you should know that identities do not do anything in terms of adding properties, or setting conditions.

The paper uses only identities up to a crucial point when one conditional is introduced.

For instance

175x^2 - 15 x + 2 = (f(x)+2)(g(x)+1)

is an identity as that's what factorizations are, like

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

is an identity, and identities are just true, not conditionally true.

If you read over my paper you can step through an argument using operations valid in the ring of algebraic integers, like I multiply both sides by 7, and re-order a bit to get

7(7(5^2)x^2 - (3)(5)x + 2) = (f(x) + 2)*(7g(x) + 7)

and I do that so that I can make a substitution using

5a_2(x) = 7g(x)

and I replied recently to a poster claiming that my argument fails because I try to use division, as he wrote:

g(x) = 5a_2(x)/7

The paper uses identities and expressions valid in the ring of algebraic integers to appear to prove a result that is not true in that ring if and only if with integer x, f(x) and g(x) are not rational.

If they are rational, then hey! It turns out that everything flows just fine and you're in the ring of algebraic integers. If they're not rational then hey! The freaking argument still says you're in the ring of algebraic integers, if you start assuming that

175x^2 - 15 x + 2 = (f(x)+2)(g(x)+1)

is true in that ring, but you can go to a field and prove that you're not in the ring.

But wait, factorizations are identities, right? So how can you be out of the ring with the factorization? Oh wait, it must be about the conditions on f(x) and g(x) as to how they're derived right? As identities do not give properties.

BUT, the conditional is given in my paper, and with it monic and clearly in the ring of algebraic integers with integer x, you are still forced out of the ring!

Hard to understand? Finding yourself confused?

That is what faux mathematics does. It is quirky, problematic and hard to grasp logically if you believe it is correct.

Mathematics abhors inconsistency.

Now I've proven my case multiple ways over a period of years and even got published, but I feel like early scientists must have felt fighting religious leaders angry at the earth supposedly not being the center of the universe.

Algebraic integers are the center of the number theory universe.

People who grew up on these mathematical ideas that are flawed, who built careers on these mathematical ideas that are flawed do NOT WANT TO ACKNOWLEDGE that their knowledge is flawed.

Any more than deeply religious people wished to accept that the earth was not at the center.

You see, they were very invested as well.

These battles keep playing out in human history, where it is about one group of people who get a vested interest in being wrong, and usually one man who is fighting for the truth, with only proof on his side.

And often with people, proof is not nearly enough, so the wasted years go by, with people fighting with a will to be wrong, so that they can hold back knowledge for just one more year if they can, or longer, as they can only see themselves and how they feel.

They only care about their own needs and cannot be bothered to care about the fate of the entire species as if they could be that great, then they wouldn't be fighting the truth in the first place!

That mathematicians around the world can continue with a demonstrated inconsistency making their efforts wasted is all about how small they are, and not at all about brilliance.

If there were any truly great mathematicians out there, they would fight to end the use of the faux math, not sit and hope no one notices, and the human race be damned.

Now I've explained and explained, yet again, as I've worked over the years against a will to be wrong from people who simply are caught in the oldest trap of humanity, needing to be wrong, to feel right.

I did not pick roots of monic polynomial with integer coefficients to build a hundred years of mathematical arguments upon.

None of you did either, but for those of you who are in number theory, the failures of that choice can have a real impact on your life, and it can be easier to be wrong, than to be brave enough to accept the truth.

The proofs I have show how you can step through an argument in the ring of algebraic integer and appear to prove one thing, but go to a field and find that you have something opposite to be the case.

That is inconsistency.

It is easily proven with the ring of algebraic integers, as if you look you will notice that every supposed rebuttal of my research relies on going to a field.

Now that is BASIC for any of you with even a little mathematical knowledge, but ignoring the basics is about a will to be wrong to hold on to a belief which is all about your human needs to feel like something that the truth makes you feel like you're not.

One mathematical journal has already died. I have sent papers to the Annals of Mathematics at Princeton University—and had them rejected. This current paper is at the Bulletin of the AMS, and I'm waiting to hear from them.

As this goes on you tear down your entire society.

To me your hatred of mathematics as a discipline is all about this story. As if you some of you think that if you hold on long enough mathematics can be trumped by human frailty.

If it could we would not be here today as Newton would have failed. Archimedes would have failed. Einstein would have failed.

Because you are failures you believe that humanity as a whole is?

Because many of you managed to keep this knowledge close for a few years hoping against hope that somehow you could betray your entire species and get away with it?

Your own children will spit upon you. The rest of the world may tear you apart.

Some of you may know that I had a number theory paper published in a now defunct electronic math journal where several sci.math posters emailed the journal claiming errors in my paper and the editors pulled my paper after publication. More on that subject is best found by doing Google searches on "SWJPAM", the initials of the journal.

I have completed a thorough rebuttal of the objections raised against my research in the area covered by the paper, but that rebuttal may be hard for some of you to understand so here is a short explanation.

The paper is a meta proof which shows a problem with the ring of algebraic integers by proceeding only in that ring with expressions valid in that ring, until a conclusion is reached which is outside the ring.

Like if you were in the ring of integers, using equations valid in the ring of integers, and found that suddenly you had fractions like 1/2. It turns out that you cannot do that with the ring of integers, nor can you do that with any other major ring or field.

The paper begins in an integral domain with a factorization of a polynomial:

175x^2 - 15 x + 2 = 2(f(x)+1)(g(x)+1)

I then proceed with some simple algebra and some substitutions to find

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

where I use 2f(x) = 5a_1(x) + 5, and 7g(x) = 5a_2(x), where the purpose is to get to a result valid in the ring of algebraic integers, as the a's are solvable, and are easily shown to be roots of

a^2 + (3x + 1 + 5Q(x))a + (49x^2 + 7Q(x)) = 0

where Q(x) is a function of x, which allows me to traverse through the full set of possible solutions for f(x) and g(x), and notably you have a monic, so that I know that with algebraic integer x, and Q(x), the a's must be algebraic integers.

That part of the meta proof I have had for some months, but recently I realized that I could derive a key expression by use of identities, and the paper shows that you can subtract

r^2 + rs - (2 + 2xt + tQ(x))st + s^2 = (2x+Q(x))t^2

from an identity found using r+s+vt = r+s+vt, where v=1+7x, and letting s=7, and t=5, it is possible to derive the same key expression used to get the a's.

But the key equation that is subtracted from the identity is only generally valid in the ring of algebraic integers, when Q(x) has specific values as only then is it monic.

Using one of those values I demonstrate a result valid in the ring of algebraic integers, with x=1, and then find another result not valid in that ring with x=2, showing that you can use only equations valid in the ring of algebraic integers, with ring operations, yet be pushed out of the ring.

Understanding meta proofs can test the limits of your mathematical abilities. In this case the meta proof has to cover an infinite ring, and show how it is flawed, where that flaw specifically is that you can be pushed out of the ring of algebraic integers using only expressions valid in the ring and operations valid within the ring.

No other major ring has the same flaw, not the ring of integers, not the ring of gaussian integers, and the major fields do not have it either.

Magidin went to the newsgroup for support.

I sent him an email with my new paper answering his charges against my research and what does he do?

He goes to this newsgroup for support.

What about rules?

If he took the effort to put up what he claims are rebuttals against my paper, and he went to the effort to email the editors of a journal that published a paper of mine, then why does he think he is above my answer?

Why does he go to this newsgroup for support of his running away from the responsibility to let me answer his claims against my research?

Is it because you people represent a mob, willing to ignore any and all rules in support of your own and Magidin is your own?

That paper clobbers the objections raised against my research.

I say, Magidin goes to you because he knows you are all he has as a weapon against responsibility for telling the truth.

That dead mathematical journal calls for accountability.

People who make claims against mathematical papers should not just be allowed to run to some mob and then walk away from mathematical papers written in answer.

Or do you people not believe in ANY RULES?

This newsgroup acted against a paper of mine published in a peer reviewed mathematical journal, and its members who did so have maintained that my paper was in error.

I have completed a rebuttal paper, which goes over the entire argument in ways meant to end their ability to easily confuse about the details of my research:

http://groups.google.com/group/extrememathematics/web/non-polynomial-factorization-paper

Those posters who went after my earlier paper should apologize both to me and the editors of the journal, which is now unfortunately a dead journal.

Civilization depends first and foremost on civilized behavior.

When angry mobs can trounce the instruments of civilization, like breaking the formal peer review process with false accusations of error in a major paper, then it is our progress as a species that is put in doubt.

I'm not a professional mathematician, so my efforts can be called recreational.

Years ago I got a paper published, and sci.math'ers went after it, and got it retracted by the journal editors with some emails. The journal later died, shutting down after one more edition.

The sci.math'ers of course have maintained that my paper was wrong, and that they were right.

Over the years I've worked at explanations of my research that take away the objections they relied upon so that I can convince that my original work was valid, and they were wrong.

I finally have that paper which I've tested in posts to this newsgroup and alt.math and alt.math.undergrad as understandably I do not like the sci.math newsgroup, but was not surprised that sci.math'ers came over to attack my paper.

And I have been gratified that their tactics have failed this time as the paper does cover everything as I need against their kind of opposition.

They use tactics that I've learned over the years are not about getting to the truth, but about convincing people that I am wrong, so I have directly answered those tactics with proof presented in such a way as to shut them down.

But what now?

Journals are wary of me now. After all, one journal keeled over and died!!!

The paper is available to the world on a Google group of mine, but that seems like a crap shoot—waiting and hoping that someone will notice this result.

And what a result!!!

I managed to use identities to get some incredible number theory analysis done.

And that's so simple of a thing, like people use identities all the time in mathematics.

e.g.

x^2 + 2xy = z^2

add y^2 to both sides to get

x^2 + 2xy + y^2 = y^2 + z^2

and you can solve for x:

x = sqrt(y^2 + z^2) - y

So I have a remarkable technique that relies on subtracting from identities, extending mathematics. Growing knowledge.

A spectacular story that even includes an entire mathematical journal imploding, and the will of a newsgroup against the foundations of mathematics itself where Usenet posters managed to trump the formal peer review process—and I'm stuck.

The real world is often about comfort, and just like inconvenient truths can get fought in other areas, so can they in mathematics.

As long as people like you allow mathematicians to not do their jobs for their own comfort against knowledge, there will not be change.

No matter what they say, mathematicians are highly political people. I think they claim to be otherwise to protect themselves.

But remember, those mathematicians at that journal that originally published my research caved with just a few emails from some sci.math'ers and consider that my research is bold, innovative, and correct, relying at its base on using identities in a very powerful way for analysis.

If you were a mathematician at a major university considering having your life significantly changed by accepting the truth, or sitting back like you just didn't know so that things could go as before, might you not be tempted if you thought you could get away with it?

Mathematicians are human beings too.

So then, what do I do? If mathematicians work to close all the doors against an inconvenient truth, what options do I have?

So the latest story is that I wrote an updated paper which specifically takes away all the objections that sci.math'ers tossed at my older paper, and it gives this really cool and amazing result with very deep number theoretic implications.

It tells you more about numbers than you could know before reading it.

But the reaction on your newsgroups has been a lot about posters ignoring the proof.

Maybe it is about all the arguing over the years and some of you may wonder, why all that arguing? Why couldn't I have been nicer? Why couldn't the discussions have been more polite?

And I say, disagreement is not a bad thing, and to the extent that people would give me nothing, and attack my ideas, as it was much better from my perspective as I need proof, and proof is not about social niceties.

And you can see the benefits of that approach, as my latest research is a continuation somewhat forced by the need for me to explain in such a way as to remove the ability of people like the sci.math'ers to credibly object.

So now you know definitely that I have proof, because I can talk about subtracting from identities not changing the ring, and you know that must be true.

Proof taken to the ultimate level of detail means that to doubt it you have to doubt the most basic axioms in mathematics, and if you find yourself wondering now if identities can change the ring then you are there.

Like how can

x=3Dx or x+y =3D x+y or x + 2xy + z =3D x + 2xy + z

change the ring?

So now I have a paper where I show the full picture showing how you can get a handle on the factorization of a quadratic such that you can cover NOT just the polynomial factors—the trivial factorization most mathematicians focus on—but ALL the possible factorizations, over infinity:

175x^2 =E2=88=92 15x + 2 =3D 2(f(x) + 1)* (g(x) + 1)

That is what's factored where I show how f(x) and g(x) can be totally defined and you can actually check through them and find when they can be algebraic integers functions with an algebraic integer x.

That is, you find when given algebraic integer x, f(x) and g(x) can give you an algebraic integer result.

The answer is enough to take away Galois Theory as it is usually taught, and change number theory itself entire as a discipline, which is more than enough for some Usenet posters to work very hard as you can see them doing, to try and hide the result.

What you are facing now is history in the making.

That means massive change.

And massive change is usually massively resisted.

My suggestion is to READ THE PAPER FOR YOURSELF as I have it in pdf format. You can get it and download it and not have to tell anyone you did.

http://groups.google.com/group/extrememathematics/web/non-polynomial-factorization-paper

Understand what all the fighting is about—please read the paper.

Ok, so for those who don't know, all the arguing that I do actually has a purpose as when I talk things out I learn better ways to explain and sometimes, I gain a better understanding myself of my own research.

That is what has happened in the last few days as yes, I do write papers, and as I've talked about how I use identities, which I call tautological spaces, and subtract from them, I've had various ideas which have gained fruition in my latest paper.

What I finally realized is that the question of what ring you are in had to ultimately rest on what was being subtracted from the identities as the identities cannot change the ring!

Identities are just like x=3Dx, or x+y =3D x+y and they are ALWAYS TRUE so they CANNOT CHANGE THE RING, so if your ring is changing then the identity cannot change it.

With that focused in my mind I could go back to an early draft of the current paper, and reverse the techniques I use to find what was being subtracted from the identities, and amazingly enough it was very particular about when it could be in the ring of algebraic integers.

Here it is:

r^2 + rs =E2=88=92 (2 + 2xt + tQ(x))st + s^2 =3D (2x + Q(x))t^2

That is the conditional that I have been using without knowing it before as I hadn't reversed to get it, which shows when you can have r, s and t all in the ring of algebraic integers with algebraic integer x.

To get the full paper go to my Extreme Mathematics group:

http://groups.google.com/group/extrememathematics/web/non-polynomial-factorization-paper

Also there is a link to my original paper that got published and retracted, where I was focused on cubics, so it's more complicated, and I didn't give how I derived key expressions, though now I can tell you I derived them from what I call tautological spaces—identities—which are true in the ring of algebraic integers.

So that's all. That closes the door. No longer can anyone argue that there is any other possibility than that the ring of algebraic integers is flawed and my research is clearly correct.

The consequences are HUGE as over a hundred years of number theory is affected.

Identities people. If you have any real mathematical bones within you, then you cannot believe that identities are controlling the ring, so what I'm subtracting from them MUST be controlling the ring, and given that you can start in the ring of algebraic integers and provably be pushed out of it, it MUST be true that the ring of algebraic integers has special problems.

No room for doubt—if you really care about what is mathematically true.

Now I've been arguing over this for years, and I'm writing papers, even got published and I'm explaining, and trying, so it's not about me. If this paper cannot convince you, then it's about you.

[A reply to someone who wrote that if the paper written by Wiles about FLT cannot convince James, then it's not about Wiles, it's about James.]

Except I've specifically addressed problems with Wiles's paper, having actually looked it over.

Posters in the past have claimed problems with my research and for years I have worked at answering them, which is what this latest paper I'm announcing does.

I've removed several areas from previous arguments where posters went after my research, eliminating talk of constant terms, values at 0, and I've derived my important polynomial expressions completely to emphasize that I subtract from identities and to show what is being subtracted from identities.

Of course identities are expressions like

x=x

or

x+y = x+y

which cannot of themselves change the ring, as they are always true.

The nature of the current paper is to directly answer any and all objections raised against this approach by giving a lot of detail.

Wiles in contrast has a paper I can tear apart in multiple ways.

His research is just wrong, as I've repeatedly explained before in talking about the errors in it.

[A reply to someone who wrote that obviously James doesn't have the necessary background knowledge to evaluate Wiles' argument.]

Well there are multiple ways including this latest paper which tears away the tools that people like Wiles use, as they rely on flawed beliefs about the ring of algebraic integers.

Now I think support of Wiles despite my latest paper is just simple hero worship.

I wrote that paper quite deliberately not only to better understand my own work, but also to shut down objectors like yourself.

Wiles does not have a proof. My own research pulls the rug out from under the techniques he relied upon and I use simple enough mathematics that the cry against my training is answered by my saying, check the math.

It is just such a good feeling to have that paper done! And I was myself surprised by a couple of things, as I deduced you should be able to reverse technique to get to the base conditional to handle the generalized argument I used, but I was surprised by what it looked like!

r^2 + rs − (2 + 2xt + tQ(x))st + s^2 = (2x + Q(x))t^2

That's not what I would have guessed was the underlying expression, which I call the conditional.

But what it does is control the factorization of

175x^2 − 15x + 2 = 2(f(x) + 1)* (g(x) + 1)

by picking the functions f(x) and g(x) by what Q(x) is chosen so that every possible factorization is covered.

The addition to the state of the art being the consideration of non-polynomial factors, as before my research, what mathematicians would think to factor that quadratic into anything other than polynomial factors?

And it is just remarkable that the full set of possible functions f(x) and g(x) are controlled by the expression

r^2 + rs − (2 + 2xt + tQ(x))st + s^2 = (2x + Q(x))t^2

as who would have guessed that?

The paper is a fascinating one, and I'm finding myself going over it again and again, which is also a good thing as I'm still doing changes to it, as I clarify and correct any errors I might find.

It is just so incredible that such complexity was wrapped up in factoring even simple quadratics, if you cover ALL possible factorizations versus being focused on polynomial ones, yet still all of the complexity—an infinite amount as there are an infinite number of factorizations—can be handled by a single expression.

The advance to mathematics is akin to when people learned to go from factoring integers into prime factors, to learning that there were non-rational factors as well.

The leap is bringing to algebra and polynomials what was brought to numbers and integers before.

Ok, yeah, I'm going on and on, but it is just so cool.

[A reply to someone who asked James to name an accepted theorem about the algebraic integers which Wiles uses and which is false.]

The flaw in the ring of algebraic integers allows a person to "prove" things that are not true based on factors results in that ring that give a false implication.

Once you realize that ring is flawed then it IS possible for the root of monic quadratic with integer coefficients to be both non-rational, and have some prime number like 7 as a factor, while the other root does not.

Once you realize that then you realize that Galois Theory tells you nothing more about non-rationals than it does about rationals, so you lose Galois Theory for any real number theory research.

So, the short answer is, my research takes away ANY value in Galois Theory for number theory research.

It just cannot tell you anything more about non-rationals than it can about rationals.

And take away Galois Theory, and you take away Wiles's argument.

Denial is not about mathematics. It's not even hard at this point to understand what's wrong with the ring of algebraic integers as the evens with 2 and 6, where 3 is excluded if you only consider evens so then 2 is coprime to 6, gives the full gist of the problem. So there is an easy example to help you grasp how the ring is flawed—it arbitrarily excludes numbers like if you say take only evens so that you find that 2 is coprime to 6.

The specifics are that the definition of algebraic integers as roots of monic polynomials with integer coefficients arbitrarily excludes numbers in a way that provably can lead to "proof" that is not actual proof.

So it's a way to believe you have proved just about anything, and Wiles is just one more person in a long line of people who have used wrong mathematical ideas and believed he had a proof when he did not, and it does not matter how many other people believed him or believe in him as the mathematics does not care.

If you think for one moment that mathematical truth cares about the misery you might feel now after reading over my paper and realizing there is no way to object then you are so far from being a real mathematician that you are getting a major favor by learning how wrong you have been before.

Mathematical discovery is about the search for truth, so sometimes people wander down the wrong path, and can do so for a while, but if you believe that truth is actually important, then you can be thankful when you are turned back.

But that is not necessarily easy.

And if you want easy, go find something else to do.

The math tells the tale. My paper closes the door on mathematical objections.

If you wish to turn to human interest ones that is a choice you can make. Worry about how you feel or how everyone should feel and curse mathematical discovery.

And then think about where our species would be if people like you ever won.

You tried to win here and hold back number theory. If people like you had won thousands of years ago we would not have algebra today.

And without algebra we would not have human civilization.

You have to lose no matter what you feel so that there IS a future.

These kind of battles are always about the fate of the species.

It's for all the marbles. When ignorance wins, we all lose.

One thing that gets lost in discussions about my research is the simple fact that identities do not add properties, as they are always true, so how can subtracting from an identity change the ring?

Like, with my now infamous paper posters seized on my starting the paper in the ring of algebraic integers when provably you get to this point where you are NOT in that ring, which is why I'd point out that they attacked my paper by seizing on its most crucial point.

So how can the ring change if all I'm doing is subtracting from an identity?

Remember an identity is something usually thought trivial, like x=x.

If I subtract x=x from x^2 + y^2 = z^2, can I change the ring?

No, I get

x^2 -x + y^2 = z^2 - x

and the ring will not care. And it cannot care.

And it CANNOT CHANGE as a result of subtracting from the identity!!!

Crucial to showing you have a mathematical intuition that is equal to the task presented by my research is understanding that I subtract from identities.

So how can the ring change?

If I start in the ring of algebraic integers, like saying that's what I'm doing in my paper, and use expressions derived by subtracting from identities, HOW CAN THE RING CHANGE?

But it does. And you know it does as that was the entire point posters who emailed the editors against my paper were making, and I myself have noted that yes, you end up outside of the ring of algebraic integers, when you start in it, and my research is about subtracting from identities.

So what gives?

Part of it is that I pulled out a piece from the full proof of Fermat's Last Theorem, so you don't see the beginning in a lot of what I posted and in what I put in my original paper, so you don't see the identities that start things, or what is subtracted from them, though posters who'd argued with me for years did see that beginning because remember, I first posted the proof of Fermat's Last Theorem, and later moved to concentrating on just one piece and that proof is all about subtracting the FLT equation from an identity, and analyzing the residue.

Those posters chose to ignore that I subtract from identities when they argued against my research.

But how can subtracting from an identity change the ring?

I'm going to leave you with that question. If you've kept up with the discussion you know that I had a paper published. If you've looked over objections to that paper you know that they center on the result not being true in the ring of algebraic integers, despite the paper starting in that ring.

But my methods involve subtracting from an identity, so how can you start in the ring of algebraic integers, only subtract from an identity, and find that you are no longer in the ring?

Any answers?

[A reply to someone who asked James whether he was talking about his paper “NonPolynomialFactorization.pdf”.]

No, I'm talking about my original paper, which readers can see at my Extreme Mathematics group.

See: http://groups.google.com/group/extrememathematics/web/non-polynomial-factorization-paper

The new paper they can also get there is the latest version of a simplified explanation where I've gone to quadratics as they are simpler, and I've now added in the identities that I call tautological spaces so that it is clear from beginning to end how it all works.

If you have read that paper then you cannot honestly continue to dispute my results.

What happened is that I was thinking to myself, why can't I reverse the process and get to the conditional from the expressions that I know are derived in a tautological space?

I do so in that paper and close the question of whether or not algebraic integer functions can exist with my original:

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

by showing they can only be algebraic integer functions with algebraic integer x, when Q(x) = -2x + 1 or Q(x) = -2x - 1.

If you are at all serious about mathematics and read that paper then all your objections have to fall away and you need to comprehend then that you have helped block the acceptance of a short proof of Fermat's Last Theorem from a young man who has been trying very, very, very hard.

Yes, I do rub people the wrong way a lot, but what does playing nice and kissing butt have to do with discovering great mathematics?

Real discovery is messy. Historians clean it all up for the history books, so don't think you know how this all should have happened.

This is how it happened.

Publication didn't matter, a fascinating and simple idea like using algebraic residues could be fought, and I keep running into an unending ability of people in math society to just lie.

Many of you are inveterate liars. Lying defines your society, so proof is just a word that you use to help you lie.

But your confidence in your lies is the confidence of those who don't understand history.

If lying were such a powerful strategy then that would dominate the world, but as George W. Bush is finding out, sometimes time catches up to you.

He probably wishes he'd lost that last election right about now, and it's going to get worse.

I promise you people this story is just beginning as it's all about time.

And the lying strategy is that of the young. Your society is immature, like the United States of America, a young country in an old world.

And you will learn—in time.

Back in December of 1999 I came up with the idea of using identities that I call tautological spaces against the problem of Fermat's Last Theorem, as I'd run out of ways to manipulate the base equations, and was desperate.

So I realized, eventually—as even after the initial idea it took me a few years to get the right approach—that I could use

x^2 + y^2 +vz^2 = x^2 + y^2 + vz^2

where I'd write that as x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2), and I could do various algebraic manipulations and eventually subtract out

x^p + y^p = z^p

and analyze the algebraic residue, as remember, I subtracted from an IDENTITY, so all the properties of what was left that were of interest would come from the FLT equation itself, as there could be no other input, since the identity is always true—which is why I call it a tautological space.

So the brilliance of this approach is in recognizing that by using an identity I could get to an algebraic residue, where I could analyze that residue with a free variable that I could make whatever I wanted, and using this approach I could first force an integer requirement for x, y and z by picking a prime factor f, of x, and using

v = -1 (mod f)

where I figured out quickly that I needed v = -1 + mf^{2j}, where m is a non-zero integer, and j is a natural number, as then I could make

x^2 + y^2 + (-1 + mf^{2j)z^2 = 0 mod f^n

where n is an arbitrary counting number, so I had a way to get the mathematics focused on x, y and z being integers!!! And it REQUIRED using my free variable v, which I only had because I was working on an algebraic residue, as if x, y and z are not all integers, then you cannot get

x^2 + y^2 + (-1 + mf^{2j)z^2 = 0 mod f^n

and given that handle I could go all the way to the complete proof of Fermat's Last Theorem, and after celebrating a bit, I began talking it out on the sci.math newsgroup, where I really believed that I'd finally be welcomed for succeeding after so many failures, but little did I know that YEARS would pass, and along the way I'd learn a lot more about math society, and a mathematical journal would even die, quietly.

So I had, after years of searching, a dramatic and simple proof of Fermat's Last Theorem found by this new technique of analyzing the algebraic residue found from subtracting from a tautology and I'm expecting cheers from mathematicians but I ended up in more heated arguments!!!

Finally trying to be smart about the situation, I pulled out one piece of the full proof, and wrote a paper, and started sending it to math journals.

An early draft went to Barry Mazur at Harvard as I sought commentary and he actually replied, wishing me luck and asking about one section. I answered him but heard nothing further.

I sent a draft to Andrew Granville for publication in the New York Journal of Mathematics, and he claimed it was out of his area and referred it to the chief editor who replied back to me that the paper was too small for the journal!!!

Looking to bank on my status as an alumnus of Vanderbilt University I got hopeful when I saw a professor from my school's math department was an editor of a journal and sent it to him, a Ralph McKenzie. He claimed to not understand it, so I forwarded him Mazur's email. After that he suggested that next time I was in Nashville I should explain it all out on the chalkboard.

So I planned a trip!!! I told him I was driving up, and set up a time, and I drove back to my alma mater for the first time in over ten years, and met with my old physics adviser as well as with Professor McKenzie with whom I talked for several hours, and explained my paper.

Nothing came of that so I sent the paper to the Southwest Journal of Pure and Applied Mathematics (SWJPAM), and they accepted it for review, and eventually published it.

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

But if you check those links you can see that the second says my paper was withdrawn, while the first talks about SWJPAM no longer being in operation.

What happened is that after my paper was published and I received more than one congratulatory email from the editors of SWJPAM, some person posted to the sci.math newsgroup that I was published, and the hate and fury that spewed out from that newsgroup was intense. They not only berated SWJPAM and its editors, maligning them repeatedly, but talked down the entire journal process, claiming that mathematical journals often publish in error.

Then some of them got the brilliant idea of emailing the editors claiming problems with my paper.

Their claims were ones I'd heard before before I ever sent the paper, which were easily resolved, but they have lied repeatedly about the mathematics in this area, and show no conscience nor regard for the truth.

I noted the posts conspiring to assault my paper by email and simply assumed that the formal peer review process would work, and if anything I'd just be questioned by the SWJPAM editors if they bothered to worry about emails from strange people asserting error.

But instead the chief editor immediately pulled my paper from the already published edition, and informed me by email that publication was a mistake and that he was going on sabbatical to Greece!!!

So not only were the rules not being followed, but I'd get no opportunity to answer.

The original paper can be found at my Extreme Mathematics group on a page that I have devoted to this story:

http://groups.google.com/group/extrememathematics/web/non-polynomial-factorization-paper

The equations in that paper are algebraic residues, though I don't explain that in the paper,. as it's not relevant to the proof.

What I did was take a slight variation on the FLT equation for p=3, and took the algebraic residue.

So what you now know is that the techniques I've talked about, using identities, what I call tautological spaces, with equation to get an algebraic residue, has had key portions formally peer reviewed.

Oh yeah, so after that capitulation to the sci.math newsgroup posters—who broke the formal peer review system with a few emails—the editors of SWJPAM managed one more edition before they just quietly shut-down.

Their school Cameron University removed ALL mention of the journal from its webpages, and they just dropped all service of the math papers within it over its entire nine years plus of existence, as did ALL of their mirror servers except EMIS.

Thankfully EMIS kept the prior papers available, for what about all those mathematicians who got published through them for years?

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

I am thankful to those Germans who were thoughtful enough to keep the history and work the journal represented available, when even its own creators were ready to just toss it and all the effort represented by it.

Think about it, as if you think this story is just about me, consider how many others who had papers published were being left hanging without a word of explanation, just by publishing in an electronic journal which one day decided to commit seppuku.

You might be one of those authors, or you could be one in the future in a math society many of you may not understand as that dead math journal is about an inconvenient truth—mathematical proof that some powerful mathematicians would just as soon not be known and accepted.

It's about people who through their actions show they actually hate mathematics, no matter what their titles, and about how difficult it can be to get the truth known—even with absolute proof—when people just do not want it.

Years have passed and so far they have won, holding back the consequences of the mathematics, stopping human intellectual progress in this area, and keeping the use of algebraic residues—out of the world's textbooks.

[A reply to someone who wanted to know whether or not “Nothing came of that” was James' way of saying that professor McKenzie had told him that he was wrong.]

No. I explained it all in detail, answered various questions he raised, and at 4:30 pm exactly he went home—along with the rest of the Vanderbilt math department. It was like a freaking whistle went off. Then I got in my car and went home, wondering a bit about the situation as I'd managed to explain everything completely in person to a leading mathematician, and answer all objections and he inexplicably just went home.

Later he emailed me thanking me for the conversation, and suggesting some books I might read, and I went off on him in reply, which, um, ended things there.

But understand, after years of arguing with very rude and obnoxious people on sci.math, and years of searching for some valid mathematical ideas of my own, with real success and the ability to explain it all in detail, face-to-face with a professional mathematician, he could just listen, and walk away.

I just couldn't take it, and maybe have learned from arguing on Usenet to too easily turn to angry responses, so I did, and that ended the conversation with McKenzie!

But later I got published in SWJPAM anyway.

During that meeting he told me I lacked "polish" which I took to mean that I didn't show formal training in terms of how I presented the mathematics, not as to whether or not it was correct, and no, I don't have formal training as a mathematician as my degree is in physics.

I DO have formal training as a physicist.

He also asked me if I was independently wealthy or had a rich uncle, apparently wondering how I was supporting my mathematical efforts.

Of course none of that sat well with me. It was like he was checking to see if I had other resources that I might apply if he just blew me off, like if I were rich, I might have pursued other means.

I learned a lot about mathematicians that day.

It was later that my paper was published in SWJPAM. And I know now what I knew then, as I'd explained everything to a leading mathematician—my research is correct, and people get away with lying about it.

That dead math journal is reality. Yes, math society can lie about reality all it wants, but make no mistake, you are lesser people if you do.

I think of you as a society that only cares what people think.

You say otherwise, but people can lie about lying, and from years of experience with your society I know that you people don't care about mathematical proof, but only about what you can convince people.

As long as people believe you care about mathematical truth, the bulk of you feel satisfied which is why this story keeps playing out as it does.

As long as you can lie to the world and be believed you don't care if you are mathematically wrong, or you would care about algebraic residues, and you couldn't so easily dismiss that dead math journal.

You people are social. Your society is purely political as long as you can get away with it, and for you, the mathematics be damned.

Years ago I came up with the idea of subtracting equations out of identities, so like you can consider

x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2

and subtract out x^3 + y^3 = z^3, and analyze what's left—the algebraic residue.

Since this approach is about residues, you use congruences so you'd have

x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)

and when I introduced this technique back in late 1999 and early 2000 on math newsgroups this approach was so new that for a long time I had arguments just over the FORM as some couldn't quite understand how

x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)

was equivalent to

x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2.

So why bother with identities and subtracting out an equation like x^3 + y^3 = z^3?

Because what's left over—the algebraic residue—is constrained by the equation you've subtracted out, so that what is true for the residue must be true for the original and vice versa.

The algebraic residue is a mathematical shadow of the original equation, but there is one key addition—another variable.

So you can analyze the residue by adjusting the free variable, and I used this technique with great success and the mathematical world did not yawn.

It blew up, as I didn't just argue on Usenet but emailed mathematicians at universities and even visited one at Vanderbilt University, and wrote a paper which was published by the now defunct Southwest Journal of Pure and Applied Mathematics aka SWJPAM.

That journal died after publishing a key paper very closely related to this area, when sci.math posters mounted a successful email campaign to break the peer review system—as my paper passed it—so that they could maintain a proof was in error.

Your society already broke, years ago, you may have just not noticed it.

The mathematical world that exists today is not the one that was here before my paper, no matter how successfully mathematicians hide that things have changed.

Why the big deal?

Because this simple idea of subtracting equations out and analyzing the residue revealed problems with previous strongly held ideas so one dead electronic mathematical journal is just the dead fish floating on the water.

One dead math journal not worth the world's news.

One of my most powerful simple ideas was to use identities against a hard math problem, as it is such a simple idea—once you consider it—to use identities and subtracting out an equation to be analyzed so that you use the algebraic residue.

For instance, consider the maybe seemingly trivial identity:

x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2

where since residues are what's important you can move to

x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2)

and now manipulate the equations in various ways, where because it is an identity, x, y, z, and v can be ANY numbers that you choose. So like you can say the ring is algebraic integers, and any algebraic integer will be ok, for x, y, z or v, because you're just manipulating an identity.

I call

x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2)

a tautological space, as it is a space in that you have variables, and in this case 4 variables so I call it a 4-dimensional space, and it's tautological in that it's always true, as you're just using an identity, where I figured out a way to use tautologies in mathematics to do detailed analysis.

For instance, consider x^3 + y^3 = z^3, and you can subtract that out of your tautological space, do some algebra and find that

(v^3+1)z^6 - 3x^2y^2(vz^2) - 2x^3y^3 = (a_1 z^2 + b_1 xy)(a_2 z^2 + b_2 xy)(a_3 z^2 + b_3 xy)

where the a's are roots of

a^3 -3va^2 + v^3+1 = 0

and for the b's you have b_1*b_2*b_3 = -2.

Lot of complexity there that just exploded out at you, right? But what I did was just go from

x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2)

to

x^2 + y^2 = -vz^2 (mod x^2 + y^2 + vz^2)

and cubed, and I took x^3 + y^3 = z^3 and squared and subtracted, to get the algebraic residue

(v^3+1)z^6 - 3x^2y^2(vz^2) - 2x^3y^3 = 0 (mod x^2 + y^2 + vz^2)

so it's not really so complicated but it is powerful, as remember, I subtracted from an identity, so in analyzing the residue I'm actually analyzing x^3 + y^3 = z^3.

But crucially v is a free variable, so I can make it whatever I wish, which is the handle given to me by this analysis technique. Now you can proceed to prove that x, y and z cannot be integers in a straightforward way that is easily extended to p odd prime.

And that is a quick introduction to tautological spaces and algebraic residues as I extended concepts began by Gauss.

Algebraic residues are the natural next step from his research where it just took a hundred years or so for the progression but that is the way mathematics actually works, as new techniques cannot be discovered until humanity is ready for them.

Here's a simple way to force an unlimited number of quadratics to have roots with the same factors in common with 7.

Start with the basic quadratic

x^2 - 6x + 35 = 0

and now let x = y + 6 + 7k, where k is an integer, and trivially you have that x is coprime to y with respect to 7.

Now then you can substitute out x, to get

(y+6+7k)^2 - 6(y+6+7k) + 35 = 0

so

y^2 + 2y(6+7k) + 36 + 84k + 49k^2 - 6y -36 -42k = 0

so

y^2 + (6y+14k)y + 42k + 49k^2 = 0

and

y = (-(6y+14k)+/-sqrt((6y+14k)^2 - 4(42k + 49k^2)))/2

and for every integer k, you have that y is coprime to x, therefore, for every integer k, you have that the roots must have the same factors in common with 7.

Let

z = (x-7)(x+7)

and

x^2 - 6x + 35 = 0, so

z = x^2 - 49, so trivially I have x = sqrt(z + 49) and can now substitute out x, to get

z + 49 - 6sqrt(z + 49) + 35 = 0, so

z + 84 = 6sqrt(z + 49)

and squaring, gives

z^2 + 168z + 7056 = 36z + 1764

so

z^2 + 132z + 5292 = 0.

And you know that each solution for z must share factors in common with 7, with x, as that's the entire point of the construction z = (x-7)(x+7).

So if you read over my research carefully and understand the implications of my discoveries, one odd thing will also reveal itself to you—people supposedly expert in these areas cannot actually be true experts!!!

So like number theorists presumed to be expert mathematicians clearly cannot be, if you understand what follows from my research.

But, um, if they're not expert, and they admit it, what then?

If you've spent your whole life being a "mathematician" and one day come across some guy on the web who has incredible research results that promptly disabuse you of the notion that you are actually one, then what would you do? Jump up and cheer? I think not.

It is actually worse for them than for people who don't think they are mathematicians because these people know a lot of mathematical stuff that is wrong, or useless that they spent a lot of time learning.

So how is that possible? Isn't mathematics important for DOING things in our modern world?

Yup, but most of that mathematics was discovered centuries ago. A lot of it was discovered thousands of years ago.

Turns out that mathematics is extremely powerful so you can do a lot with a little, so there is now a bubble where these people can live, doing nothing of value at all because nothing they are discovering is actually true!!!

Nothing. Like zero. Total waste.

So they look at my research and it's just pushed on them that nothing they're doing is of any importance but they are known as they world's experts so hey!

They can finally get something right!!!

They can just ignore valid research that threatens their position as the one solution they have—which actually works.