JSH: Mystery increases

I was actually very surprised at the arguments that ensued over my solution to the factoring problem.

It is a very simple argument with a rather basic proof, so why were posters so diligent in throwing up distractions around it, or in making false statements?

After all, it is the factoring problem.

Crucial to me was getting help and it looks like one poster has stepped up in a huge way, but remarkably posters who have argued with me are acting almost as if that thread is invisible.

I have YEARS of having had major mathematical discoveries and learned a long time ago that proof wasn't enough to convince people in the mathematical community, but I didn't realize just how bad it truly is.

Mathematical proof has not only routinely been denied, people have behaved as if it would always be, and even now with the factoring problem itself solved they have continued.

What is the explanation for this behavior?

How are any of you justifying doing these things? I mean, you pretend to be interested in mathematics. But you show behavior that indicates almost a complete contempt for it, what gives?

The mystery is well beyond the bizarre. Like you people destroyed a mathematical journal. You've ignored incredible and dramatic proofs.

And now with the factoring problem solved the entire Internet will be affected, but some of you STILL continue with the same behavior?

Do any of you actually believe in mathematical proof?

JSH: Let's play a game

For this game I need you to as they say in literature, movies or other fictional settings: suspend disbelief

For the sake of argument, let's say I'm right: I have solved the factoring problem. I found a proof of Fermat's Last Theorem years ago. I found a devastating error in number theory.

But the world has not recognized any of it. Why?

Is there ANY explanation—beyond the obvious that I'm wrong, remember the beginning of the game is the suspension of disbelief—for that happening?

I mean, we have a world of people who should be excited about any one of those things, how could someone have incredible discoveries on such a scale and not have them accepted?

Is it possible that the human species is devolving? Or could there be an alien influence?

It's brainstorming time. Any and all ideas within the parameters of the game are welcome.

Is the human race getting stupider on a worldwide scale?

Or are possibly aliens insuring the demise of the species by acting to prevent its growth in science and technology by blocking continued development of mathematics?

Or something else?

JSH: Welcome to the real world

With some very easy algebra I can prove to those of you who are intellectually honest, how little you understand of how knowledge is accepted today.

I found a connection between the equation commonly called Pell's equation

x^2 - Dy^2 = 1

and an expression which I like to say is a discrete ellipse or it is a Pythagorean Triple with integer x and y, if D-1 is a square:

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

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

That expression connects rational solutions of the first equation to rational solutions of the second, which may not seem like a big deal until you see that a factorization of the D in the first is connected to a factorization of D-1 in the second:

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

and

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

and it doesn't take long to wonder, if you think about it, if you can factor efficiently using that remarkable relation.

That's the easy part. Convincing people that the factoring problem could go down so easily—now that is hard.

I've done more to explain in other posts so I'll step away from that to emphasize to you: mathematical proof, easy algebra, major importance in the factoring problem.

And let you ponder your own actions which I assume are to do nothing.

(If you are doing something, YEAH!!! This post does not apply to you.)

We live in a structured world with a LOT of assumptions. People around the world are already seeing many of those assumptions shattered with the financial crisis.

I'm hoping some of you read this post, and if you do nothing, you go to bed tonight, and tomorrow think about why you do nothing.

And then understand HOW we have financial crises and so many other crises in our world as you realize how hard it can be, to accept absolute mathematical proof.

Absolute mathematical proof.

Now think about what happens when people don't even have that, and your world makes a lot more sense.

It is harder than most people realize to do the right thing, even when you have absolute knowledge of what the truth must be.

JSH: One more explanation, factoring solution

With posters yet again chortling victory against my research despite their failure to disprove anything about my solution to the factoring problem it's worth explaining again.

What I did was exploit a rational connection.

With rational solutions to

x^2 - Dy^2 = 1

I noticed you have rationals solutions to

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

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

And that's it. 99% of the mathematics is right there in front of you, and covered with that initial statement. So how do I solve for x and y directly?

Well, I have TWO FACTORIZATIONS available:

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

and

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

So what I do is generally factor the second and I also found I needed to split up j, so I add variables: u and v

(x+y - (j+/-1)) = f_1*u

(x+y + (j+/-1)) = f_2*u*v^2

where f_1*f_2 = D-1.

And that's how v comes into the picture. Now recap: for EVERY rational solution to x^2 - Dy^2 = 1, you have a rational solution to:

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

I note that if x = r/t and y = s/t, I have (r-t)(r+t) = Ds^2, and can consider a solution that factors D if it is an odd composite and g_1 and g_2 are non-trivial factors with:

r-t = g_1 and r+t = g_1, as then r= (g_1 + g_2)/2 and t = (g_1 - g_2)/2, and s = 1 or -1.

So rational solutions to r, s and t EXIST at a point that will factor D non-trivially.

One set of posters has repeatedly claimed they do not. With at least one claiming to have disproven that using the quadratic formula.

Now to guarantee non-trivial factorization of D, it suffices with non-zero r, s and t, for

abs(r-t) < D and abs(r+t) < D

and you'll notice I already showed at least one example of that case which must exist!

One set of posters have routinely claimed that both conditions cannot be simultaneously true.

Now I've noted that now you have a calculus problem of minimizing to find r, s and t as functions of v, such that you meet those conditions, and I've given ONE possible answer while to practically factor it may never be the case that you even need the s=1 or -1 case. But the proof of its existence shows that rational v is available in the desired range.

Now here are the explicit solutions for x and y:

y = [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

and

x = +/-(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v) - [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

where again f_1*f_2 = D-1, and the f's are non-zero integer factors, while v is a free variable.

To recap: what I did was exploit a connection between the factorization of D and the factorization of D-1.

Remarkably they are connected through Pell's Equation and an equation I derived from Pell's Equation using my Quadratic Diophantine Theorem (Google it).

Now for years I've claimed that math society has been ignoring major proofs of mine to hold on to the status quo as my research upsets HUGE swaths of established number theory, and now you have clear and irrefutable evidence in front of you of how far they will go in that denial.

Pell's Equation is one of the most famous in mathematical history.

The factoring problem is being used to supposedly secure the Internet.

With every security breach you read in the news, consider the possibility that factoring has been used, and that practitioners in various specialties are lying about it being broken, just like posters on these newsgroups lie about the efficacy of the equations above.

They do so to preserve their BELIEFS about the world in a way that makes them most comfortable without realizing the consequences of their belief system can be catastrophic. They are—religious about mathematics.

Factoring always had an easy answer: connect factoring one number to factoring another.

People just got it wrong for a while and now the truth is out. People make mistakes. That's not news. But please don't make the bigger mistake of continued denial to try and hold on to math ideas that just do not work.

Mathematics is a heartless discipline.

I know many of you have invested huge amounts of time and energy and years of your lives to learn mathematical ideas that if you're honest you'll have to realize are wrong.

But holding on to them will never make them right.

JSH: Your Google check on the Hammer

Those of you wondering how to check off the newsgroup on the progress of the Hammer can just do a search in Google.

The Hammer just emerged into the top 10 and is now at #9 last time I checked.

So the world is already voting against sci.math posters who keep claiming error.

When the Hammer reaches #1 and upsets the RSA page, then get ready for the aftermath.

I know, I know, none of you believe in Google searches as relevant, right?

Let's see how long that arrogance lasts.

JSH: Problem of positions

Maybe some of you need an explanation as to how it is possible that mathematicians as a group, around the world, in number theory can be wrong, and even have it proven to them, and keep being wrong, and I say it goes back to European history.

Feudal societies built first around powerful individuals but later institutionalized worth by birth, so a noble could be someone not noble at all!

In those societies people learned that some things were just about position: some people regardless of ability just had these positions and you accepted it for the good of society.

It's easy for us to look down on classed societies from a distance but I doubt peasants spent most of their days thinking about overthrowing their social order. Most probably thought of themselves as good and loyal subjects who accepted things as they are.

While modern dominant societies claim to have escaped arbitrary positional rules, repeatedly reality shows that they have not, like the recent disastrous presidency of George W. Bush, where it is difficult to understand how he ever even became president except for him being a part of a dynastic family. He was, quite simply, born into a family which allowed him to achieve high position.

Clearly, feudal realities still remain in modern societies!

Modern mathematicians have a SOCIAL position, and people are loathe to undermine the social order unless they are pushed extremely hard, as most people in modern societies I think see themselves as good and loyal citizens—they support their societies.

And I think there is an unconscious continuation of feudal behavior, which has put me in the difficult position of facing social structures with mathematical proofs.

The factoring problem unfortunately represents the one area in mathematics where there is the potential to shock people out of the feudal behavior, forcing them to see that there is a major problem in the mathematical field, just like financial woes have been necessary to force people to accept problems with the financial system.

Note that a feudal class had started to evolve in finance as well, where huge amounts of wealth were had by people who not only we later found were not doing anything commensurate with their level of compensation, but worse, they were actually a net negative!

We paid people billions of dollars to wreck our financial systems.

We REWARDED them for failing, which I see as feudal vestiges—unconscious but powerful.

The rich were the nobles: with positions just because, given special rights and dispensations just because.

Feudal behavior in modern societies.

The proof of the easy solution to the factoring problem will indubitably have a massive impact on the world, and it will in time force what other mathematical proofs could not—a shift in the view of mathematicians and probably academics in general, just as there is an ongoing shift of the view of people in the finance industry.

The sad thing is that catastrophe or near catastrophe is the only way to change social inertia on this scale.

We are approaching potential catastrophe with Internet security, as the only way to remove more feudal behavior, and force an upgrade of the academic system—worldwide.

So what finance people are going through now, academics will face next.

JSH: Best guess as to what happened

For years I've been dealing with a problem where mathematical proofs I've discovered are not being accepted by the mathematical community, and tracing it back, I found that over a hundred years ago a devastating error entered into the math field, which is so devastating that it removes most of the research done in number theory over a hundred plus years.

Seeing that error I was not terribly surprised that a simple solution to the factoring problem might exist as the same practitioners who came into the math field under the error, were making the claims about the factoring problem being difficult.

But you see, factoring is like practical non "pure math" results in that you can either factor a number or you can't.

So success requires math that works. While a lot of currently established number theory is covered by the error so it does NOT work, but that is obscured because the results are with non-rationals.

So modern mathematicians who came in under the error aren't actually that good at mathematics, so to them, factoring was a hard problem, but it's actually not.

Now I've found the simple solution to factoring and posted it, but the people who are in charge in the math field are the people brought in under the error!!!

And it's increasingly clear that they realize what they have invested in the error: their careers.

So even with a simple proof of the factoring problem, with all that entails, they are trying to just ignore it like they did with my proof of the error and my other mathematical research including my proof of Fermat's Last Theorem.

But the very reason I picked the factoring problem to work on was to end the efficacy of that strategy.

However to my surprise there has been a delay that has gone on far longer than I thought possible.

It is not clear that the simple solution is not being exploited. One worst case scenario is that practitioners in various field related to Internet security are simply making stuff up about breaches to hide the fact that the current methods are no longer working.

If so, that can only go on for so long though. So we're in a delicate and dangerous period where as far as I'm concerned the Internet no longer has security, but is wide open for anyone with the factoring solution, while the people who are supposed to alert the world are not doing so as they realize that after the world knows the story about factoring the full reality of the devastating error will be revealed, ending the free ride that people have gotten with false mathematics.

Which would probably push a tremendous number of people out of the mathematical field.

A hundred years with a math error of this type can wreak a lot of havoc.

Here the entire field of number theory worldwide is full of people who are, unfortunately, probably actually incompetent at mathematics. They convinced the world that factoring was a hard problem on which to base Internet security. I've proven an error in number theory, and also solved the factoring problem trivially. These people are for the moment not acknowledging the result.

It remains to be seen what the full consequences of their behavior will be.

JSH: Critiquing factoring solution proof

I will say I've been surprised by a willingness of posters to use the same tactics against my solution to the factoring problem as they've used against my other research. You might say that before I didn't think they'd have the nerve!

But with posters already talking victory as if they had shot down a remarkably simple proof I thought it worth it to step out a critique of the actual proof, and then you can ponder how posters were so confident they could block knowledge of such a thing.

Ok. Field is rationals. First thing is I use Pell's Equation:

x^2 - Dy^2 = 1

but I SOLVE Pell's Equation, giving x and y explicitly as functions of a variable I call v, and factors of D-1:

y = [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

and

x = +/-(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v) - [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

where f_1*f_2 = D-1, and the f's are non-zero integer factors, while v is a free variable.

Note there is no argument about the correctness of those equations. If you plug in some rational v, then you will get x and y such that x^2 - Dy^2 = 1. That's a mathematical absolute, so don't worry about if they are right or not (or go check and you'll see they are).

And remarkably while there is a factoring dependency—familiar to anyone who may have tried to solve for x and y directly who probably faced factoring D—it is for D-1, which is a good thing!!! If it were D, then that would be it. This approach could not be a solution to the factoring problem.

So that is a HUGE thing which allows everything that follows which is why I belabor it: yes there is a factoring dependency but it is on D-1, not D.

But integer factorization is about integers!!! Not rationals, so I move in the direction of integers with a simple step,

by considering x and y as ratios of integer functions r, s, and t:

x = r(v)/t(v), y = s(v)/t(v)

gives

(r(v) - t(v))*(r(v) + t(v)) = D*(s(v))^2

and that is not a big step though it might seem controversial to some. Quite simply I only care about integer values for r, s and t, as I'm trying to factor D, and integers would be nice!!!

Now the proof is almost done!!! It is remarkably short considering it is a proof of a solution to the factoring problem.

The penultimate step is to realize simple conditions on r(v) and t(v) such that you MUST factor D non-trivially if it is a composite.

That is important. We're now worrying about what mathematical conditions can be satisfied which will give us the desired non-trivial factorization, which is a very big step. But it turns out it is easy as well!

Well, if abs(r(v) - t(v)) < D and abs(r(v) + t(v)) < D, then of course neither can have D itself as a factor!

And for some of you there should be this excited rush of amazement and awe that it can be that simple!!!

ALL we have to do is determine r(v) and t(v) such that the they are below a minimum value.

And, um, from where I'm from, that's calculus!

Somehow, someway the much vaunted factoring problem has just turned into a calculus problem.

But I said the penultimate step. So what is the final one?

Existence of course!!!

We need to know that rational v exists such that the conditions above can be met, and I can start you on the answer.

Let D = g_1*g_2 where the g's are integer factors and D is an odd composite, and now let

r - t = g_1 and r + t = g_2

where I leave off v because this time we're going BACKWARDS and are going to find v, so now v is the dependent variable, as I'm setting the functions.

So r = (g_1 + g_2)/2 and t = (g_1 - g_2)/2, and then s = 1.

That means I have x and y, and it turns out I can now go off and find v by a rather straightforward method, proving it exists, which completes the proof.

I've leave that out for the moment and see what happens in reply to this post.

JSH: Remarkably odd

Do any of you realize that I actually do have a remarkably simple proof of a method that solves the factoring problem through use of Pell's Equation? Or do you all believe posters who post with claims it does not work?

I'm curious.

One question I've often wondered is, are you people serious or are you playing some weird games?

Here I can step rather succinctly through one of the most beautiful mathematical proofs in human history which happens to solve the factoring problem.

I've argued about it a bit and watched in amazement as posters have made false statements about the argument, or ignored key pieces, and I find it remarkably odd.

Is ANYONE aware of the proof itself or its outlines? Any of you have a clue how the proof goes? I've talked about it quite a bit now, and it is on my math blog if you want to go read it again.

Anyone know it? Or are you all clueless for real? Or all lying in a remarkable group effort, as if it matters.

There is no stopping the Hammer. It cannot be stopped.

JSH: Hammer falls, Pell's Equation used to solve factoring problem

Well, the factoring problem is solved easily enough with Pell's Equation by a direct solution with x and y functions of an independent variable:

x^2 - Dy^2 = 1

y = [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

and

x = +/-(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v) - [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

where f_1*f_2 = D-1, and the f's are non-zero integer factors, while v is a free variable.

What makes this solution so remarkable is that there is a factoring dependency in it, but it's for D-1, not D, but you can go after factoring D by considering x and y as ratios of integer functions r, s, and t:

x = r(v)/t(v), y = s(v)/t(v)

gives

(r(v) - t(v))*(r(v) + t(v)) = D*(s(v))^2

which means oddly enough that just playing around, like running through some values of v, you might accidentally factor D, but the mathematics also allows you to be very serious and just directly find minima.

So why minima?

Well, if abs(r(v) - t(v)) < D and abs(r(v) + t(v)) < D, then of course neither can have D itself as a factor!

So those conditions cannot even occur if D is prime, but MUST occur at some value for v if it's not, and then you have a non-trivial factorization of D from the gcd's.

Absolutely. With no mathematical doubt.

So the factoring problem becomes a calculus problem. Calculus. Yup. It becomes a freaking calculus problem.

Kind of like a homework one too, as the highest power is 2.

So finally you get to see the Hammer.

Your best course as a newsgroup at this point is to acknowledge the solution. Start accepting correct mathematics in general.

And apologize for all the years of abuse, lying about my research, and the math journal you killed.

Just start apologizing and wait for someone to tell you to stop.

Factoring with Pell's Equation

One of the weirder, possibly ironic twists lately has been my discovery of a direct solution to Pell's Equation with x and y functions of an independent variable:

x^2 - Dy^2 = 1

y = [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

and

x = +/-(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v) - [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

where f_1*f_2 = D-1, and the f's are non-zero integer factors, while v is a free variable.

What makes this solution so remarkable is that there is a factoring dependency in it, but it's for D-1, not D, but you can go after factoring D by considering x and y as ratios of integer functions r, s and t:

x = r(v)/t(v), y = s(v)/t(v), gives

(r(v) - t(v))*(r(v) + t(v)) = D*(s(v))^2

which means oddly enough that just playing around, like running through some values of v, you might accidentally factor D, but the mathematics also allows you to be very serious and just directly find minima.

So why minima?

Well, if abs(r(v) - t(v)) < D and abs(r(v) + t(v)) < D, then of course neither can have D itself as a factor! So those conditions cannot even occur if D is prime, but MUST occur at some value for v if it's not, and then you have a non-trivial factorization of D from the gcd's.

So the factoring problem becomes a calculus problem. Calculus. Yup. It becomes a freaking calculus problem.

Kind of like a homework one too, as the highest power is 2.

I've been talking about this solution for a while so I guess for some reason people are not getting it, possibly listening to people claiming there must be some catch, as hey, it's the factoring problem!

Sorry. No catch. The mathematics is fairly straightforward and easily verified to be correct. There IS an inherent factoring dependency, but it is for factoring D-1, not D itself. The reality of minima is as simple as I explained above. So yeah, the factoring problem is solved using Pell's Equation in a remarkable twist on thousands of years of math history in this area.

But why the lack of news?

Easiest answer is that people don't believe me. It IS the factoring problem, so there is a lot of expectations built up around it, and maybe it's kind of a letdown for the much vaunted problem to be handled so easily. Like the tiger turned out to be a tabby.

It was never REALLY hard, with computers at least, it's just no one knew how to solve it, so it felt hard until you see the answer. Like a puzzle with a simple answer that becomes too easy when you see it worked out.

Still it's weird for it to be so easy, so maybe that's a lot of it. But it's so damn strange for things to be so quiet. I will admit to not having expected this situation, and I've been waiting to talk to people, like government people, about this solution and this situation, but it has been all quiet.

Who knew? You could put out a solution to the factoring problem, on a blog, talk about it on newsgroups, and have NOTHING happen for an entire week and counting.

That sure kills all the Hollywood stories about how things would work out. Truth is way stranger than fiction.

JSH: Denial does not change the equations

Here is where the story once again becomes sad. I like to think that deep down most of you are decent people.

However, I have years now with denial from people like you who are supposed to act when there are important research finds.

The general solution to Pell's Equation is somewhat accidental in that I stumbled across it, but it's also important because I worry that it leads to a general method for factoring integers.

If that is true then no matter what you may believe, it is quite naive to think no one will notice if you do nothing, say nothing, pretend to see nothing.

I am making this post because I fear some of you DO think that if you are just very quiet, nothing will happen, and already I am at a loss as to how things can have been so quiet with such a major find.

But the equations will not change.

All your denial can do is make the aftermath very, very, very bad.

If the path to factoring is now revealed and mainstream people get quiet, that will only leave the people who do not have egos to protect, who do not care where the information comes from, who do not care if the result is published in some established journal.

People who will only care if it works.

And now more of the dark side, as I fear some of you are hoping now for a literal end of the world. As if in death you can go holding on to a worldview you cherished above all else, as long as you take the rest of the world with you.

I assure you that will not happen. This world will not end. You may.

Things can get rather bad, but the world will keep on turning, humanity will keep going, and eventually the knowledge will be accepted for what it is, and questions will be raised about the community of experts who kept silent.

You owe it to the world to tell the truth. You owe it to your future to wake from your stupor and get a sense of freaking self-preservation.

The dark side of the potential of this result is not nice. And there will be no time-out if things go badly.

And no saying you're just an academic or you had no clue what to do, or no idea how bad it could get.

I'm telling you how bad it can get now. Your life may be in the balance NOW.

Your future may depend on making the right decision. TELL THE TRUTH.

Solving Pell's Equation, hidden variables and implications

Rather remarkably I think I have found a way to generally solve for Pell's Equation, which brings up the issue yet again of "hidden variables", and raises severe implications ranging from mathematical viewpoints all the way to Internet security (possibly unfortunately).

First it helps to see the result!!!

General solution to Pell's Equation

Given

x^2 - Dy^2 = 1

I have proven:

y = [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

and

x = +/-(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v) - [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

where f_1*f_2 = D-1, and the f's are non-zero integer factors, while v is a free variable.

As Pell's Equation is normally considered in integers as a Diophantine equation note that you find rational v such that x and y are integers, which gives the 'why' of Pell's Equation. For instance, for D=2, f_1*f_2 = 1, so I have:

y = [+/-4v/(1 - v^2 - 2v) +/- 1 -/+(1 + v^2)/(1 - v^2 - 2v)]

and

x = +/-(1 + v^2)/(1 - v^2 - 2v) - [+/-4v/(1 - v^2 - 2v) +/- 1 -/+(1 + v^2)/(1 - v^2 - 2v)]

where I notice an easy case to give integer solutions with v = -2 (but v can be a fraction as well so that is just what worked for this example), as I have then:

y = [-/+8/(1 - 4 + 4) +/- 1 -/+(1 + 4)/(1 - 4 + 4)] =[-/+8 +/- 1 -/+ 5]

and

x = +/-(1 + 4)/(1 - 4 + 4) - [-/+8/(1 - 4 + 4) +/- 1 -/+(1 + 4)/(1 - 4 + 4)] = +/-5 - [-/+8 +/- 1 -/+ 5]

So I get several solutions with that choice.

For instance, y = 8 - 1 + 5 = 12 is a solution, with x = 5 + 12 = 17, as 17^2 - 2(12)^2 = 1.

Derivation of the result was done using research found by doors opened by my Quadratic Diophantine Theorem, my use of it against Pell's Equation, and my research result linking Pell's Equation to discrete ellipses and Pythagorean Triples.

The weird thing here then is that you can just solve for x and y in Pell's Equation in terms of an additional variable, which is otherwise nowhere to be found in x^2 - Dy^2 = 1. So an additional previously hidden variable allows functional solutions that encompass all behavior of Pell's Equation, whether people realize it is there or not.

But now x and y are functions of v, so I can have x = r(v)/t(v), and y = s(v)/t(v), so that I can solve to find:

(r(v) - t(v))(r(v) + t(v)) = D(s(v))^2

and remarkably enough, turn factoring into a calculus problem of finding minima when D is a composite you are trying to factor!!!

So the reality of the result has HUGE implications in lots of areas, and it should be of interest I'd think, how I even found the solution.

I've previously posted today on some math newsgroups to get some independent verification which I've gotten so the equations are correct.

That changes the game. We can solve problems in new ways. And in so doing get direct answers even in areas where people thought they had all the answers, as I note that continued fractions have been considered the way to go with Pell's Equation, while this research opens the door to an entirely new way to solve with reasons for why it can be more difficult for certain values of D that continued fractions could NOT explain.

It answers the 'why' of Pell's Equation—by introducing an additional previously hidden variable dependency.

Quite simply it is a revolutionary discovery as to how to approach certain types of problems.

General solution to Pell's Equation

General solution to Pell's Equation

Given

x^2 - Dy^2 = 1

I have proven:

y = [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

and

x = +/-(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v) - [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

where f_1*f_2 = D-1, and the f's are non-zero integer factors, while v is a free variable.

As Pell's Equation is normally considered in integers as a Diophantine equation note that you find rational v such that x and y are integers, which gives the 'why' of Pell's Equation. For instance, for D=2, f_1*f_2 = 1, so I have:

y = [+/-4v/(1 - v^2 - 2v) +/- 1 -/+(1 + v^2)/(1 - v^2 - 2v)]

and

x = +/-(1 + v^2)/(1 - v^2 - 2v) - [+/-4v/(1 - v^2 - 2v) +/- 1 -/+(1 + v^2)/(1 - v^2 - 2v)]

where I notice an easy case to give integer solutions with v = -2, as I have then:

y = [-/+8/(1 - 4 + 4) +/- 1 -/+(1 + 4)/(1 - 4 + 4)] = [-/+8 +/- 1 -/+ 5]

and

x = +/-(1 + 4)/(1 - 4 + 4) - [-/+8/(1 - 4 + 4) +/- 1 -/+(1 + 4)/(1 - 4 + 4)] = +/-5 - [-/+8 +/- 1 -/+ 5]

So I get several solutions with that choice.

For instance, y = 8 - 1 + 5 = 12 is a solution, with x = 5 + 12 = 17, as 17^2 - 2(12)^2 = 1.

Derivation of the result was done using research found by doors opened by my Quadratic Diophantine Theorem, my use of it against Pell's Equation, and my research result linking Pell's Equation to discrete ellipses and Pythagorean Triples.

The value of basic research is shown by how somewhat disparate results came together to answer a classical problem that is over 2000 years old in an entirely new way, shattering beliefs I have come across within the modern mathematical community that there are not new answers to be found in seemingly well-worked areas.

Mathematics is an infinite subject. That is a good thing. Human beings limit mathematics out of intellectual ignorance.

Discovery never ends unless people decide for ignorance over knowledge.

JSH: Have proof, now what?

I'm at a loss and need some advice. I now have the full proof that I've found this wacky way to solve the factoring problem using Pell's Equation, but I don't know what to do next.

Should I write a paper, and send it where? Or what?

I will NOT implement, so do not even post that I should. Theory is one thing, but implementation is a dangerous tool.

Besides I have a rather simple mathematical proof. Implementation is just for the people who don't understand mathematical proof.

There is no room for doubt. Factoring problem is trivially solved. Only question now is how long it takes the world to find out, and how it finds out.

Thanks!

JSH: Situation beyond weird

So I figure out how to connect Pell's Equation to factoring in a direct solution. Explain it all with simple algebra, and unbelievably posters still come out of the woodwork to lie about it. Unbelievable. Nothing sacred.

So, ok, why not implement and prove with a monster factorization?

And tell who? Oh, I know, US Government people who will send me where? Oh I know, the NSA.

And what is the NSA?—one of the world's largest collections of mathematicians.

Besides I can prove mathematically that it's a direct solution to the factoring problem. Prove that it is about the calculus and calculating minima, and what good has that been?

I often wonder what goes through your heads. How any of you can still put up the energy to do what you call mathematics, and ignore even the most basic mathematical proofs that I present.

Double-think.

Or, maybe you are all in on the scam. Maybe when you start college as undergrads your professors pull you aside and make sure you understand that it's all fake that you don't believe in mathematical proof and that as long as you play ball you can live comfortably on the stupidity of the world.

JSH: Pell's Equation, factoring, and two way method

I will admit I've been a bit puzzled by recent false claims about my latest research linking Pell's Equation to a discrete ellipse and in so doing to integer factorization. I've heard of academic jealousy but what's going on here is almost beyond belief.

To understand why, here are the equations again in rationals:

x^2 - Dy^2 = 1

you have solutions for an ellipse or Pythagorean triples with

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

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

I use rationals so that it is all more compact.

The result above links solving the Pell's Equation to solving a discrete ellipse—a much easier problem!!!

One approach I've given is to let j = uv, introducing yet another

degree of freedom, and use the discrete ellipse:

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

so

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

and using f_1*f_2 = D-1 you can factor with:

(x+y - (uv+/-1)) = f_1*u

and

(x+y - (uv+/-1)) = f_2u*v^2

That's just one way to go, but notice I can now solve for u in terms of v (or vice versa but I think it's easier) and solve for x+y, and then solve for x and y directly as functions of v.

So how can I evaluate claims against this method using those equations above?

They go in BOTH directions.

So you can actually factor a composite D, get x and y, in rationals, and solve for j, and then have a choice of u and v.

What I've done is turn the factoring problem into a calculus problem of finding minima.

There are posters who unbelievably are making false claims about this research, but all you have to do is go backwards from Pell's Equation.

For instance, if x = r/c and y = s/c, then you can just let s=1, and use r+c = g_1, and r-c = g_2, where g_1*g_2 = D, and solve for r and c, and then find j and consider available u and v.

For some of you that may be the best way to approach this research. Pick some easy composites for your D. Find r and c, and then get j, and consider values for u and v that will work with it.

You will find minima are what work. Guaranteed.

Now the factoring problem is trivially solved. Posters arguing against it on newsgroups won't change that, and the equations will work no matter what any of you say.

But do any of you believe in mathematics at all?

I mean, a solution to the factoring problem using Pell's Equation, of all things, which brings in the calculus turning the problem into a minima problem is just very cool!

To hate this result because you don't like me or are jealous of me or want to protect your jobs or because you're just terribly dense is just sad. Very, very sad.

It doesn't get much bigger than this result. It's one of the hugest finds in all of mathematical history.

And there are people freaking lying about it??!!! What gives?

What's wrong with you people?

JSH: Long process

Wow, seems no one can shoot down this approach of using Pell's Equation to factor, and I'll admit that when I first came up with it, I posted in certain ways just in case. You never know, you know?

But now that it looks like it's a viable path for basic research and nothing is happening in the world with my research as usual—I'm so ignored—the process takes over, which is rather long.

I may write another paper, and may submit to a journal. I'll think about what ways this approach might fail and meditate on the implications as well as just how freaking cool it is. Months can easily go by, as I ponder it and other things as I'm about to move on.

And if need be I'll post looking for people shooting holes in it and nothing else.

In case you've all missed prior postings where I talk about what I do: newsgroups are for brainstorming. Or entertainment.

I'm only interested in people finding out error. So I antagonize people because if you don't like me, you're better at finding things that are wrong. Sorry but that's reality. When people are all chummy they are much worse at finding mistakes, but when it's angry, people jump on them.

So where does that leave things with the social issues of factoring?

Well, I've often worried about the impact of finding a solution to the factoring problem on the rest of the world, but, um, I think this approach is my SECOND solution to it, and mostly the world just ignores all of it, so those fears are unfounded.

Truth is stranger than fiction.

People can believe all kind of things and can miss things just because they refuse to see them.

I have what I think is my second solution to the factoring problem which is just simpler than my first. After my first I found a general TSP algorithm, so I proved P=NP, and then I was playing around with discrete stuff and stumbled on this Pell's Equation thing.

Now, most of you don't believe any of that which oddly enough seems to mean you cannot see it, which is something that used to bug me, but increasingly I don't care.

You live in your simple world of primitive mathematics that is woefully too complicated as well as often horribly wrong, and I'll live in the real one.

I know P=NP. I have the solution to TSP and understand why it works. I have two solutions to the factoring problem to add to my proof of Fermat's Last Theorem, my prime counting function, and my proof of the spherical packing problem among other major discoveries.

The more I pile on the discoveries the crazier you all get in denial, which I have to admit is starting to just seem freaking cool, as in weird as hell.

But I didn't make this world. I just live in it.

Maybe there is something about the human brain in groups when trained in modern mathematical teachings which breaks something. Or maybe not. Even my understanding of reality may be far more primitive than even I like to admit.

You people know nothing about reality and what's possible. But, I fear, I may only know a little bit more.

JSH: Why factoring solution must work

Turns out you can prove through mathematical logic that given a solution like:

(r(v) - c(v))(r(v) + c(v)) = D(s(v))^2

where r(v), c(v) and s(v) are non-zero integer functions of v, if D is a composite, you MUST have a non-trivial factorization if

abs(r(v) - c(v)) < D and abs(r(v) + c(v)) < D.

So if you can FIND such function that smoothly traverse through all possible solutions in integers, then you know that factoring D is just a minima problem.

It goes without saying then, since minima problems are calculus problems and I think I would have heard if the factoring problem were considered to be an area of the calculus that NO ONE prior in all of human history has a known result doing the above.

But here is one now, and ironically it comes from Pell's Equation.

In rationals—I'll explain more on that later—given

x^2 - Dy^2 = 1

you have solutions for an ellipse or Pythagorean triples with

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

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

I use rationals so that it is all more compact.

The result above links solving the Pell's Equation to solving a discrete ellipse—a much easier problem!!!

One approach I've given is to let j = uv, introducing yet another degree of freedom, and use the discrete ellipse:

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

so

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

and using f_1*f_2 = D-1 you can factor with:

(x+y - (uv+/-1)) = f_1*u

and

(x+y - (uv+/-1)) = f_2u*v^2

That's just one way to go, but notice I can now solve for u in terms of v (or vice versa but I think it's easier) and solve for x+y, and then solve for x and y directly as functions of v.

Which turns the factoring problem into a minima problem, which makes it work for the calculus.

I've already had some flak on newsgroups from posters who spent YEARS talking down my research where I noted often enough that they lie, but of course, with a lot of them saying that I was lying, while I was saying they were lying, things didn't really move much.

But now, if you believe them, you have to dismiss some easy mathematical logic, some easy algebra, and even then, just wait for other people to exploit the new mathematical find, and try to explain why you didn't do anything.

Kind of like sitting quietly while gasoline is poured on you and someone begins striking a match—or I am in error.

Break the logical chain in this post, prove me wrong, or wait for authorities to come and take some of you to jail when they find out I am correct and start looking at prosecutions.

You are not in a dream. What is happening is not a joke. What you do not do for some of you can get you prosecuted at this point.

And your life will be over. The world will go on. This result can have negatives but I am certain we will be ok.

But for some of you, the last days of your happy lives outside of prison are ticking away, and it may be for what you do NOT do.

Which may not seem fair, but, prove me wrong! The argument I have is in this post.

If no one can break the logic here then you DO SOMETHING RIGHT FOR ONCE, or wait. When I know you'll whine and cry as they lead you off to prison saying you didn't do anything—but that will be why you get prosecuted.

JSH: Pondering demographics

Here's a link to QuantCast data about readers of my three blogs:

http://www.quantcast.com/p-89GNpWgpweHjg

I find the data puzzling.

JSH: Frustrating situation

Solving the factoring problem has long been the last thing I wanted to do. Almost every time I'd buy something online I'd tell myself, why mess this up? And ask myself if there wasn't another way.

Now the factoring problem IS solved (again) and watching the reaction of the newsgroups and the continued lack of proper reaction by the cryptological and mathematical communities, I realize it was, unfortunately necessary.

We could still have days to go while the bizarre and stupid denial continues. Which begs for explanation.

So why'd you do it? Why did so many of you lie about so many mathematical discoveries for so many years, and ignore warnings that if necessary I'd solve the factoring problem and now act like I didn't when I did (for the second time)?

My analysis: women

My analysis is that for many of you your egos are wrapped up in your supposed discoveries and accomplishments as mathematicians, and my research takes that away from you, so to keep from looking into the eyes of your wives or girlfriends, and seeing the disappointment or worse, you just lied, and lied, and have kept lying.

I guess it's evolutionary, but so weird to have \$5 trillion U.S. plus wiped from the world economy because some dudes didn't want to disappoint their women.

Such is life.

JSH: What have you been looking for?

As I take the world down the path I've been trying to avoid for years I am really curious about what would have mattered say for readers of this group to believe that there was a massive error in number theory, or that I had fairly simple mathematical research that indicated big things in multiple areas from the Riemann Hypothesis, to, yes the factoring problem.

My guess is nothing but validation from the mathematical establishment, while some of you may claim a demonstration factorization would have done it (you're lying, it'd take too long to explain why I believe that is true).

I consider the situation to have been a Catch-22: I couldn't prove to any of you a problem with the very establishment you wanted to tell you of any problem.

Is that assessment correct?

Control is now out of your hands, but information is still of interest to me. As I make your life change, I want to give you one more opportunity to explain to me why it is necessary for this path.

Was there anything else I could have done?

The clock is ticking. Unless my analysis is very wrong, the world as you know it will be gone in less than 24 hours.

Here is a chance to chat until that change, whether you believe it or not. By the time you know it's true, you will understand that I had an impossible decision to make, where a decision had to be made, so I made one.

[A reply to someone who wondered if James was in an alcoholic death spiral.]

Hey, try to imagine you're in my shoes. Assume just for the sake of argument that you are an amateur at mathematics despite a B.Sc. in physics, but you come across some major mathematical problem—and find that math people rip on you rather than acknowledge it.

You even get a result published—but the freaking journal dies.

With easy results using basic algebra you prove your case without any reasonable doubt left, but the experts in the field do not do what they're supposed to do.

You even go back to your people—physics people.

And cannot get anywhere there either.

This thread is an opportunity for the sci.physics newsgroup to say what would work.

My guess is, nothing other than mathematicians telling you that the research is correct.

That is the Catch-22 as they are in error, so they will not.

If that is the answer, then fine. It's the one I have concluded is the correct one already.

If you disagree, give another way.

You have less than 24 hours to answer in this thread. Before the world changes.

JSH: Using Usenet

The point of posting what looks to me like a very simple solution to the factoring problem on Usenet is not to convince the newsgroups of anything. It's to project the result around the world in a way that presumably cannot be blocked.

The reason for that is that everything has been REALLY WEIRD where I can prove major things and have nothing happen, and I have past experience with journals where nothing happened and one of them even freaking died when I did get something briefly published.

I also know that number theorists have this massive error in their field which I can prove easily enough and I can't get anything going there either.

But that means mathematicians are massively leveraged against my research being correct.

The careers of some fairly important people around the world could be shattered by a simple solution to the factoring problem, where these are mathematicians in extremely powerful positions.

Best thing then is to project the information worldwide first, and then either let someone else break the news, or come back later and do it myself, ending the possibility say, of some organization like the NSA simply locking me up away, hiding this result, along with my other research, so that mathematicians can keep up the lies.

The result I've posted is trivially a simple solution to the factoring problem.

None of you really need to do anything though, as if that is correct, I can assure you that it will be known, oh, in about 24 hours.

There may be serious network disruptions around the world, so you can prepare for that worst case.

The Internet as you've known it is about to change. I didn't want to do it this way.

But given the bizarre things that have happened up until now, I see no there way.

Once stability is restored governments can fully investigate how mathematicians around the world did whatever they did as I'm not sure now either what has been going on.

Then, of course, they can be questioned under oath.

The answers will be revealed. The consequence though is the end of the Internet as you have known it.

I apologize to the people of the world for any serious disruptions or inconvenience.

I saw no other way.

Factoring problem trivially solved

We're now firmly in the danger zone. I have a simple solution to the factoring problem. I will give it in a moment but I need to emphasize to you all that there is no longer any room for playing games with this result nor for denial.

I've found in rationals that there is a simple relation connecting what is commonly called Pell's Equation to a discrete ellipse:

Given x^2 - Dy^2 = 1

it must be true that

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

where

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

That allows you to solve for the variables by factoring:

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

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

so

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

and as you have j as a free variable, you can let j = mn, where m and n are rationals, f_1*f_2 = (D-1), where f_1 and f_2 are integers and generally solve with:

x+y + mn+/-1 = f_1*m

x+y - (mn +/-1) = f_2*mn^2

so you can, for instance, solve out m, and choose n to be some integer, and then solve for x+y, and then solve for x and y directly with

j = mn = ((x+Dy)-/+1)/D, so you have two simultaneous equations

which gives you a solution to

x^2 - Dy^2 = 1.

And you have available an infinity of such solutions, as many as you wish, simply by cycling through values for n. Or you could solve out n, and cycle through values for m.

Let D equal a target composite T to be factored, then it is shown that you can trivially find an infinite number of solutions in rationals to

x^2 - Ty^2 = 1.

Letting x=r/c and y=s/c, you have then a simple technique for finding integers r, s and c, such that

(r-c)(r+c) = Ts^2

which strongly indicates the factoring problem is simply solved.

The technique allows you to easily, casually, with trivial algebra, generate as many solutions to

x^2 = y^2 mod T

as you wish! Without regard or even consideration to the size of the target composite T.

You DO not have the choice to just play games with this result.

Unfortunately mathematicians thought factoring is a hard problem when it is a trivial one.

Factoring is a trivial problem, solved with easy algebra.

You cannot sit on this result, lie about this result, or just ignore this result.

The old games are done. The lies without consequence are over. You have to properly acknowledge this result.

It changes how the Internet must do security. And that change is immediate. 