### Saturday, March 31, 2007

## Factoring more beautiful now

The reality of the mathematics of what I've recently discovered is so simply beautiful. Succinctly for EVERY composite factorization by a difference of squares you have an alternate factorization automatically given as well:

x^2 = y^2 mod T

algebraically forces

S = (α + 1)k^2 mod T

where

α*k = 2x mod T

so α*k is set by your factorization, but you have an infinity of alternates available for α and k individually.

That is not in debate, and it is beautiful mathematics in and of itself which expands the factoring problem outside of what was previously known as mathematicians of the 20th century primarily labored developing various techniques to find a difference of squares where the most advanced known today is the Number Field Sieve.

I've mostly focused on the second factorization as a way to find the first, calling the technique surrogate factoring, but the mathematics offers more than one way to use this knowledge, and just yesterday I figured out yet another way to use the connectedness of factoring.

With that way I use a difference of squares of the surrogate, and then search for

(α + 1)k^2 mod S = 0 mod T

where I found a very easy way to do that, so that now you can deliberately set out to find a difference of squares, easily, with a straightforward method, for any composite that you choose.

My previous post on the subject was flawed—I'd just come up with it so mistakes are not a surprise—so to get the latest corrected the best thing is to go to my Extreme Mathematics group:

http://groups.google.com/group/extrememathematics/web/another-page-more-surrogate-factoring

It is such beautiful and simple mathematics that it is a shame that there is so much other surrounding this and I freely admit that I came to factoring primarily to force mathematicians to acknowledge my mathematical discoveries.

I found a proof of Fermat's Last Theorem, THE prime counting function, which not only offers a way to disprove the Riemann Hypothesis, but also tells 'why' prime numbers distribute as they do. I've given the definition of mathematical proof. And I cleaned up some areas in logic, explaining how logical form is crucial.

My research is defined like this latest factoring research by its simplicity.

In contrast modern mathematicians have work that is often complex, difficult to understand, and that is a technique to hide the reality that much of it is wrong.

Their research is so abstruse not because mathematics forces it so, but because by making it so, they can hide that it is not correct.

I have come to the sad conclusion that much of that is deliberate.

Because I am someone who breaks their way to make money, modern mathematicians are quite willing to ignore my research and try to continue their scam.

Mathematical research is difficult. It is easier for groups of people to play at being mathematicians than to be real ones. People like me emerge rarely.

But the greatest proof of the lack of real intellectual ability of those in the mathematical community who fight me is that fight.

They can't win. No one has won against someone like me. If they had our world would not be as it is, if Newton had been beaten, or Archimedes, or Einstein.

So factoring is the way to beat them. They will fight still because they're not smart enough to quit and later they will beg the world for mercy.

But remember, they sought to cut the jugular of the world. They sought to end the progress of the human species itself.

They sought to end us all no matter what they claim later.

They tried to make your life meaningless, to take away the future of your children, and to make the sacrifices and efforts of all those who came before us…meaningless.

[A reply to someone who wanted to know more about how James had explained “how logical form is crucial”.]

"Liar's Paradox". I showed how it's about form and is just about a need for a 3 logic, as the "law of the excluded middle" is no law at all.

I think it telling though that modern math people can turn successes like mine into a joke, or claim I do, because they have figured out a style.

Today people think they know what important mathematics looks like, and they think they know how people with correct mathematics should act.

But it's not a big deal now as that's why I went to the factoring problem, where the latest results are just so beautiful and amazing.

IN ways I feel sorry for modern "mathematicians" many of whom were brought into the field after it was corrupted (assuming that really happened in the latter 20th) and taught style as substance and trained to believe that being correct in mathematics is about being believed to be correct by your peers.

It's almost ludicrous to look back at how often these people would call me names or claim that no one believed me as if they had trumped me with the maximum!!!

To think, not being believed, how could I go on? LOL.

Well because most people don't know squat about math and lots of people get even basic things wrong, and, oh yeah, people do lie.

These losers just lie about math. From the proper perspective they just join a long line of people who pretended to be something they are not for personal gain, at the expense of others.

x^2 = y^2 mod T

algebraically forces

S = (α + 1)k^2 mod T

where

α*k = 2x mod T

so α*k is set by your factorization, but you have an infinity of alternates available for α and k individually.

That is not in debate, and it is beautiful mathematics in and of itself which expands the factoring problem outside of what was previously known as mathematicians of the 20th century primarily labored developing various techniques to find a difference of squares where the most advanced known today is the Number Field Sieve.

I've mostly focused on the second factorization as a way to find the first, calling the technique surrogate factoring, but the mathematics offers more than one way to use this knowledge, and just yesterday I figured out yet another way to use the connectedness of factoring.

With that way I use a difference of squares of the surrogate, and then search for

(α + 1)k^2 mod S = 0 mod T

where I found a very easy way to do that, so that now you can deliberately set out to find a difference of squares, easily, with a straightforward method, for any composite that you choose.

My previous post on the subject was flawed—I'd just come up with it so mistakes are not a surprise—so to get the latest corrected the best thing is to go to my Extreme Mathematics group:

http://groups.google.com/group/extrememathematics/web/another-page-more-surrogate-factoring

It is such beautiful and simple mathematics that it is a shame that there is so much other surrounding this and I freely admit that I came to factoring primarily to force mathematicians to acknowledge my mathematical discoveries.

I found a proof of Fermat's Last Theorem, THE prime counting function, which not only offers a way to disprove the Riemann Hypothesis, but also tells 'why' prime numbers distribute as they do. I've given the definition of mathematical proof. And I cleaned up some areas in logic, explaining how logical form is crucial.

My research is defined like this latest factoring research by its simplicity.

In contrast modern mathematicians have work that is often complex, difficult to understand, and that is a technique to hide the reality that much of it is wrong.

Their research is so abstruse not because mathematics forces it so, but because by making it so, they can hide that it is not correct.

I have come to the sad conclusion that much of that is deliberate.

Because I am someone who breaks their way to make money, modern mathematicians are quite willing to ignore my research and try to continue their scam.

Mathematical research is difficult. It is easier for groups of people to play at being mathematicians than to be real ones. People like me emerge rarely.

But the greatest proof of the lack of real intellectual ability of those in the mathematical community who fight me is that fight.

They can't win. No one has won against someone like me. If they had our world would not be as it is, if Newton had been beaten, or Archimedes, or Einstein.

So factoring is the way to beat them. They will fight still because they're not smart enough to quit and later they will beg the world for mercy.

But remember, they sought to cut the jugular of the world. They sought to end the progress of the human species itself.

They sought to end us all no matter what they claim later.

They tried to make your life meaningless, to take away the future of your children, and to make the sacrifices and efforts of all those who came before us…meaningless.

[A reply to someone who wanted to know more about how James had explained “how logical form is crucial”.]

"Liar's Paradox". I showed how it's about form and is just about a need for a 3 logic, as the "law of the excluded middle" is no law at all.

I think it telling though that modern math people can turn successes like mine into a joke, or claim I do, because they have figured out a style.

Today people think they know what important mathematics looks like, and they think they know how people with correct mathematics should act.

But it's not a big deal now as that's why I went to the factoring problem, where the latest results are just so beautiful and amazing.

IN ways I feel sorry for modern "mathematicians" many of whom were brought into the field after it was corrupted (assuming that really happened in the latter 20th) and taught style as substance and trained to believe that being correct in mathematics is about being believed to be correct by your peers.

It's almost ludicrous to look back at how often these people would call me names or claim that no one believed me as if they had trumped me with the maximum!!!

To think, not being believed, how could I go on? LOL.

Well because most people don't know squat about math and lots of people get even basic things wrong, and, oh yeah, people do lie.

These losers just lie about math. From the proper perspective they just join a long line of people who pretended to be something they are not for personal gain, at the expense of others.

### Friday, March 30, 2007

## JSH: You are SO dumb

I'm sitting here laughing my ass off as eventually you just have to laugh.

That is just such a STUPID IDEA. You people are just such total freaking fools.

LOL. None of you really know any real mathematics, do you?

You are just such fools.

BILLIONS of people on earth, and you fools can get so far with such rank stupidity about basic math.

We're not talking complicated mathematics here people.

It's baby math. And you're just such simple fools.

Easy algebra, oh, but most people hate algebra. The fools. That's how you fooled them.

They hate easy math, so you can screw up easy math.

You are just the stupidest people EVER.

Mathematics is an adult game, not a game for babies.

You people wish you had a clue about numbers. LOL. I can't stop laughing.

You have TRILLIONS OF DOLLARS running around the world supposedly protected by your moronic idea.

TRILLIONS.

It is a world of fools.

That is just such a STUPID IDEA. You people are just such total freaking fools.

LOL. None of you really know any real mathematics, do you?

You are just such fools.

BILLIONS of people on earth, and you fools can get so far with such rank stupidity about basic math.

We're not talking complicated mathematics here people.

It's baby math. And you're just such simple fools.

Easy algebra, oh, but most people hate algebra. The fools. That's how you fooled them.

They hate easy math, so you can screw up easy math.

You are just the stupidest people EVER.

Mathematics is an adult game, not a game for babies.

You people wish you had a clue about numbers. LOL. I can't stop laughing.

You have TRILLIONS OF DOLLARS running around the world supposedly protected by your moronic idea.

TRILLIONS.

It is a world of fools.

## JSH: Problem with cons, they never quit

I think that the latest with surrogate factoring is just the stupidest answer yet. So damn simple.

So nincompoops propose to the world one of the dumbest ideas in the history of the world for security, saying that factoring is a hard problem, and hey, even when R, S and A first put up their little test saying it would take umpteen years like some significant fraction of the lifetime of the Universe—it was solved in a few years.

Um, that to me is what one calls in literary circles, a clue.

Now it is finally revealed that for EVERY factorization with a difference of square you have a gimme alternate factorization, and what do these "beautiful minds" wish to do with the knowledge?

Why, ignore it of course!!!

Because they are freaking con artists who got over for years and years because, hey, yes, let's admit it, the world is stupid.

It is a stupid world. Look at what it let George W. Bush do. What it still let's him do.

It is a a STUPID WORLD.

It is just one of the dumbest worlds ever, and people are just on the whole, remarkably stupid.

Yeah, you know it. Admit it. Deep down you know you're stupid. That's why some of you protest so much that you're really smart, because you know you are STUPID.

Most of you are brainless but you get to pretend to be smart people so you will keep at it, and I understand it now and just wish to tell you that, hey, you can keep pretending. It's ok.

Because, hey, you're right, one day, you're just going to die any way. Why not act like you have a real brain along the way?

So I do a few extra things with the surrogate factoring theory that has been around for a few months now, and there is NO DOUBT that the supposedly intelligent mathematical community will act like nothing happened, and whistle along wishing that the entire freaking world continues to be so stupid that no one in that world exploits the latest revelations from the mathematical theory.

Or, they just don't care. I think they just don't care as long as they can pretend for one more day.

LOL. They are so stupid. I chuckle to myself at the dumbness of the con and a world that could be convinced with something that could be broken so easily.

I am more and more convinced that exploits are happening worldwide as I see more activity from McAfee and Microsoft with their updates, and as more and more news headlines are about reasons for why credit card numbers are all over the place, as liars go to lying, I believe, thinking that maybe if they just keep lying long enough, the world will keep using a system that has more holes than a sieve.

Hey, lying works, just ask George W. Bush.

Here's to R, S and A! They learned the real secret to success—lie to a stupid world about mathematics and just keep lying.

In a stupid world that can take you a long way. And yup, this IS a stupid world.

Stupid world. Stupid world. Stupid world.

So nincompoops propose to the world one of the dumbest ideas in the history of the world for security, saying that factoring is a hard problem, and hey, even when R, S and A first put up their little test saying it would take umpteen years like some significant fraction of the lifetime of the Universe—it was solved in a few years.

Um, that to me is what one calls in literary circles, a clue.

Now it is finally revealed that for EVERY factorization with a difference of square you have a gimme alternate factorization, and what do these "beautiful minds" wish to do with the knowledge?

Why, ignore it of course!!!

Because they are freaking con artists who got over for years and years because, hey, yes, let's admit it, the world is stupid.

It is a stupid world. Look at what it let George W. Bush do. What it still let's him do.

It is a a STUPID WORLD.

It is just one of the dumbest worlds ever, and people are just on the whole, remarkably stupid.

Yeah, you know it. Admit it. Deep down you know you're stupid. That's why some of you protest so much that you're really smart, because you know you are STUPID.

Most of you are brainless but you get to pretend to be smart people so you will keep at it, and I understand it now and just wish to tell you that, hey, you can keep pretending. It's ok.

Because, hey, you're right, one day, you're just going to die any way. Why not act like you have a real brain along the way?

So I do a few extra things with the surrogate factoring theory that has been around for a few months now, and there is NO DOUBT that the supposedly intelligent mathematical community will act like nothing happened, and whistle along wishing that the entire freaking world continues to be so stupid that no one in that world exploits the latest revelations from the mathematical theory.

Or, they just don't care. I think they just don't care as long as they can pretend for one more day.

LOL. They are so stupid. I chuckle to myself at the dumbness of the con and a world that could be convinced with something that could be broken so easily.

I am more and more convinced that exploits are happening worldwide as I see more activity from McAfee and Microsoft with their updates, and as more and more news headlines are about reasons for why credit card numbers are all over the place, as liars go to lying, I believe, thinking that maybe if they just keep lying long enough, the world will keep using a system that has more holes than a sieve.

Hey, lying works, just ask George W. Bush.

Here's to R, S and A! They learned the real secret to success—lie to a stupid world about mathematics and just keep lying.

In a stupid world that can take you a long way. And yup, this IS a stupid world.

Stupid world. Stupid world. Stupid world.

### Thursday, March 29, 2007

## Question on integer solutions to a circle

Here's something where I was working on one thing, did something wrong, so I came up with this conjecture, and later realized that it didn't follow from what I was working on originally so I dropped it, but find myself curious enough to pose it as a question.

Given a circle

a^2 + b^2 = r^2

with an integer radius what is the frequency with which you will have non-zero integer solutions?

Here's an example to understand the question as consider the first five cases:

3^2 + 4^2 = 5^2

6^2 + 8^2 = 10^2

5^2 + 12^2 = 13^2

9^2 + 12^2 = 15^2

and

8^2 + 15^2 = 17^2

So you have 5 of the first 17 naturals, and originally I had thought the ratio would be about 1/3, but some posters replying to an original post of mine on this subject claimed otherwise.

What is the actual ratio?

It's the kind of question that I'd think would have been asked before, so alternatively, can someone provide a link to an answer?

Oh yeah, secondary would be what is the ratio for primitive solutions i.e. taking away the multiples of another solution like 6^2 + 8^2 = 10^2?

Given a circle

a^2 + b^2 = r^2

with an integer radius what is the frequency with which you will have non-zero integer solutions?

Here's an example to understand the question as consider the first five cases:

3^2 + 4^2 = 5^2

6^2 + 8^2 = 10^2

5^2 + 12^2 = 13^2

9^2 + 12^2 = 15^2

and

8^2 + 15^2 = 17^2

So you have 5 of the first 17 naturals, and originally I had thought the ratio would be about 1/3, but some posters replying to an original post of mine on this subject claimed otherwise.

What is the actual ratio?

It's the kind of question that I'd think would have been asked before, so alternatively, can someone provide a link to an answer?

Oh yeah, secondary would be what is the ratio for primitive solutions i.e. taking away the multiples of another solution like 6^2 + 8^2 = 10^2?

## Reversal with surrogate factoring

Abstracting to the gist of surrogate factoring I realized it says that for every factorization of a composite T, with difference of squares

x^2 = y^2 mod T

which of course is the same as

(x-y)*(x+y) = 0 mod T

and is for that reason a factorization, there is an alternate factorization of an integer S, where

S = (α + 1)k^2 mod T

and α and k are defined by

α*k = 2x mod T

as then trivially you can multiply both sides of that relation by k, add back to the original difference of squares, add k^2 to both sides and find

(x+k)^2 = y^2 + (α + 1)k^2 mod T

explaining the second factorization, as now you have a factorization of this alternate number using the same x, y and this extra variable k:

(x+k - y)*(x+k+y) = (α + 1)k^2 mod T

So for EVERY composite factorization of T using difference of square that algebra tosses a factorization of ANOTHER composite (α + 1)k^2 mod T back at you!!!

It's just been there all the time. There has just been another factorization out there all along with difference of squares and I call the second factorization the surrogate factorization.

But hey, that goes both ways! I like to use T as the target, and S as the surrogate, but what if you reverse the equations so that you pick some surrogate and factor it to see how easily you can factor the target T?

Equations then become

x^2 = y^2 mod S

and

α*k = 2x mod S

gives you

(x+k)^2 = y^2 + (α + 1)k^2 mod S

and now, of course, you'd want to choose

(α + 1)k^2 mod S = 0 mod T

and you can go explicit to start solving everything, so I have

α*k = 2x + u_1*S,

so α = (2x + u_1*S)/k, and

((2x + u_1*S)/k + 1)k^2 + u_2*S = 0 mod T, so

2xk + u_1*S*k + k^2 + u_2*S = 0 mod T

so you can collect to get

k^2 + (2x + u_1*S)k + u_2*S = 0 mod T

and now complete the square to get

(k + (2x + u_1*S)/2)^2 = (2x + u_1*S)^2/4 + u_2*S mod T

and now you have a fairly trivial way to solve for k, as u_1 and u_2 are integers of your choice, so you can, intriguingly enough, simply CHOOSE a quadratic residue of T, pick some S, factor it, get x, and then find u_1 and u_2 that will work!

What makes this reversal fascinating to me is that it turns the factoring problem on its head, where instead of searching for quadratic residues modulo T, like most major factoring methods work to do to solve the congruence of squares, you just pick one. You pick a quadratic residue of your target T, and use it and the surrogate factorization to get a difference of squares with your target, so then you should have a 50% chance of factoring your target with that choice, as then you just plug your solutions for k and α back into

(x+k)^2 = y^2 + (α + 1)k^2 mod S

as then also you will have an alternate factorization, the surrogate factorization

(x+k)^2 = y^2 mod T

as shown above with trivial algebra.

Remember people very trivial algebra that turns the factoring problem upside down.

If my suspicions are correct because many modern mathematicians willfully lie to the public, they will try their best to ignore this result, like they have with the previous surrogate factoring theory.

Or hey, they can surprise me—and do a press release—and then I'll apologize for saying they are willful con artists.

I don't think I'll be surprised, as I fear they will fight to the bitter end and the Math Wars will end nastily, unfortunately.

Do the press release people, and notify your governments. Or try the waiting game one last time, and prove to the world that every nasty thing I've said about you is true.

x^2 = y^2 mod T

which of course is the same as

(x-y)*(x+y) = 0 mod T

and is for that reason a factorization, there is an alternate factorization of an integer S, where

S = (α + 1)k^2 mod T

and α and k are defined by

α*k = 2x mod T

as then trivially you can multiply both sides of that relation by k, add back to the original difference of squares, add k^2 to both sides and find

(x+k)^2 = y^2 + (α + 1)k^2 mod T

explaining the second factorization, as now you have a factorization of this alternate number using the same x, y and this extra variable k:

(x+k - y)*(x+k+y) = (α + 1)k^2 mod T

So for EVERY composite factorization of T using difference of square that algebra tosses a factorization of ANOTHER composite (α + 1)k^2 mod T back at you!!!

It's just been there all the time. There has just been another factorization out there all along with difference of squares and I call the second factorization the surrogate factorization.

But hey, that goes both ways! I like to use T as the target, and S as the surrogate, but what if you reverse the equations so that you pick some surrogate and factor it to see how easily you can factor the target T?

Equations then become

x^2 = y^2 mod S

and

α*k = 2x mod S

gives you

(x+k)^2 = y^2 + (α + 1)k^2 mod S

and now, of course, you'd want to choose

(α + 1)k^2 mod S = 0 mod T

and you can go explicit to start solving everything, so I have

α*k = 2x + u_1*S,

so α = (2x + u_1*S)/k, and

((2x + u_1*S)/k + 1)k^2 + u_2*S = 0 mod T, so

2xk + u_1*S*k + k^2 + u_2*S = 0 mod T

so you can collect to get

k^2 + (2x + u_1*S)k + u_2*S = 0 mod T

and now complete the square to get

(k + (2x + u_1*S)/2)^2 = (2x + u_1*S)^2/4 + u_2*S mod T

and now you have a fairly trivial way to solve for k, as u_1 and u_2 are integers of your choice, so you can, intriguingly enough, simply CHOOSE a quadratic residue of T, pick some S, factor it, get x, and then find u_1 and u_2 that will work!

What makes this reversal fascinating to me is that it turns the factoring problem on its head, where instead of searching for quadratic residues modulo T, like most major factoring methods work to do to solve the congruence of squares, you just pick one. You pick a quadratic residue of your target T, and use it and the surrogate factorization to get a difference of squares with your target, so then you should have a 50% chance of factoring your target with that choice, as then you just plug your solutions for k and α back into

(x+k)^2 = y^2 + (α + 1)k^2 mod S

as then also you will have an alternate factorization, the surrogate factorization

(x+k)^2 = y^2 mod T

as shown above with trivial algebra.

Remember people very trivial algebra that turns the factoring problem upside down.

If my suspicions are correct because many modern mathematicians willfully lie to the public, they will try their best to ignore this result, like they have with the previous surrogate factoring theory.

Or hey, they can surprise me—and do a press release—and then I'll apologize for saying they are willful con artists.

I don't think I'll be surprised, as I fear they will fight to the bitter end and the Math Wars will end nastily, unfortunately.

Do the press release people, and notify your governments. Or try the waiting game one last time, and prove to the world that every nasty thing I've said about you is true.

### Wednesday, March 28, 2007

## String theory is a symptom of a deep problem

Mathematicians lie. They lie routinely but they do it in a way that works to keep them in business by lying in areas where they cannot easily be checked objectively—so they can just claim arguments are correct even when they are not—and they're seeding their corruption of intellectual activity into physics, as consider the following.

Factoring is kind of important. People rely on the idea that factoring big numbers is hard for security, but consider some basic math.

Start with the well-known congruence of squares

x^2 = y^2 mod T

introduce another congruence with a new variable,

k = 2x mod T so 2x = k mod T, multiply both sides by k, to get

2xk = k^2 mod T

and now add back to the original congruence of squares to get

x^2 + 2xk = y^2 + k^2 mod T

and then add k^2 to both sides and get

x^2 + 2xk + k^2 = y^2 + 2k^2 mod T, so finally you have

(x+k)^2 = y^2 + 2k^2 mod T

and I just showed how you relate EVERY factorization of one composite to a factorization of some other composite, which I call the surrogate.

Actually trying to use this idea to factor you need explicit equations:

(x+k)^2 = y^2 + 2k^2 + omega*T

And you can factor your target T by instead factoring 2k^2 + omega*T.

And it turns out that with a big T you still get big numbers to factor, but instead of one, you have as many as you wish.

IF that simple mathematics says to you what it says to me then you know that mathematicians have the world using a system where factoring one number can be transferred to factoring some other number, and if you do detailed analysis, you find that for an arbitrary non-zero omega and a non-zero k, as long as

2k^2 + omega*T

is over a certain size limit—that size depends on the size of T—then you can factor your target 50% of the time with some combination of factors of 2k^2 + omega*T.

If that sounds really complicated, just think of it as a two-way street where you can go either way factoring numbers and it's rather neat, and it's worth worrying about but mathematicians will NOT talk about it because it works, and they lie.

They know me. If they acknowledge this result I'll push forward all my other mathematical results as I'm using factoring to break them. Yup, I'm deliberately researching this area to PROVE they lie, so they can't talk about this research as then they just help me break them.

But why do they lie? I say because doing real mathematics is harder than faking it, and it's easier for groups of people to lie than most people accept, which is why they get away with it.

And that brings us to string "theory" with a lot of complexity but somehow, it doesn't give you much to experiment with, as mathematicians teach their tricks to the physics community.

Eventually you can have physics that doesn't work, is not correct, and no one knows or cares because physics people will, like many modern mathematicians, produce papers that sound physicky, with a lot of complexity, and there will be people who will cheer and claim stuff is correct, when it's just wrong.

Of course, you can think I'm wrong, but I just gave some basic math.

And, if I'm not, then the simple mathematics for breaking security systems is now out there as it has been for a few months as I talk about this, and I am currently working on coding an implementation to prove my point.

And mathematicians won't talk about it as I'm using this to get them, and you may end up collateral damage.

If that algebra above is correct then the world as you know it is about to change.

But consider, many of you probably know lots of complex mathematics and can check the algebra in your sleep, but a corrupted mathematical field means you will hold and wait, despite the implications because they are that good.

They are that good at lying.

And you are that intellectually weak, in actuality, no matter how brilliant you have deluded yourself into thinking you are.

If you can't trust some basic algebra, then how smart can you really be?

Factoring is kind of important. People rely on the idea that factoring big numbers is hard for security, but consider some basic math.

Start with the well-known congruence of squares

x^2 = y^2 mod T

introduce another congruence with a new variable,

k = 2x mod T so 2x = k mod T, multiply both sides by k, to get

2xk = k^2 mod T

and now add back to the original congruence of squares to get

x^2 + 2xk = y^2 + k^2 mod T

and then add k^2 to both sides and get

x^2 + 2xk + k^2 = y^2 + 2k^2 mod T, so finally you have

(x+k)^2 = y^2 + 2k^2 mod T

and I just showed how you relate EVERY factorization of one composite to a factorization of some other composite, which I call the surrogate.

Actually trying to use this idea to factor you need explicit equations:

(x+k)^2 = y^2 + 2k^2 + omega*T

And you can factor your target T by instead factoring 2k^2 + omega*T.

And it turns out that with a big T you still get big numbers to factor, but instead of one, you have as many as you wish.

IF that simple mathematics says to you what it says to me then you know that mathematicians have the world using a system where factoring one number can be transferred to factoring some other number, and if you do detailed analysis, you find that for an arbitrary non-zero omega and a non-zero k, as long as

2k^2 + omega*T

is over a certain size limit—that size depends on the size of T—then you can factor your target 50% of the time with some combination of factors of 2k^2 + omega*T.

If that sounds really complicated, just think of it as a two-way street where you can go either way factoring numbers and it's rather neat, and it's worth worrying about but mathematicians will NOT talk about it because it works, and they lie.

They know me. If they acknowledge this result I'll push forward all my other mathematical results as I'm using factoring to break them. Yup, I'm deliberately researching this area to PROVE they lie, so they can't talk about this research as then they just help me break them.

But why do they lie? I say because doing real mathematics is harder than faking it, and it's easier for groups of people to lie than most people accept, which is why they get away with it.

And that brings us to string "theory" with a lot of complexity but somehow, it doesn't give you much to experiment with, as mathematicians teach their tricks to the physics community.

Eventually you can have physics that doesn't work, is not correct, and no one knows or cares because physics people will, like many modern mathematicians, produce papers that sound physicky, with a lot of complexity, and there will be people who will cheer and claim stuff is correct, when it's just wrong.

Of course, you can think I'm wrong, but I just gave some basic math.

And, if I'm not, then the simple mathematics for breaking security systems is now out there as it has been for a few months as I talk about this, and I am currently working on coding an implementation to prove my point.

And mathematicians won't talk about it as I'm using this to get them, and you may end up collateral damage.

If that algebra above is correct then the world as you know it is about to change.

But consider, many of you probably know lots of complex mathematics and can check the algebra in your sleep, but a corrupted mathematical field means you will hold and wait, despite the implications because they are that good.

They are that good at lying.

And you are that intellectually weak, in actuality, no matter how brilliant you have deluded yourself into thinking you are.

If you can't trust some basic algebra, then how smart can you really be?

### Tuesday, March 27, 2007

## Integer solutions for a circle, conjecture from surrogate factoring

Working through the mathematics for surrogate factoring I found that for circles

a^2 + b^2 = r^2

with an integer radius greater than 4, there will be at least one integer solution for non-zero a and b, only about 1/3 of the time. As an example to understand the conjecture consider the first four cases:

3^2 + 4^2 = 5^2

6^2 + 8^2 = 10^2

5^2 + 12^2 = 13^2

and

8^2 + 15^2 = 17^2

the first four such solutions over the space from 5 to 17, which is 13 numbers, so you have 4/13 and an almost perfect 1/3 of the cases.

That follows from surrogate factoring as adding one more congruence to a congruence of squares gives the following:

x^2 = y^2 mod T

you can let k = 2x mod T so

2x = k mod T, multiply both sides by k, to get

2xk = k^2 mod T

and now add back to the original congruence of squares to get

x^2 + 2xk = y^2 + k^2 mod T

and then add k^2 to both sides and get

(x+k)^2 = y^2 + 2k^2 mod T

so you relate every factorization of one composite a factorization of some other composite, which I call the surrogate.

Actually trying to use this idea to factor you need explicit equations:

(x+k)^2 = y^2 + 2k^2 + omega*T

And you can factor your target T by instead factoring 2k^2 + omega*T which I call the surrogate.

Notice that for ANY difference of squares the surrogate factorization exists.

Now the obvious question since you have to pick k and omega is, how likely are you to factor a given target composite T with your arbitrary choice of k and omega?

You have three possibilities with the two main congruences:

x^2 = y^2 mod T and k^2 = 2xk mod T

And with no reason to pick either possibility over the other, you get that last possibility about 1/3 of the time, which gives the circle conjecture mentioned above, which I just say is a conjecture to be fair.

It seems to me that the proof is obvious enough at this point, but for now I'll call it a conjecture.

a^2 + b^2 = r^2

with an integer radius greater than 4, there will be at least one integer solution for non-zero a and b, only about 1/3 of the time. As an example to understand the conjecture consider the first four cases:

3^2 + 4^2 = 5^2

6^2 + 8^2 = 10^2

5^2 + 12^2 = 13^2

and

8^2 + 15^2 = 17^2

the first four such solutions over the space from 5 to 17, which is 13 numbers, so you have 4/13 and an almost perfect 1/3 of the cases.

That follows from surrogate factoring as adding one more congruence to a congruence of squares gives the following:

x^2 = y^2 mod T

you can let k = 2x mod T so

2x = k mod T, multiply both sides by k, to get

2xk = k^2 mod T

and now add back to the original congruence of squares to get

x^2 + 2xk = y^2 + k^2 mod T

and then add k^2 to both sides and get

(x+k)^2 = y^2 + 2k^2 mod T

so you relate every factorization of one composite a factorization of some other composite, which I call the surrogate.

Actually trying to use this idea to factor you need explicit equations:

(x+k)^2 = y^2 + 2k^2 + omega*T

And you can factor your target T by instead factoring 2k^2 + omega*T which I call the surrogate.

Notice that for ANY difference of squares the surrogate factorization exists.

Now the obvious question since you have to pick k and omega is, how likely are you to factor a given target composite T with your arbitrary choice of k and omega?

You have three possibilities with the two main congruences:

x^2 = y^2 mod T and k^2 = 2xk mod T

- Neither is satisfied.
- They are both true for one prime factor or only some prime factors of T.
- They are both satisfied.

And with no reason to pick either possibility over the other, you get that last possibility about 1/3 of the time, which gives the circle conjecture mentioned above, which I just say is a conjecture to be fair.

It seems to me that the proof is obvious enough at this point, but for now I'll call it a conjecture.

### Monday, March 26, 2007

## JSH: Properly understanding the situation

Part of the problem for people who don't know the full story here would be wondering why it'd be so devastating for me to be right. After all, why would some new factoring technique be such a crusher that people might commit suicide if the full truth came out as a big deal around the world?

Because I'm trying to use factoring to bring attention to the full story, which says that for quite a few mathematicians around the world, likely NOTHING they have ever produced is valid. That's why.

Which goes back to that published paper of mine and the mathematical journal later dying—after it yanked the paper under pressure.

NOTHING valid. That's why these people fight so hard and why it could be so devastating.

I can find a simple new way to factor that somehow escaped these people because I can do these kinds of things, and have done them before, so it's not a surprise. But they lie about the math to protect themselves. So I went to something where I could force the situation and in so doing prove they lie, to everyone.

I can force the situation and take away the lies, but in so doing, take away every supposed accomplishment they have in mathematics over their entire careers.

I can reduce leading mathematicians to people who thought they knew math but actually had NO mathematical accomplishments.

Because they don't. It's not like the mathematics changes. AT this time many of these people have no actual mathematical accomplishments—but they have the belief of others that they do.

And I can take that away.

Because I'm trying to use factoring to bring attention to the full story, which says that for quite a few mathematicians around the world, likely NOTHING they have ever produced is valid. That's why.

Which goes back to that published paper of mine and the mathematical journal later dying—after it yanked the paper under pressure.

NOTHING valid. That's why these people fight so hard and why it could be so devastating.

I can find a simple new way to factor that somehow escaped these people because I can do these kinds of things, and have done them before, so it's not a surprise. But they lie about the math to protect themselves. So I went to something where I could force the situation and in so doing prove they lie, to everyone.

I can force the situation and take away the lies, but in so doing, take away every supposed accomplishment they have in mathematics over their entire careers.

I can reduce leading mathematicians to people who thought they knew math but actually had NO mathematical accomplishments.

Because they don't. It's not like the mathematics changes. AT this time many of these people have no actual mathematical accomplishments—but they have the belief of others that they do.

And I can take that away.

## JSH: Psyching myself up

That previous post was more about convincing myself than anyone else as I now have charge of the situation. When I began coding an implementation of surrogate factoring I moved beyond needing anyone else to do much as I took control of the situation, and saw that it worked.

I just need to get it working a little better.

The math IS simple.

A key equation in the coding is 2f_1+/- k where you check the gcd with the target T, and f_1 is a factor of 2k^2 + omega*T. Such simple mathematics to give me the final keys to ending the Math Wars.

Two things bother me as I get ready to try out another idea and maybe put forward a surrogate factoring program finally at my coding site that I want to put out there as proof of everything that is happening—proof to the world of how it has been lied to—is that I see what many of you keep denying may happen:

I have done my best to protect the world, and not just some people who keep making very wrong decisions in an attempt to control mathematics for their benefit, and to the harm of everyone else.

I am not responsible for the world. I have some good math ideas. That's all.

I just need to get it working a little better.

The math IS simple.

A key equation in the coding is 2f_1+/- k where you check the gcd with the target T, and f_1 is a factor of 2k^2 + omega*T. Such simple mathematics to give me the final keys to ending the Math Wars.

Two things bother me as I get ready to try out another idea and maybe put forward a surrogate factoring program finally at my coding site that I want to put out there as proof of everything that is happening—proof to the world of how it has been lied to—is that I see what many of you keep denying may happen:

- Suicides as mathematicians who are trapped against a wall finally lose the "Math Wars" and decide they cannot live with the world knowing the truth.
- An impact on world stock markets and maybe worse, a rapid growth in hostile use of surrogate factoring around the world, and exploits against systems.

I have done my best to protect the world, and not just some people who keep making very wrong decisions in an attempt to control mathematics for their benefit, and to the harm of everyone else.

I am not responsible for the world. I have some good math ideas. That's all.

## Surrogate factoring, top to bottom

The best way to understand surrogate factoring may be to to start with

x^2 = y^2 mod T

where T is the target composite to be factored, so you just have the familiar congruence of squares, also called the difference of squares, where you MAY factor T non-trivially by checking for prime factors in common between it and x-y or x+y.

Now for some solution set {x,y} let

k = 2x mod T

so

2x = k mod T, and you can multiply both sides by k, to get

2xk = k^2 mod T

and now you can add back to the original congruence of squares to get

x^2 + 2xk = y^2 + k^2 mod T

and, of course, you can complete the square and simplify to get

(x+k)^2 = y^2 + 2k^2 mod T

and that gives you most of surrogate factoring as you now know that for ANY solutions with a difference of squares you have a solution to that congruence.

So, trivially, every factorization of any given target composite T is connected to a factorization of some other composite, as of course, looking explicitly—introducing the integer omega—I have

(x+k)^2 = y^2 + 2k^2 + omega*T

and y is determined by the factorization of 2k^2 + omega*T, with again, trivial algebra.

Note then that y is determined by two factorizations as also with the original difference of squares, again going explicit, this time using alpha, I have

x^2 = y^2 + alpha*T

and here y is defined by a factorization of T, and SIMULTANEOUSLY y is defined by the factorization of omega*T, which means that at a very fundamental level, factorizations of numbers are bound by some very basic congruence relationships.

So it becomes a simple step to ask, what if I start with k, can I factor T from the surrogate factorization of 2k^2 + omega*T?

And the answer is, yes, you can.

In terms of actually trying to factor, you have to pick k, which forces the residue modulo T of x, and you also have to pick omega, and that is it.

You can only pick k and omega.

And you know that for any x you can pick there is a k and omega that have to work, but maybe the two don't match well somehow? Maybe your pick of k and omega on your end is unlikely to find a welcoming x and y on the other?

Well, good questions, and shouldn't there be answers? Shouldn't there be some mathematics that would tell you if that were true?

I've checked. It's trivial algebra, and the mathematics does not say that at all.

It says that for a given k and omega, outside of a narrow size range where k is too small and the surrogate is too small to give factors, you have a 50% chance of factoring T, with SOME combination of factors of the surrogate 2k^2 + omega*T.

(If I am wrong, it'd be nice for someone to post a nice analysis of the very trivial algebra that DOES show that it's hard to pick k and omega.)

So then if this idea might be worth something, why wouldn't it be a big deal by now?

Well you have to factor SOMETHING. If you can't factor that well, then you can't use surrogate factoring as you still need to factor 2k^2 + omega*T, and that's why this method could be a stealth one that takes the world by surprise because to check it against really big numbers, you need to be able to factor some really big numbers.

Easy math then that says that you can factor one number through another, and on the "pure math" side, easy proof of a connection between composites where otherwise it might not be clear that a connection existed.

For the "pure math" fun of it, what if you play around with the equations? Can you find more congruence relations?

Well, I've done that and you can do some other stuff, but it doesn't seem to change much, like you can have

ak = 2x mod T

and get to

(x+k)^2 = y^2 + (a+1)k^2 mod T

which is a variable I like enough to add to my own main research page on the subject, but that doesn't change much…hmmm…what about something like

k^2 = 3xk mod T?

Well, that can be shown to be equivalent to what I already have (do the math for yourself if you wish).

Ok then, what about something like

k^2 = 2xk + 3 mod T

or something else of that nature?

Well, that isn't always true, as it's not necessarily the case that you have a solution for k, so I haven't played with anything like that, and if you just go all out to add another variable, something weird happens, like with

k^2 = 2xk + z mod T

you completely de-couple T from the equations anyway and it behaves like random.

It's like some kind of quantum mechanics thing, where if you add another variable, you blow up surrogate factoring as the mathematics kind of throws up its hands and asks, how do I know what T is now?

You know, it's like you collapse the wave function or something, and it doesn't work any more.

That is freaky if you see it, so I suggest you play with these equations.

The other way to de-couple T, is, of course, to have k = 0, or have omega = 0, as then you just get random.

So yes, the mathematics here may relate back to how our very reality works, but the mathematics here is VERY troubling for mathematicians out there who have made it their business to ignore my research.

You think they give a damn if somehow composites and factorizations could be related to physics?

Nope, not if it helps me they don't. These are what I call the Math Wars for a reason.

And like with any war you do not help the enemy, no matter what.

The human species can rot and die before any of those people will lift a finger to support any of my research as they rightly see it as an issue of survival. Their survival against the Progress of the species, and given that choice they keep picking themselves.

After all, I say they are fakes and cons. If I win, they are probably out of jobs.

If you were a Ph.D in mathematics with decades of living rather well, would you really accept having to go find a new job, maybe working at McDonald's or whatever you could get just to tell the truth about some basic mathematics?

Just for the fate of the human species? Like anybody even gives a damn any more.

It's a cold, cruel world, and we're all doomed anyway, right?

Make no mistake, each day while I chat about this to the world through this medium while the "mathematicians" ignore it, while you may notice, still promoting themselves and their research, is another day they choose themselves against Progress, and the future.

They hate everything, but protecting narrow interests, and deep down, they hate mathematics.

x^2 = y^2 mod T

where T is the target composite to be factored, so you just have the familiar congruence of squares, also called the difference of squares, where you MAY factor T non-trivially by checking for prime factors in common between it and x-y or x+y.

Now for some solution set {x,y} let

k = 2x mod T

so

2x = k mod T, and you can multiply both sides by k, to get

2xk = k^2 mod T

and now you can add back to the original congruence of squares to get

x^2 + 2xk = y^2 + k^2 mod T

and, of course, you can complete the square and simplify to get

(x+k)^2 = y^2 + 2k^2 mod T

and that gives you most of surrogate factoring as you now know that for ANY solutions with a difference of squares you have a solution to that congruence.

So, trivially, every factorization of any given target composite T is connected to a factorization of some other composite, as of course, looking explicitly—introducing the integer omega—I have

(x+k)^2 = y^2 + 2k^2 + omega*T

and y is determined by the factorization of 2k^2 + omega*T, with again, trivial algebra.

Note then that y is determined by two factorizations as also with the original difference of squares, again going explicit, this time using alpha, I have

x^2 = y^2 + alpha*T

and here y is defined by a factorization of T, and SIMULTANEOUSLY y is defined by the factorization of omega*T, which means that at a very fundamental level, factorizations of numbers are bound by some very basic congruence relationships.

So it becomes a simple step to ask, what if I start with k, can I factor T from the surrogate factorization of 2k^2 + omega*T?

And the answer is, yes, you can.

In terms of actually trying to factor, you have to pick k, which forces the residue modulo T of x, and you also have to pick omega, and that is it.

You can only pick k and omega.

And you know that for any x you can pick there is a k and omega that have to work, but maybe the two don't match well somehow? Maybe your pick of k and omega on your end is unlikely to find a welcoming x and y on the other?

Well, good questions, and shouldn't there be answers? Shouldn't there be some mathematics that would tell you if that were true?

I've checked. It's trivial algebra, and the mathematics does not say that at all.

It says that for a given k and omega, outside of a narrow size range where k is too small and the surrogate is too small to give factors, you have a 50% chance of factoring T, with SOME combination of factors of the surrogate 2k^2 + omega*T.

(If I am wrong, it'd be nice for someone to post a nice analysis of the very trivial algebra that DOES show that it's hard to pick k and omega.)

So then if this idea might be worth something, why wouldn't it be a big deal by now?

Well you have to factor SOMETHING. If you can't factor that well, then you can't use surrogate factoring as you still need to factor 2k^2 + omega*T, and that's why this method could be a stealth one that takes the world by surprise because to check it against really big numbers, you need to be able to factor some really big numbers.

Easy math then that says that you can factor one number through another, and on the "pure math" side, easy proof of a connection between composites where otherwise it might not be clear that a connection existed.

For the "pure math" fun of it, what if you play around with the equations? Can you find more congruence relations?

Well, I've done that and you can do some other stuff, but it doesn't seem to change much, like you can have

ak = 2x mod T

and get to

(x+k)^2 = y^2 + (a+1)k^2 mod T

which is a variable I like enough to add to my own main research page on the subject, but that doesn't change much…hmmm…what about something like

k^2 = 3xk mod T?

Well, that can be shown to be equivalent to what I already have (do the math for yourself if you wish).

Ok then, what about something like

k^2 = 2xk + 3 mod T

or something else of that nature?

Well, that isn't always true, as it's not necessarily the case that you have a solution for k, so I haven't played with anything like that, and if you just go all out to add another variable, something weird happens, like with

k^2 = 2xk + z mod T

you completely de-couple T from the equations anyway and it behaves like random.

It's like some kind of quantum mechanics thing, where if you add another variable, you blow up surrogate factoring as the mathematics kind of throws up its hands and asks, how do I know what T is now?

You know, it's like you collapse the wave function or something, and it doesn't work any more.

That is freaky if you see it, so I suggest you play with these equations.

The other way to de-couple T, is, of course, to have k = 0, or have omega = 0, as then you just get random.

So yes, the mathematics here may relate back to how our very reality works, but the mathematics here is VERY troubling for mathematicians out there who have made it their business to ignore my research.

You think they give a damn if somehow composites and factorizations could be related to physics?

Nope, not if it helps me they don't. These are what I call the Math Wars for a reason.

And like with any war you do not help the enemy, no matter what.

The human species can rot and die before any of those people will lift a finger to support any of my research as they rightly see it as an issue of survival. Their survival against the Progress of the species, and given that choice they keep picking themselves.

After all, I say they are fakes and cons. If I win, they are probably out of jobs.

If you were a Ph.D in mathematics with decades of living rather well, would you really accept having to go find a new job, maybe working at McDonald's or whatever you could get just to tell the truth about some basic mathematics?

Just for the fate of the human species? Like anybody even gives a damn any more.

It's a cold, cruel world, and we're all doomed anyway, right?

Make no mistake, each day while I chat about this to the world through this medium while the "mathematicians" ignore it, while you may notice, still promoting themselves and their research, is another day they choose themselves against Progress, and the future.

They hate everything, but protecting narrow interests, and deep down, they hate mathematics.

### Friday, March 23, 2007

## JSH: World is complicated

I think the problem for some of you is that you think you are very smart.

I AM very smart.

I am smarter on a scale you cannot really comprehend and there is the problem.

So I can warn you of what may happen but you are not smart enough to understand that what I say CAN really happen, so like so many before you, you will keep doing what you're doing up until the point that you are literally torn apart by an angry mob.

And there you can see my problem, it doesn't help me much if I can't convince you before you get torn apart by an angry mob, and it isn't much help for me if I can only convince you at the point that you are about to be torn apart by an angry mob.

How about this?

Global warming is likely to kill 90% of the human population within the next 50 years, but most people are too damn dumb to appreciate that fact which is fine with me. I say, let evolution take its course.

But, then again, maybe one could argue that if you have information you should share and at least give those people a chance.

I can assure you of some crucial things: there will not be over a billion people left alive on this planet in 50 years. That is a FACT.

In the meantime, most of you will not have a snowball's chance in hell of surviving because you are geek idiots who don't have a clue about how to survive in a hostile world that is not giving you just about everything you need to survive.

Many of you will die miserably, crying for your mommies, and suddenly, finally, believing in God, just before you die, and it will not matter.

On the other hand I am a true, living super genius who can possibly save some of your dumb asses if you give me a reason, or I can just sit here, wait, until you die, and then get to business.

And you know what?

I like these messages because they salve my conscience.

You people have no clue about how reality works. Nothing has changed.

Thousands of years pass and you people do not evolve. Some of you have devolved.

I can smell your weak genes.

The real world is far more cruel than you can ever imagine, and the pause is just a measure of that cruelty.

Most of you will die in pain, misery, and in despair, thinking about what might have happened if you had considered for an instant that maybe you should side with the most intelligent person versus those idiots you think have the power because for the moment they have convinced a few other idiots.

And I will watch you die.

But that's why the world stays interesting.

Most of you are too damn stupid to do anything else but lose.

I AM very smart.

I am smarter on a scale you cannot really comprehend and there is the problem.

So I can warn you of what may happen but you are not smart enough to understand that what I say CAN really happen, so like so many before you, you will keep doing what you're doing up until the point that you are literally torn apart by an angry mob.

And there you can see my problem, it doesn't help me much if I can't convince you before you get torn apart by an angry mob, and it isn't much help for me if I can only convince you at the point that you are about to be torn apart by an angry mob.

How about this?

Global warming is likely to kill 90% of the human population within the next 50 years, but most people are too damn dumb to appreciate that fact which is fine with me. I say, let evolution take its course.

But, then again, maybe one could argue that if you have information you should share and at least give those people a chance.

I can assure you of some crucial things: there will not be over a billion people left alive on this planet in 50 years. That is a FACT.

In the meantime, most of you will not have a snowball's chance in hell of surviving because you are geek idiots who don't have a clue about how to survive in a hostile world that is not giving you just about everything you need to survive.

Many of you will die miserably, crying for your mommies, and suddenly, finally, believing in God, just before you die, and it will not matter.

On the other hand I am a true, living super genius who can possibly save some of your dumb asses if you give me a reason, or I can just sit here, wait, until you die, and then get to business.

And you know what?

I like these messages because they salve my conscience.

You people have no clue about how reality works. Nothing has changed.

Thousands of years pass and you people do not evolve. Some of you have devolved.

I can smell your weak genes.

The real world is far more cruel than you can ever imagine, and the pause is just a measure of that cruelty.

Most of you will die in pain, misery, and in despair, thinking about what might have happened if you had considered for an instant that maybe you should side with the most intelligent person versus those idiots you think have the power because for the moment they have convinced a few other idiots.

And I will watch you die.

But that's why the world stays interesting.

Most of you are too damn stupid to do anything else but lose.

### Tuesday, March 20, 2007

## What surrogate factoring theory now says

I want to emphasize that there can be a new factoring method that just takes a while to be fully engineered, as there is the theory and there is the engineering into a practical solution.

Like Isaac Newton knew difference of squares, but he didn't have the Number Field Sieve.

Surrogate factoring theory says that you can turn factoring a hard target T, into a problem of factoring an indefinite number of surrogates S_1, S_2, S_3, … making the problem potentially tractable.

My own target for my research is factoring an RSA sized number—of any bit length feasible—within ten minutes on a home computer.

That has been my research target for years now. The theory says that once the engineering is figured out that is achievable.

You can personally check the very simple underlying mathematics yourself.

(Web search on surrogate factoring, stay away from the old failed stuff though.)

Mainly I just added one more congruence to the difference of squares.

So it's not like the algebra is hard, or it's difficult to follow.

But just like with just the difference of squares, figuring out a practical solution could take a while, where I don't think it'll take centuries like with difference of squares to the NFS.

I am in the process of trying to turn what could take years of research from lots of people around the world into months or days of research where I am the primary engine, but I could fail, and others could succeed.

So, say, Russia could succeed. Or China could succeed. Or Iran could succeed. Or, maybe even North Korea could succeed.

Would my own country the United States?

Sure, but history says that people here might not bother because we're on the top of the heap.

People at the top tend to ignore "crackpot" ideas.

But I could be wrong, right? Lots of math people say I'm a crackpot and I've been babbling about surrogate factoring for YEARS, including in the past having said that I'd solved the factoring problem, when I hadn't.

Yup. I've failed a lot. I admit it. But I've succeeded a lot, and "mathematicians" won't admit it.

Roll the dice and the fate of the world could change.

That's how it's happened before…people like you under-rate the power of ideas despite thinking you're idea people, and you ignore something you DECIDE is dinky and worthless, and civilization itself changes.

If that didn't happen, no dominant country would ever lose that position. We might be under the Persian Empire, or the Roman or the Egyptian or some other if people just learned not to underestimate the power of ideas.

Then again, I could be wrong. I don't think I am, but I have been wrong before.

But hey, it's mathematics!!! I say, don't trust me. I don't trust you, or I wouldn't be making this post. I think most of you are complacent idiots who would let the world go up in flames because you're too small-minded and maybe corrupt to really care, and I don't trust you.

Go with the math.

If I'm right, it says I'm right. If I'm wrong, it says I'm wrong.

If it says I'm right, and you think you can just play the odds that no one in the world will figure this out, not Russia, not China, not anybody, and you're wrong…well, welcome then to a Brave New World, and yet another example that history repeats…

[A reply to someone who wrote that “there can be a new factoring method that never becomes practical no matter how much engineering you put into it”.]

The equations are fundamental being congruence of squares plus one more congruence:

x^2 = y^2 mod T

and

α*k^2 = 2xk mod T

with T the target composite, where the second congruence is necessary to introduce surrogate factoring, and now it can be seen that prior factoring techniques can be said to be using k=0.

So the two main factoring congruences encompass all prior known factoring techniques that are based on congruence of squares so the Number Field Sieve among others is just part of the smaller mathematical knowledge base.

And now in the 21st century we know more.

One way to look at it is that humanity was not quite as advanced with its mathematics in this area as it could have been, where it is easy to prove that the second congruence while including EVERYTHING known with just the previous also gives you a few extras.

It took hundreds of years to develop congruence of squares, while I'm working at developing the full theory, as I now know that there are no more useful factoring congruences, in just a few months.

There are only two main factoring congruences. For centuries only one was known, and now the second is known as well.

Hundreds of years were needed with just the one, but in the 21st century, it will be a lot faster with the two, I hope.

As then the Math Wars will be over, and the world will finally understand how close it came to losing Progress in its most important discipline, and maybe the future of the human race as well.

Like Isaac Newton knew difference of squares, but he didn't have the Number Field Sieve.

Surrogate factoring theory says that you can turn factoring a hard target T, into a problem of factoring an indefinite number of surrogates S_1, S_2, S_3, … making the problem potentially tractable.

My own target for my research is factoring an RSA sized number—of any bit length feasible—within ten minutes on a home computer.

That has been my research target for years now. The theory says that once the engineering is figured out that is achievable.

You can personally check the very simple underlying mathematics yourself.

(Web search on surrogate factoring, stay away from the old failed stuff though.)

Mainly I just added one more congruence to the difference of squares.

So it's not like the algebra is hard, or it's difficult to follow.

But just like with just the difference of squares, figuring out a practical solution could take a while, where I don't think it'll take centuries like with difference of squares to the NFS.

I am in the process of trying to turn what could take years of research from lots of people around the world into months or days of research where I am the primary engine, but I could fail, and others could succeed.

So, say, Russia could succeed. Or China could succeed. Or Iran could succeed. Or, maybe even North Korea could succeed.

Would my own country the United States?

Sure, but history says that people here might not bother because we're on the top of the heap.

People at the top tend to ignore "crackpot" ideas.

But I could be wrong, right? Lots of math people say I'm a crackpot and I've been babbling about surrogate factoring for YEARS, including in the past having said that I'd solved the factoring problem, when I hadn't.

Yup. I've failed a lot. I admit it. But I've succeeded a lot, and "mathematicians" won't admit it.

Roll the dice and the fate of the world could change.

That's how it's happened before…people like you under-rate the power of ideas despite thinking you're idea people, and you ignore something you DECIDE is dinky and worthless, and civilization itself changes.

If that didn't happen, no dominant country would ever lose that position. We might be under the Persian Empire, or the Roman or the Egyptian or some other if people just learned not to underestimate the power of ideas.

Then again, I could be wrong. I don't think I am, but I have been wrong before.

But hey, it's mathematics!!! I say, don't trust me. I don't trust you, or I wouldn't be making this post. I think most of you are complacent idiots who would let the world go up in flames because you're too small-minded and maybe corrupt to really care, and I don't trust you.

Go with the math.

If I'm right, it says I'm right. If I'm wrong, it says I'm wrong.

If it says I'm right, and you think you can just play the odds that no one in the world will figure this out, not Russia, not China, not anybody, and you're wrong…well, welcome then to a Brave New World, and yet another example that history repeats…

[A reply to someone who wrote that “there can be a new factoring method that never becomes practical no matter how much engineering you put into it”.]

The equations are fundamental being congruence of squares plus one more congruence:

x^2 = y^2 mod T

and

α*k^2 = 2xk mod T

with T the target composite, where the second congruence is necessary to introduce surrogate factoring, and now it can be seen that prior factoring techniques can be said to be using k=0.

So the two main factoring congruences encompass all prior known factoring techniques that are based on congruence of squares so the Number Field Sieve among others is just part of the smaller mathematical knowledge base.

And now in the 21st century we know more.

One way to look at it is that humanity was not quite as advanced with its mathematics in this area as it could have been, where it is easy to prove that the second congruence while including EVERYTHING known with just the previous also gives you a few extras.

It took hundreds of years to develop congruence of squares, while I'm working at developing the full theory, as I now know that there are no more useful factoring congruences, in just a few months.

There are only two main factoring congruences. For centuries only one was known, and now the second is known as well.

Hundreds of years were needed with just the one, but in the 21st century, it will be a lot faster with the two, I hope.

As then the Math Wars will be over, and the world will finally understand how close it came to losing Progress in its most important discipline, and maybe the future of the human race as well.

### Monday, March 19, 2007

## Surrogate factoring works very well

Nothing like coding a potential factoring solution to see whether or not it works or not.

The latest code I have is in Java and is available to the world at

http://code.google.com/p/surrogate-factoring/

and it is optimized for small numbers less than 10,000,000 as that's what I experiment with.

But that is just about the k/T ratio. If you figure out how to adjust it you can use surrogate factoring for any size number.

What the code can show you though even with small numbers is a method that is not at all random. And the mathematics behind it shows that it doesn't really care about number size.

But those are just the facts.

We live in a world of lies. And powerful people who call themselves mathematicians created the necessity of me finding surrogate factoring and putting it forward because they decided that human progress was not important to them.

They decided that humanity had discovered enough mathematics.

No need for any more they decided. No need for any other major discoveries in mathematics because it was inconvenient for them.

Yes, you can continue to ignore this information. But my progress continues on coding, and the theory is done anyway:

http://groups.google.com/group/extrememathematics/web/surrogate-factoring

As it is so simple it makes you want to cry, as someone should have figured it out before, but the people playing at being mathematicians are like actors playing at being doctors.

But they are like actors who took over a hospital when no one was looking and then tried to lock out the real doctors, and make no mistake, these people now are fighting for their lives.

Their fake lives.

So no, Andrew Wiles did not prove Fermat's Last Theorem. Taylor and Ribet are probably worse at mathematics than your average 12 year old if you take them out of what they know by rote memory.

And none of them are worth a damn to humanity.

And yes, the facts can shake stock markets no matter what people make up to explain it as somewhere some people seem to understand who I actually am, and know what the situation actually is, but stupidly seem to believe that the "mathematicians" currently in power actually are intelligent people who can control things versus being dumb actors caught up in a powerful drama way beyond their intellects.

THEY ARE ACTORS.

I have tried various ideas without success before but hey that is normal problem solving.

Failures are over. Surrogate factoring is here. Anyone implementing the simple theory with a will can probably do some serious factoring.

And that means that nations implementing it can possibly do some serious factoring.

And the United States is the LEAST likely to know that as it is top dog right now, so has no interest in supposedly lame ideas from a supposed crackpot.

These kind of stories are how the fate of the world changes and how dominant nations can be thrown on the crap heap.

Do any of you know your history?

Did you know that in World War II Adolph Hitler was so certain of German superiority that he refused to believe that the German codes had been broken, even though they had been?

Rommel had a great effort in Africa, but the Allies had information, and the will to keep a secret.

A very big secret…

They killed people to keep that secret. They let their own people get killed to keep that secret.

And the world keeps on turning, life goes on, people keep getting born, and dying, and preventable catastrophes keep occurring simply because of willful human stupidity.

People in the end want pain, misery, death and failure because there is a Death Instinct in the species.

You people want the worst case. You want the world to hurt.

The newest revision to my surrogate factoring program is now in FLASH, and is available at

http://code.google.com/p/surrogate-factoring/

it is optimized for numbers larger than 100,000,000, which is an improvement from the java program. The world will see that surrogate factoring is the wave of the future, and that you have been fed lies!

[A reply to someone who wrote that the antique Unix

Problem may be that k needs to be around sqrt(T), which kind of makes sense for multiple reasons.

That is, with k near sqrt(T), so the k/T ratio is approximately 1/sqrt(T), you minimize BOTH k and the size of the factors of the surrogate needed to find the prime factors of T.

I think that might be the key, and of course I'll need to check it to be sure.

At this point I have things settled to focusing on k, so there is not a lot of room to fiddle with things.

The latest code I have is in Java and is available to the world at

http://code.google.com/p/surrogate-factoring/

and it is optimized for small numbers less than 10,000,000 as that's what I experiment with.

But that is just about the k/T ratio. If you figure out how to adjust it you can use surrogate factoring for any size number.

What the code can show you though even with small numbers is a method that is not at all random. And the mathematics behind it shows that it doesn't really care about number size.

But those are just the facts.

We live in a world of lies. And powerful people who call themselves mathematicians created the necessity of me finding surrogate factoring and putting it forward because they decided that human progress was not important to them.

They decided that humanity had discovered enough mathematics.

No need for any more they decided. No need for any other major discoveries in mathematics because it was inconvenient for them.

Yes, you can continue to ignore this information. But my progress continues on coding, and the theory is done anyway:

http://groups.google.com/group/extrememathematics/web/surrogate-factoring

As it is so simple it makes you want to cry, as someone should have figured it out before, but the people playing at being mathematicians are like actors playing at being doctors.

But they are like actors who took over a hospital when no one was looking and then tried to lock out the real doctors, and make no mistake, these people now are fighting for their lives.

Their fake lives.

So no, Andrew Wiles did not prove Fermat's Last Theorem. Taylor and Ribet are probably worse at mathematics than your average 12 year old if you take them out of what they know by rote memory.

And none of them are worth a damn to humanity.

And yes, the facts can shake stock markets no matter what people make up to explain it as somewhere some people seem to understand who I actually am, and know what the situation actually is, but stupidly seem to believe that the "mathematicians" currently in power actually are intelligent people who can control things versus being dumb actors caught up in a powerful drama way beyond their intellects.

THEY ARE ACTORS.

I have tried various ideas without success before but hey that is normal problem solving.

Failures are over. Surrogate factoring is here. Anyone implementing the simple theory with a will can probably do some serious factoring.

And that means that nations implementing it can possibly do some serious factoring.

And the United States is the LEAST likely to know that as it is top dog right now, so has no interest in supposedly lame ideas from a supposed crackpot.

These kind of stories are how the fate of the world changes and how dominant nations can be thrown on the crap heap.

Do any of you know your history?

Did you know that in World War II Adolph Hitler was so certain of German superiority that he refused to believe that the German codes had been broken, even though they had been?

Rommel had a great effort in Africa, but the Allies had information, and the will to keep a secret.

A very big secret…

They killed people to keep that secret. They let their own people get killed to keep that secret.

And the world keeps on turning, life goes on, people keep getting born, and dying, and preventable catastrophes keep occurring simply because of willful human stupidity.

People in the end want pain, misery, death and failure because there is a Death Instinct in the species.

You people want the worst case. You want the world to hurt.

The newest revision to my surrogate factoring program is now in FLASH, and is available at

http://code.google.com/p/surrogate-factoring/

it is optimized for numbers larger than 100,000,000, which is an improvement from the java program. The world will see that surrogate factoring is the wave of the future, and that you have been fed lies!

[A reply to someone who wrote that the antique Unix

`factor`utility correctly factored the number 53762053 in a small fraction of a second, as a reply to someone else who noticed that James' code was unable to factor that number (which is equal to 6599×8147).]Problem may be that k needs to be around sqrt(T), which kind of makes sense for multiple reasons.

That is, with k near sqrt(T), so the k/T ratio is approximately 1/sqrt(T), you minimize BOTH k and the size of the factors of the surrogate needed to find the prime factors of T.

I think that might be the key, and of course I'll need to check it to be sure.

At this point I have things settled to focusing on k, so there is not a lot of room to fiddle with things.

### Wednesday, March 14, 2007

## JSH: Sobering invention, surrogate factoring

Years ago I thought to myself as I argued with mathematicians who were willing to ignore my findings that it might take figuring out something practical that they couldn't get away with ignoring.

Now, it's odd to have an invention of my own factoring method, where for years I just had a concept, and question: Could one number be factored by instead factoring another?

The answer is, yes.

There is not a doubt at this point of impact from this new invention. I see evidence of the same tactics from the mathematical community of ignore, ignore, ignore, but they don't matter here because it's just a matter of the theory moving further into practicality.

I remember at times in the past debating with myself about why I would do such a thing, and find such a thing, and I'd give myself the example of liking to do a lot of transactions myself on-line, and I scared myself into believing that I personally could take that all away, but I got a little more faith in humanity, less fears about my ability to change the world, and now figure that if necessary new ways can be found.

The way I see it, people love the Internet, they love the conveniences it offers, and new systems can be used, if this idea is really as potent as of course I'd like to to be potent if only for my own research, to break through against mathematicians I see as con artists, pretending to be someone like me.

But the recent coincidental gyrations of the world's stock markets can maybe give some of you some sense of the fears that use to stop me cold in my research in this area. I used to post about those fears that went all the way to worrying about ending modern civilization as we know it, by collapsing the global economy. But the gyrations are a coincidence as I can't believe that people could be acting on the

knowledge of this situation and not just step in, after all, the mathematicians are lying about my research, so it'd be a matter of just stopping them. They are just con artist, not that intelligent, not brilliant, not worth losing everything over.

And now I say, it's like the researchers who invented the atomic bomb, who worried for a while that the damn thing might ignite the atmosphere and kill all life on the planet—overblown worries, no one is that big, the world will be just fine, no matter what I do.

The atomic bomb went off, since then lots of nuclear weapons have gone off, and our atmosphere is still here. We are still here.

I settled down and was able to continue my research, and code an implementation of it because I realized that I am just not that big. There is just no way that I can, by myself, take down civilization, so I needed to deflate the ego, just do the research and get it over with.

Now, it's odd to have an invention of my own factoring method, where for years I just had a concept, and question: Could one number be factored by instead factoring another?

The answer is, yes.

There is not a doubt at this point of impact from this new invention. I see evidence of the same tactics from the mathematical community of ignore, ignore, ignore, but they don't matter here because it's just a matter of the theory moving further into practicality.

I remember at times in the past debating with myself about why I would do such a thing, and find such a thing, and I'd give myself the example of liking to do a lot of transactions myself on-line, and I scared myself into believing that I personally could take that all away, but I got a little more faith in humanity, less fears about my ability to change the world, and now figure that if necessary new ways can be found.

The way I see it, people love the Internet, they love the conveniences it offers, and new systems can be used, if this idea is really as potent as of course I'd like to to be potent if only for my own research, to break through against mathematicians I see as con artists, pretending to be someone like me.

But the recent coincidental gyrations of the world's stock markets can maybe give some of you some sense of the fears that use to stop me cold in my research in this area. I used to post about those fears that went all the way to worrying about ending modern civilization as we know it, by collapsing the global economy. But the gyrations are a coincidence as I can't believe that people could be acting on the

knowledge of this situation and not just step in, after all, the mathematicians are lying about my research, so it'd be a matter of just stopping them. They are just con artist, not that intelligent, not brilliant, not worth losing everything over.

And now I say, it's like the researchers who invented the atomic bomb, who worried for a while that the damn thing might ignite the atmosphere and kill all life on the planet—overblown worries, no one is that big, the world will be just fine, no matter what I do.

The atomic bomb went off, since then lots of nuclear weapons have gone off, and our atmosphere is still here. We are still here.

I settled down and was able to continue my research, and code an implementation of it because I realized that I am just not that big. There is just no way that I can, by myself, take down civilization, so I needed to deflate the ego, just do the research and get it over with.

### Monday, March 12, 2007

## Surrogate factoring, potential

I ended up with a longer post than I would have liked as I went over the functional equation behind surrogate factoring—like, if someone wanted to implement an algorithm without worrying about theory they could just go by that post.

But some may still wonder why this idea might have potential and, how would they know?

Well, I invented a factoring method that relies on VERY fundamental equations:

x^2 = y^2 mod T

and

k^2 = 2xk mod T

and you can solve to get

(x+k)^2 = y^2 + 2k^2(mod T)

which is then the fundamental factoring congruence.

And you have as a surrogate to factor 2k^2 - wT, where I subtract just because I like to do so, as w is a non-zero integer.

So factoring a target T is turned into factoring a series of surrogates, S_1, S_2, S_3, ... and checking with each one to see if you've factored which may seem like a lot, but it doesn't have to be, and it is a direct assault against methods like RSA encryption because it allows you to go after smaller numbers that might be easily

factored to get your target, and you get as many tries as you want where if you fully factor your surrogates you have a 50% chance of getting your target as well, versus just banging your head against one big number.

Example:

T = 732367903, k=24412263

Surrogate: 1191915704826532 = ( 4 )( 7 )( 73 )( 583129014103 )

f_1 = 7/2 and f_2 = 85136836059038

and 4f_1*f_2 = 2k^2 - 2T

y=-170273672118069/2 and x=170273623293557/2

so, x+y=-24412256, which has 223 as a factor.

T = 732367903 = (223)(3284161)

Iterations: 1

Notice that with the surrogate factored there was not even a lot of iterating through combinations of those factors, as it took one try.

That example is the potential, as it could be a much larger target, where the primary issue is factoring the surrogate.

With the surrogate factored, looping through all the iterations of factors the theory says you have a 50% chance of factoring, so, if I got the mathematical theory right, anybody with some serious factoring power already could convert maybe even an RSA sized number into several surrogates, go at it, and just walk away with a factorization, just like that.

It is a fascinating invention to me, because it is amazing that such simple mathematics could have been missed, while it is so simple it's hard to see where the math can sneak in something that ruins it, like actually the only thing is if w's that work get scarcer as T increases in size, where there is no mathematical indication that they do, as you are multiplying T, and there are an infinity of w's that must work. So how do you concentrate that infinity?

How do you force mathematically a smaller infinite subset of working w's against a bigger infinite majority that don't?

Do the math. There is no way. So you get the same factoring potential as is already known for the difference of squares—50%—as that is just a variation with k=0 anyway.

And if I'm right, people with the know-how all over the world can immediately start factoring much larger numbers than they did before. Immediately, all over the world, with some very simple mathematics that is inexplicably still just out there, with barely a murmur from the mathematical community.

Maybe they just want to wait and see what happens…

But some may still wonder why this idea might have potential and, how would they know?

Well, I invented a factoring method that relies on VERY fundamental equations:

x^2 = y^2 mod T

and

k^2 = 2xk mod T

and you can solve to get

(x+k)^2 = y^2 + 2k^2(mod T)

which is then the fundamental factoring congruence.

And you have as a surrogate to factor 2k^2 - wT, where I subtract just because I like to do so, as w is a non-zero integer.

So factoring a target T is turned into factoring a series of surrogates, S_1, S_2, S_3, ... and checking with each one to see if you've factored which may seem like a lot, but it doesn't have to be, and it is a direct assault against methods like RSA encryption because it allows you to go after smaller numbers that might be easily

factored to get your target, and you get as many tries as you want where if you fully factor your surrogates you have a 50% chance of getting your target as well, versus just banging your head against one big number.

Example:

T = 732367903, k=24412263

Surrogate: 1191915704826532 = ( 4 )( 7 )( 73 )( 583129014103 )

f_1 = 7/2 and f_2 = 85136836059038

and 4f_1*f_2 = 2k^2 - 2T

y=-170273672118069/2 and x=170273623293557/2

so, x+y=-24412256, which has 223 as a factor.

T = 732367903 = (223)(3284161)

Iterations: 1

Notice that with the surrogate factored there was not even a lot of iterating through combinations of those factors, as it took one try.

That example is the potential, as it could be a much larger target, where the primary issue is factoring the surrogate.

With the surrogate factored, looping through all the iterations of factors the theory says you have a 50% chance of factoring, so, if I got the mathematical theory right, anybody with some serious factoring power already could convert maybe even an RSA sized number into several surrogates, go at it, and just walk away with a factorization, just like that.

It is a fascinating invention to me, because it is amazing that such simple mathematics could have been missed, while it is so simple it's hard to see where the math can sneak in something that ruins it, like actually the only thing is if w's that work get scarcer as T increases in size, where there is no mathematical indication that they do, as you are multiplying T, and there are an infinity of w's that must work. So how do you concentrate that infinity?

How do you force mathematically a smaller infinite subset of working w's against a bigger infinite majority that don't?

Do the math. There is no way. So you get the same factoring potential as is already known for the difference of squares—50%—as that is just a variation with k=0 anyway.

And if I'm right, people with the know-how all over the world can immediately start factoring much larger numbers than they did before. Immediately, all over the world, with some very simple mathematics that is inexplicably still just out there, with barely a murmur from the mathematical community.

Maybe they just want to wait and see what happens…

## Pragmatic take on surrogate factoring

Having finally done some experimentation versus just play with theory, I have a better handle now on the theory and think I can give a succinct explanation, finally, that should be, well, perfect which covers the practical implementation of surrogate factoring.

Practically surrogate factoring can be considered to be finding prime factors of a target composite using the factors of what I call the surrogate where the surrogate is

2k^2 - wT

where k is a non-zero integer, and w is a non-zero integer.

(For an explanation of the 'why' of surrogate factoring please go to my Extreme Mathematics group:

http://groups.google.com/group/extrememathematics/web/surrogate-factoring)

With factors

4f_1*f_2 = 2k^2 - wT

then it's trivial to find that functionally using surrogate factoring involves checking

2f_1 +/- k and 2f_2 +/- k

for prime factors in common with T, as

x+y = 2f_1 +/- k

and

x - y = 2f_2 +/- k

gives

x^2 = y^2 mod p, where p is a prime factor of the target T, when this method works.

So when does it work?

Theory indicates that for a given w, k pair, you will get a non-trivial factorization for SOME f_1 and f_2 as defined above, 50% of the time.

So, for instance, if you hold k constant, and increment w, then the probability goes as

(1 - (1/2)^i)*100%

so, for instance, with a single k, and using -1, - 2 and -3 in succession for w, the probability that you will non-trivially factor T, with one of your combinations of factors f_1 and f_2 is

(1 - 1/8)*100% = 87.5%

but that requires that EACH of the surrogates is fully factored, which means that you need to be able to factor rather well first, before you start with surrogate factoring, though then you get an additional boost to any factoring method, according to the theory, and my limited experiments with small numbers have held to that theory.

When the surrogates are fully factored, the method works extremely well, factoring numbers very few iterations, as I've pointed out on my math sites.

If this simple theory was done correctly by me, it means that an approach to an RSA sized number would require use of heavy factoring machinery already known, like the Number Field Sieve to factor the surrogates, as they'd be very large.

But if a particular surrogate is hard to factor—you can go to another.

And that's how surrogate factoring could be a very powerful and practical method, as given any hard target, you can turn it into multiple targets where methods may be found to make them a lot easier to factor, and you have a high probability of success once you do so, though you may loop through a lot of combinations as you find every

possible f_1 and f_2 to check, which is where you can get into heavy iterations.

Most of the above is easy to figure out from the mathematics itself, which is very trivial algebra, and from playing around with actual factoring using this technique.

The reason it has interesting properties like the 50% probability with every w, k has to do with the fundamental nature of the equations used, as astute readers may note that if you take away the restriction that k be non-zero, and let k=0, then you just find the well-known difference of squares.

Surrogate factoring can be described as yet another way to get a difference of squares.

The mathematics is easy to the point of triviality.

There is no mathematical doubt that this method works as described.

Practically surrogate factoring can be considered to be finding prime factors of a target composite using the factors of what I call the surrogate where the surrogate is

2k^2 - wT

where k is a non-zero integer, and w is a non-zero integer.

(For an explanation of the 'why' of surrogate factoring please go to my Extreme Mathematics group:

http://groups.google.com/group/extrememathematics/web/surrogate-factoring)

With factors

4f_1*f_2 = 2k^2 - wT

then it's trivial to find that functionally using surrogate factoring involves checking

2f_1 +/- k and 2f_2 +/- k

for prime factors in common with T, as

x+y = 2f_1 +/- k

and

x - y = 2f_2 +/- k

gives

x^2 = y^2 mod p, where p is a prime factor of the target T, when this method works.

So when does it work?

Theory indicates that for a given w, k pair, you will get a non-trivial factorization for SOME f_1 and f_2 as defined above, 50% of the time.

So, for instance, if you hold k constant, and increment w, then the probability goes as

(1 - (1/2)^i)*100%

so, for instance, with a single k, and using -1, - 2 and -3 in succession for w, the probability that you will non-trivially factor T, with one of your combinations of factors f_1 and f_2 is

(1 - 1/8)*100% = 87.5%

but that requires that EACH of the surrogates is fully factored, which means that you need to be able to factor rather well first, before you start with surrogate factoring, though then you get an additional boost to any factoring method, according to the theory, and my limited experiments with small numbers have held to that theory.

When the surrogates are fully factored, the method works extremely well, factoring numbers very few iterations, as I've pointed out on my math sites.

If this simple theory was done correctly by me, it means that an approach to an RSA sized number would require use of heavy factoring machinery already known, like the Number Field Sieve to factor the surrogates, as they'd be very large.

But if a particular surrogate is hard to factor—you can go to another.

And that's how surrogate factoring could be a very powerful and practical method, as given any hard target, you can turn it into multiple targets where methods may be found to make them a lot easier to factor, and you have a high probability of success once you do so, though you may loop through a lot of combinations as you find every

possible f_1 and f_2 to check, which is where you can get into heavy iterations.

Most of the above is easy to figure out from the mathematics itself, which is very trivial algebra, and from playing around with actual factoring using this technique.

The reason it has interesting properties like the 50% probability with every w, k has to do with the fundamental nature of the equations used, as astute readers may note that if you take away the restriction that k be non-zero, and let k=0, then you just find the well-known difference of squares.

Surrogate factoring can be described as yet another way to get a difference of squares.

The mathematics is easy to the point of triviality.

There is no mathematical doubt that this method works as described.

### Wednesday, March 07, 2007

## Factoring has two legs

The theoretical side of my latest research is to show one and only one more remaining fundamental factoring congruence, showing you could say, that factoring has two legs:

With T the target integer to be factored

x^2 = y^2 mod T

and

k^2 = 2xk mod T

and, of course, the first has been known for centuries and a highly refined development of it is the modern Number Field Sieve, while the second is my addition, for what I call surrogate factoring.

The second congruence—the second fundamental leg of factoring—represents the mathematics that brings in what I christened surrogate factoring, where you can factor your target T, by solving for the two congruences, with a chosen non-zero integer k, using explicit equations.

There is no debate about whether or not it can be done, but a general consensus I've gathered that it is not important, and research using the second leg cannot be made practical, seems to be the opinion of the mathematical community.

But it is the only other congruence that extends factoring, and I'm sure some would try to debate that, as I make a theoretical point, so I leave it as an open challenge—knowing I've often lost these!!!—for someone to show that it is not.

My own research into this second leg, the mathematics that gives you surrogate factoring, indicates that it is not picky, as there must exist solutions for any integer k, so there are solutions for the difference of squares, and that the methods that gives you solutions cannot be shown to be picky either, so that hey, it could work well enough with the Number Field Sieve to greatly extend the range of

numbers that can be factored.

That is, bringing the two factoring legs together allows you to go much further.

But, who knows. At this point it's early research, as while I've tossed the idea of surrogate factoring around for years, I just realized how simple the surrogate factoring piece actually is—back around August of last year.

For mathematical research that's like, just found, so it's rather new, but kind of cool from what I've tested with factoring numbers, and from what I've seen with small numbers—the easier for me to test—the results follow theory.

Those who don't understand how I use Usenet need to know that I tend to post on Usenet to announce, while previously I posted as well to work out theory, and often to grouse about non-acceptance of my mathematical research.

This post is primarily a continuation of me announcing.

I have factoring research. Here it is in terms of the basics. I say, hey, other people should care, but if they don't, what then?

I don't know. But if it is important, somehow, someway, then, of course, I figure that will come out eventually.

With T the target integer to be factored

x^2 = y^2 mod T

and

k^2 = 2xk mod T

and, of course, the first has been known for centuries and a highly refined development of it is the modern Number Field Sieve, while the second is my addition, for what I call surrogate factoring.

The second congruence—the second fundamental leg of factoring—represents the mathematics that brings in what I christened surrogate factoring, where you can factor your target T, by solving for the two congruences, with a chosen non-zero integer k, using explicit equations.

There is no debate about whether or not it can be done, but a general consensus I've gathered that it is not important, and research using the second leg cannot be made practical, seems to be the opinion of the mathematical community.

But it is the only other congruence that extends factoring, and I'm sure some would try to debate that, as I make a theoretical point, so I leave it as an open challenge—knowing I've often lost these!!!—for someone to show that it is not.

My own research into this second leg, the mathematics that gives you surrogate factoring, indicates that it is not picky, as there must exist solutions for any integer k, so there are solutions for the difference of squares, and that the methods that gives you solutions cannot be shown to be picky either, so that hey, it could work well enough with the Number Field Sieve to greatly extend the range of

numbers that can be factored.

That is, bringing the two factoring legs together allows you to go much further.

But, who knows. At this point it's early research, as while I've tossed the idea of surrogate factoring around for years, I just realized how simple the surrogate factoring piece actually is—back around August of last year.

For mathematical research that's like, just found, so it's rather new, but kind of cool from what I've tested with factoring numbers, and from what I've seen with small numbers—the easier for me to test—the results follow theory.

Those who don't understand how I use Usenet need to know that I tend to post on Usenet to announce, while previously I posted as well to work out theory, and often to grouse about non-acceptance of my mathematical research.

This post is primarily a continuation of me announcing.

I have factoring research. Here it is in terms of the basics. I say, hey, other people should care, but if they don't, what then?

I don't know. But if it is important, somehow, someway, then, of course, I figure that will come out eventually.

### Saturday, March 03, 2007

## JSH: They lie

Some of you know that a mathematician named Anatoly Plotnikov claims to have proven that P=NP, about a decade ago.

What if he is right?

Then the bulk of mathematicians have sold the world on factoring as a hard problem despite a proof that P=NP, a proof they have willfully ignored for over a decade.

In "pure math" areas it is just the word of some people, a system perfect for lying, with a clear benefit for those who take it over, if they have critical mass then they can support each other without ever having to worry about having real mathematical proofs.

You claim error? They say you are wrong. What can you do then?

It's their word, against yours.

But they stepped out of "pure math"—mathematical research with no practical application—into the real world, with a method that supposedly protects the world's information.

Yet Rivest, Shamir, and Adelman when first proposing the idea that factoring was a hard problem, put up a code that was rather quickly broken, despite their initial claims of how long it should stand. (Know how long they said?)

Despite the reality of rapidly advancing research that kept breaking bigger and bigger keys, mathematicians have managed to keep in place a system that many say would fall anyway with the advent of quantum computers.

That is wacky. It is insane.

You tell me that you have given a system that will be broken any day, and make everything I've encrypted completely open to just about anyone with a powerful enough computer system, and that's ok?

It's ok in a world that trusts experts who inexplicably say it's ok.

And how dare you challenge them? Start asking real questions about the brilliance of the "beautiful minds" who supposedly are the world's mathematicians, and you quickly get shot down.

They will question your sanity; they will insult you into silence.

Sounds quite brilliant, eh? But people get used to abuse from supposedly highly intelligent people and think it just normal that if you seriously question mathematicians they will act like complete asses in taking you down.

And people trust them when for most of them, their careers are built on research that has only been checked by other members of their own group—looking it over.

I know they lie the hard way. I found important mathematical proofs that change number theory from top to bottom, and in the process found major errors, huge enough to shatter the careers of many of them, and they have tried now for years to ignore me.

I got a paper published. Social pressure got it pulled. The mathematical journal died.

That paper has to be held back by current math society because it highlights a technique that proves that major tools currently being taught in "pure math" areas, do not actually work.

As a leading researcher challenged by a corrupted field, with players who are willing to lie so easily, and can get away with it, even to the point that a dead math journal means nothing, I am pushed to try and find something that they can't just lie about, and that is factoring.

Those of you who more than pretend to care about your field should do the fact checking. Like check on Plotnikov's research. See if your field is not willfully ignoring proof that P=NP, so that they can sell the world on a system that is about keeping some con artists in power, at the expense of the world.

[A reply to someone who told James that Anatoly Plotnikov doesn't claim anymore that he solved the P = NP problem.]

Assuming you are telling the truth, I withdraw my prior speculations on why Plotnikov's work was not accepted. The rest of the post stands as written.

What if he is right?

Then the bulk of mathematicians have sold the world on factoring as a hard problem despite a proof that P=NP, a proof they have willfully ignored for over a decade.

In "pure math" areas it is just the word of some people, a system perfect for lying, with a clear benefit for those who take it over, if they have critical mass then they can support each other without ever having to worry about having real mathematical proofs.

You claim error? They say you are wrong. What can you do then?

It's their word, against yours.

But they stepped out of "pure math"—mathematical research with no practical application—into the real world, with a method that supposedly protects the world's information.

Yet Rivest, Shamir, and Adelman when first proposing the idea that factoring was a hard problem, put up a code that was rather quickly broken, despite their initial claims of how long it should stand. (Know how long they said?)

Despite the reality of rapidly advancing research that kept breaking bigger and bigger keys, mathematicians have managed to keep in place a system that many say would fall anyway with the advent of quantum computers.

That is wacky. It is insane.

You tell me that you have given a system that will be broken any day, and make everything I've encrypted completely open to just about anyone with a powerful enough computer system, and that's ok?

It's ok in a world that trusts experts who inexplicably say it's ok.

And how dare you challenge them? Start asking real questions about the brilliance of the "beautiful minds" who supposedly are the world's mathematicians, and you quickly get shot down.

They will question your sanity; they will insult you into silence.

Sounds quite brilliant, eh? But people get used to abuse from supposedly highly intelligent people and think it just normal that if you seriously question mathematicians they will act like complete asses in taking you down.

And people trust them when for most of them, their careers are built on research that has only been checked by other members of their own group—looking it over.

I know they lie the hard way. I found important mathematical proofs that change number theory from top to bottom, and in the process found major errors, huge enough to shatter the careers of many of them, and they have tried now for years to ignore me.

I got a paper published. Social pressure got it pulled. The mathematical journal died.

That paper has to be held back by current math society because it highlights a technique that proves that major tools currently being taught in "pure math" areas, do not actually work.

As a leading researcher challenged by a corrupted field, with players who are willing to lie so easily, and can get away with it, even to the point that a dead math journal means nothing, I am pushed to try and find something that they can't just lie about, and that is factoring.

Those of you who more than pretend to care about your field should do the fact checking. Like check on Plotnikov's research. See if your field is not willfully ignoring proof that P=NP, so that they can sell the world on a system that is about keeping some con artists in power, at the expense of the world.

[A reply to someone who told James that Anatoly Plotnikov doesn't claim anymore that he solved the P = NP problem.]

Assuming you are telling the truth, I withdraw my prior speculations on why Plotnikov's work was not accepted. The rest of the post stands as written.

### Thursday, March 01, 2007

## JSH: So how could they win?

There is one problem that some of you may be considering, as if I did figure out my own factoring method—primarily to force mathematicians to acknowledge my other mathematical research—and if so much of my mathematical research is so huge that it could upend so much, how could they ever think they'd win?

Why would anyone fight a battle they couldn't win? Why the Math Wars if mathematicians to some extent realized I was a person with major mathematical results who had pushed the frontiers of human knowledge in several areas?

My analysis—speculation—is that there were two key rationalizations:

So if they waited long enough, I might solve myself as a problem.

But they don't understand someone like me.

History has lots of stories of melt-downs by mathematical people and it has long been considered that people who have deep insights into mathematics are often mentally unstable, and likely to suffer tremendous melt-downs and take themselves out.

But no major mathematical discoverer at my level has ever failed.

Not in all of human history, and they didn't understand that I would not allow myself to be the first.

And I am more than capable of changing the entire world if necessary to make certain that mathematics progresses.

After all, that is why I was born in the first place.

If mathematicians maintain enough power to continue to block as they have in the past, then I will simply push forward the factoring research, or just find something else.

Why would anyone fight a battle they couldn't win? Why the Math Wars if mathematicians to some extent realized I was a person with major mathematical results who had pushed the frontiers of human knowledge in several areas?

My analysis—speculation—is that there were two key rationalizations:

- As long as they kept status, woke up each day to go in front of students who believed in them, and the world believed in them, mathematicians felt the end justified the means.
- They tried to figure out what they would do, if they were me. If they had major mathematical results, but noone believed them, and they concluded that they would steadily break-down, go insane or even might commit suicide.

So if they waited long enough, I might solve myself as a problem.

But they don't understand someone like me.

History has lots of stories of melt-downs by mathematical people and it has long been considered that people who have deep insights into mathematics are often mentally unstable, and likely to suffer tremendous melt-downs and take themselves out.

But no major mathematical discoverer at my level has ever failed.

Not in all of human history, and they didn't understand that I would not allow myself to be the first.

And I am more than capable of changing the entire world if necessary to make certain that mathematics progresses.

After all, that is why I was born in the first place.

If mathematicians maintain enough power to continue to block as they have in the past, then I will simply push forward the factoring research, or just find something else.