### Tuesday, June 27, 2006

## JSH: Latest factoring idea is crap

Ok, I accept it. My latest idea just goes in circles like all the others.

That is all.

[A reply to someone who said that at last James admited that.]

Hey, I always admit it when I'm proven wrong and realize it.

I don't see the point in holding on to wrong ideas.

In this case, the approach was just kind of thrilling, and I was just so sure that using the square root ambiguity would work, but everything I was doing was a BFC—big freaking circle.

But you know, this is fun to me. And if you realized how much fun this is you'd understand why I just love going after these problems in this way.

So my initial approach collapsed—yuck, felt terrible. Really questioned myself, and what I was doing. Posted it was crap.

Walked away, determined to move on to other things, and of course, I get another idea.

Unfortunately, I've found that's what works!!!

I need to put something out there, fight for it, be really wrong, and then, as everything collapses, and I start a pity party, wonder what is the meaning of life, etc., I get another idea.

It's a hard way to discover things, but it is the way that works for me.

It's kind of like living in Hell. No it is living in Hell. I actually think this is Hell and I'm being punished for something.

I am serious.

[A reply to someone who asked James whether or not he is a crackpot.]

Hey, I think most of this is entertainment and I say so, and have said so over years.

I AM serious about a lot of my mathematical ideas, but I also have a sense of humor and perspective about life that many of you seem to lack.

And in the big scheme of things, some guy musing about various math ideas, where most of them are wrong, even though he can get VERY convinced at times wanting something to be true that's not, is just not something that most would consider this big, bad deal.

I think for some reason many of you put on as if I should be ashamed of myself, or as if I'm this horrible person, but hey, I'm just some guy coming up with math ideas and posting about them on Usenet.

Sure I really don't like a lot of math people, as I've seen how they behave and born the brunt of a lot of abusive behavior from that community, but who would?

If you'd had math people after you like they've been after me for years, would you like them?

I'm just trying to have my little bit of fun in cyberspace, with a whole horde of mad math nincompoop people after me year after year.

Of course I don't like the bastards.

[A reply to someone who said that if James would stop posting at sci.math, then the sci.mathers wouldn't go after him.]

Censorship has always been the aim of many of you—in direct contradiction with the spirit of Usenet.

So you insult me and stalk my postings to try and get me to stop putting up my ideas on what you think are your forums.

But they are public forums. You people are no different from hoodlums hanging out in a public park attacking people to protect "their territory".

That is all.

[A reply to someone who said that at last James admited that.]

Hey, I always admit it when I'm proven wrong and realize it.

I don't see the point in holding on to wrong ideas.

In this case, the approach was just kind of thrilling, and I was just so sure that using the square root ambiguity would work, but everything I was doing was a BFC—big freaking circle.

But you know, this is fun to me. And if you realized how much fun this is you'd understand why I just love going after these problems in this way.

So my initial approach collapsed—yuck, felt terrible. Really questioned myself, and what I was doing. Posted it was crap.

Walked away, determined to move on to other things, and of course, I get another idea.

Unfortunately, I've found that's what works!!!

I need to put something out there, fight for it, be really wrong, and then, as everything collapses, and I start a pity party, wonder what is the meaning of life, etc., I get another idea.

It's a hard way to discover things, but it is the way that works for me.

It's kind of like living in Hell. No it is living in Hell. I actually think this is Hell and I'm being punished for something.

I am serious.

[A reply to someone who asked James whether or not he is a crackpot.]

Hey, I think most of this is entertainment and I say so, and have said so over years.

I AM serious about a lot of my mathematical ideas, but I also have a sense of humor and perspective about life that many of you seem to lack.

And in the big scheme of things, some guy musing about various math ideas, where most of them are wrong, even though he can get VERY convinced at times wanting something to be true that's not, is just not something that most would consider this big, bad deal.

I think for some reason many of you put on as if I should be ashamed of myself, or as if I'm this horrible person, but hey, I'm just some guy coming up with math ideas and posting about them on Usenet.

Sure I really don't like a lot of math people, as I've seen how they behave and born the brunt of a lot of abusive behavior from that community, but who would?

If you'd had math people after you like they've been after me for years, would you like them?

I'm just trying to have my little bit of fun in cyberspace, with a whole horde of mad math nincompoop people after me year after year.

Of course I don't like the bastards.

[A reply to someone who said that if James would stop posting at sci.math, then the sci.mathers wouldn't go after him.]

Censorship has always been the aim of many of you—in direct contradiction with the spirit of Usenet.

So you insult me and stalk my postings to try and get me to stop putting up my ideas on what you think are your forums.

But they are public forums. You people are no different from hoodlums hanging out in a public park attacking people to protect "their territory".

## SF: Tandem factorization back, new approach

Sorry but this is how the development process works with me.

I went down one path before, found out it didn't work like I thought it did, so I called crap crap and started feeling sorry for myself. Then I thought of another approach, so here we go, again.

The idea here is to use a tandem factorization of two numbers where

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

which means that

2*sqrt(x)*k_1 = f_1 + g_1

2*sqrt(y)*k_2 = f_1 - g_1

2*sqrt(x)*k_3 = f_2 + g_2

2*sqrt(y)*k_4 = f_2 - g_2

and the initial approach I put forward was with a target composite T to factor you pick some surrogate S, and I had various ideas for how you then find all the other variables—when for that idea to work, you need to already know the factorization of S and T.

So that was when I was ready to toss all of this and declared it crap.

However, why pick S?

Using the first two equations I have

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

and focusing on that second equation if I let

k_1*k_3 = A

k_2*k_4 = B

and pick squares for x and y, then I determine S, if T is the target factorization:

S = 2*k_1*k_3*x + 2*k_2*k_4*y - T

so

S = 2*A*x + 2*B*y - T

and you can use those equations to solve out two of the k's, and substitute out S, to relate the remaining two k's in an equation that I hope will give the approach that works.

>From that equation the idea is you figure out how to pick A and B such that everything is an integer, and that gives you S, with S, you have integer k's and an incidental factorization of T, if it works.

I'll have to work through the equations and see if it is another crapshoot.

Ok, so making that substitution into the first equation I have

2A*x + 2*B*y - 2T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and now I can solve out two of the k's, and divide by 2 to get

A*x + B*y - T =(k_2*(A/k_1) + k_1*(B/k_2))*sqrt(xy)

where there were a couple of possible ways to substitute out and I just picked one.

Now multiplying both sides by k_1*k_2 gives

(A*x + B*y - T)*k_1*k_2 = (A*k_2^2 + B*k_1^2)*sqrt(xy)

so I have

A*sqrt(xy)*k_2^2 - (A*x + B*y - T)*k_1*k_2 + B*sqrt(xy)*k_1^2 = 0

and completing the square with respect to k_2, I have

A*sqrt(xy)*k_2^2 - (A*x + B*y - T)*k_1*k_2 + (A*x + B*y - T)^2*k_1^2/(4A*sqrt(xy)) + B*sqrt(xy)*k_1^2 = (A*x + B*y - T)^2*k_1^2/(4A*sqrt(xy))

multiplying both sides by 4*A*sqrt(xy), and simplifying a bit gives

(A*sqrt(xy)*k_2 - (A*x + B*y - T)*k_1)^2 + 4*A* B*(xy)*k_1^2 = (A*x + B*y - T)^2*k_1^2

which is

(A*sqrt(xy)*k_2 - (A*x + B*y - T)*k_1)^2 = ((A*x - B*y)^2 - 2T(A*x+B*y) + T^2)*k_1^2

which yuck, shows that I need to know the factorization of T, to know how to pick A and B, so that

((A*x - B*y)^2 - 2T(A*x+B*y) + T^2)

and you need the factorization of B and T, or A and T to go further so it's another crap idea.

Went in a big freaking circle, again.

Maybe, maybe not. After all, I have

S = 2*A*x + 2*B*y - T

so what if I just decide that S is the target?

Then I need to solve out further with

(A*sqrt(xy)*k_2 - (A*x + B*y - T)*k_1)^2 = ((A*x - B*y)^2 - 2T(A*x+B*y) + T^2)*k_1^2

as that is

(A*sqrt(xy)*k_2 - (A*x + B*y - T)*k_1)^2 = ((A*x - B*y - T)^2 - 4T*B*y)*k_1^2

and if I just pick B and T, as surrogates, and assume I know the factorizations

a_1 * a_2 = B*y, and b_1*b_2 = T, then I have

A*x = B*y + T + a_1*b_1 + a_2*b_2

and I can now substitute to find that S is given by

S = 4*B*y + T + 2*a_1*b_1 + 2*a_2*b_2

and making my substitutions that is

S = 4*y*a_1 * a_2 + b_1*b_2 + 2*a_1*b_1 + 2*a_2*b_2

and let's collect with respect to the a's, so I have

S = 4*y*a_1 * a_2 + 2*a_1*b_1 + b_1*b_2 + 2*a_2*b_2

and simplify to get

S - b_1*b_2 - 2*a_2*b_2 = (4*y* a_2 + 2*b_1)*a_1

so I have finally that

a_1 = (S - b_1*b_2 - 2*a_2*b_2)/(4*y* a_2 + 2*b_1)

and it looks like I can just pick b_1 and b_2, and then I just need to look for integer values for a_2 such that a_1 is an integer.

One last thing to do then, with

4*y* a_2 + 2*b_1 = h_1

then

4*y* a_2 + 2*b_1 = 0 mod h_1

S - b_1*b_2 - 2*a_2*b_2 = 0 mod h_1

so multiplying the first by -2*b_2 and the second by 4*y, I have

-8*y*b_2* a_2 - 4*b_1*b_2 = 0 mod h_1

4*y*S - 4*y* b_1*b_2 - 8*y*a_2*b_2 = 0 mod h_1

and subtracting the top one from the bottom one I get

4*y*S - 4*y* b_1*b_2 + 4*b_1*b_2= 0 mod h_1

which is a surpise to me, as it looks like if I go ahead and use b_1*b_2 = T, I just have

4*y*S + 4*T*(y-1)= 0 mod h_1

whci is the kind of result I don't believe in (thinking I made an algebra mistake somewhere) as it says that if y = 1, then you need the factorization of S, but otherwise, you can factor this other thing.

Going back over it to find someplace where I left off y…

Didn't see it on a quick run back, but I guess I'll find something later. In any event, will post in the meantime!

Of course, if by some chance I did the algebra right, then you have this odd result that you need to pick a square for y other than 1, like y=4, and then the damn thing will work!

But that seems too arbitrary, so I must have made a mistake somewhere?

That was dumb. That's why y is in at the end, as I put it in here when it was substituted out.

Corrected it goes as follows.

and let's collect with respect to the a's, so I have

S = 4*a_1 * a_2 + 2*a_1*b_1 + b_1*b_2 + 2*a_2*b_2

and simplify to get

S - b_1*b_2 - 2*a_2*b_2 = (4* a_2 + 2*b_1)*a_1

so I have finally that

a_1 = (S - b_1*b_2 - 2*a_2*b_2)/(4a_2 + 2*b_1)

and it looks like I can just pick b_1 and b_2, and then I just need to look for integer values for a_2 such that a_1 is an integer.

One last thing to do then, with

4a_2 + 2*b_1 = h_1

then

4a_2 + 2*b_1 = 0 mod h_1

S - b_1*b_2 - 2*a_2*b_2 = 0 mod h_1

so multiplying the first by -2*b_2 and the second by 4, I have

-8b_2* a_2 - 4*b_1*b_2 = 0 mod h_1

4S - 4b_1*b_2 - 8*y*a_2*b_2 = 0 mod h_1

and subtracting the top one from the bottom one I get

4S = 0 mod h_1

so not surprisingly I need the factorization of S to go further so it's another big freaking circle.

I went down one path before, found out it didn't work like I thought it did, so I called crap crap and started feeling sorry for myself. Then I thought of another approach, so here we go, again.

The idea here is to use a tandem factorization of two numbers where

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

which means that

2*sqrt(x)*k_1 = f_1 + g_1

2*sqrt(y)*k_2 = f_1 - g_1

2*sqrt(x)*k_3 = f_2 + g_2

2*sqrt(y)*k_4 = f_2 - g_2

and the initial approach I put forward was with a target composite T to factor you pick some surrogate S, and I had various ideas for how you then find all the other variables—when for that idea to work, you need to already know the factorization of S and T.

So that was when I was ready to toss all of this and declared it crap.

However, why pick S?

Using the first two equations I have

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

and focusing on that second equation if I let

k_1*k_3 = A

k_2*k_4 = B

and pick squares for x and y, then I determine S, if T is the target factorization:

S = 2*k_1*k_3*x + 2*k_2*k_4*y - T

so

S = 2*A*x + 2*B*y - T

and you can use those equations to solve out two of the k's, and substitute out S, to relate the remaining two k's in an equation that I hope will give the approach that works.

>From that equation the idea is you figure out how to pick A and B such that everything is an integer, and that gives you S, with S, you have integer k's and an incidental factorization of T, if it works.

I'll have to work through the equations and see if it is another crapshoot.

Ok, so making that substitution into the first equation I have

2A*x + 2*B*y - 2T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and now I can solve out two of the k's, and divide by 2 to get

A*x + B*y - T =(k_2*(A/k_1) + k_1*(B/k_2))*sqrt(xy)

where there were a couple of possible ways to substitute out and I just picked one.

Now multiplying both sides by k_1*k_2 gives

(A*x + B*y - T)*k_1*k_2 = (A*k_2^2 + B*k_1^2)*sqrt(xy)

so I have

A*sqrt(xy)*k_2^2 - (A*x + B*y - T)*k_1*k_2 + B*sqrt(xy)*k_1^2 = 0

and completing the square with respect to k_2, I have

A*sqrt(xy)*k_2^2 - (A*x + B*y - T)*k_1*k_2 + (A*x + B*y - T)^2*k_1^2/(4A*sqrt(xy)) + B*sqrt(xy)*k_1^2 = (A*x + B*y - T)^2*k_1^2/(4A*sqrt(xy))

multiplying both sides by 4*A*sqrt(xy), and simplifying a bit gives

(A*sqrt(xy)*k_2 - (A*x + B*y - T)*k_1)^2 + 4*A* B*(xy)*k_1^2 = (A*x + B*y - T)^2*k_1^2

which is

(A*sqrt(xy)*k_2 - (A*x + B*y - T)*k_1)^2 = ((A*x - B*y)^2 - 2T(A*x+B*y) + T^2)*k_1^2

which yuck, shows that I need to know the factorization of T, to know how to pick A and B, so that

((A*x - B*y)^2 - 2T(A*x+B*y) + T^2)

and you need the factorization of B and T, or A and T to go further so it's another crap idea.

Went in a big freaking circle, again.

Maybe, maybe not. After all, I have

S = 2*A*x + 2*B*y - T

so what if I just decide that S is the target?

Then I need to solve out further with

(A*sqrt(xy)*k_2 - (A*x + B*y - T)*k_1)^2 = ((A*x - B*y)^2 - 2T(A*x+B*y) + T^2)*k_1^2

as that is

(A*sqrt(xy)*k_2 - (A*x + B*y - T)*k_1)^2 = ((A*x - B*y - T)^2 - 4T*B*y)*k_1^2

and if I just pick B and T, as surrogates, and assume I know the factorizations

a_1 * a_2 = B*y, and b_1*b_2 = T, then I have

A*x = B*y + T + a_1*b_1 + a_2*b_2

and I can now substitute to find that S is given by

S = 4*B*y + T + 2*a_1*b_1 + 2*a_2*b_2

and making my substitutions that is

S = 4*y*a_1 * a_2 + b_1*b_2 + 2*a_1*b_1 + 2*a_2*b_2

and let's collect with respect to the a's, so I have

S = 4*y*a_1 * a_2 + 2*a_1*b_1 + b_1*b_2 + 2*a_2*b_2

and simplify to get

S - b_1*b_2 - 2*a_2*b_2 = (4*y* a_2 + 2*b_1)*a_1

so I have finally that

a_1 = (S - b_1*b_2 - 2*a_2*b_2)/(4*y* a_2 + 2*b_1)

and it looks like I can just pick b_1 and b_2, and then I just need to look for integer values for a_2 such that a_1 is an integer.

One last thing to do then, with

4*y* a_2 + 2*b_1 = h_1

then

4*y* a_2 + 2*b_1 = 0 mod h_1

S - b_1*b_2 - 2*a_2*b_2 = 0 mod h_1

so multiplying the first by -2*b_2 and the second by 4*y, I have

-8*y*b_2* a_2 - 4*b_1*b_2 = 0 mod h_1

4*y*S - 4*y* b_1*b_2 - 8*y*a_2*b_2 = 0 mod h_1

and subtracting the top one from the bottom one I get

4*y*S - 4*y* b_1*b_2 + 4*b_1*b_2= 0 mod h_1

which is a surpise to me, as it looks like if I go ahead and use b_1*b_2 = T, I just have

4*y*S + 4*T*(y-1)= 0 mod h_1

whci is the kind of result I don't believe in (thinking I made an algebra mistake somewhere) as it says that if y = 1, then you need the factorization of S, but otherwise, you can factor this other thing.

Going back over it to find someplace where I left off y…

Didn't see it on a quick run back, but I guess I'll find something later. In any event, will post in the meantime!

Of course, if by some chance I did the algebra right, then you have this odd result that you need to pick a square for y other than 1, like y=4, and then the damn thing will work!

But that seems too arbitrary, so I must have made a mistake somewhere?

That was dumb. That's why y is in at the end, as I put it in here when it was substituted out.

Corrected it goes as follows.

and let's collect with respect to the a's, so I have

S = 4*a_1 * a_2 + 2*a_1*b_1 + b_1*b_2 + 2*a_2*b_2

and simplify to get

S - b_1*b_2 - 2*a_2*b_2 = (4* a_2 + 2*b_1)*a_1

so I have finally that

a_1 = (S - b_1*b_2 - 2*a_2*b_2)/(4a_2 + 2*b_1)

and it looks like I can just pick b_1 and b_2, and then I just need to look for integer values for a_2 such that a_1 is an integer.

One last thing to do then, with

4a_2 + 2*b_1 = h_1

then

4a_2 + 2*b_1 = 0 mod h_1

S - b_1*b_2 - 2*a_2*b_2 = 0 mod h_1

so multiplying the first by -2*b_2 and the second by 4, I have

-8b_2* a_2 - 4*b_1*b_2 = 0 mod h_1

4S - 4b_1*b_2 - 8*y*a_2*b_2 = 0 mod h_1

and subtracting the top one from the bottom one I get

4S = 0 mod h_1

so not surprisingly I need the factorization of S to go further so it's another big freaking circle.

## JSH: Stepping it out

So yes, I'm giving more details, and some of you should realize by now that I have found a new factoring method which shows that factoring is not a hard problem.

I kind of find it fascinating watching the flurry of hostile postings to my new thread going over much of the theory.

You people hate mathematics.

You're a social group, and as you hear the drumbeat come closer you show ever more clearly your true colors—your hatred of what is mathematically correct.

How much longer? Looks like now the full details should be out tonight or tomorrow.

And then what will you do?

I suspect some of you will simply post with more hostility, more anger, more demonstration of your hatred of civilization, logic and mathematics.

Hey, I do not make what is mathematically true, true.

It was always true.

Some of you are simply not equipped to handle a world where what is believed to be true is not connected to your own self-image.

I kind of find it fascinating watching the flurry of hostile postings to my new thread going over much of the theory.

You people hate mathematics.

You're a social group, and as you hear the drumbeat come closer you show ever more clearly your true colors—your hatred of what is mathematically correct.

How much longer? Looks like now the full details should be out tonight or tomorrow.

And then what will you do?

I suspect some of you will simply post with more hostility, more anger, more demonstration of your hatred of civilization, logic and mathematics.

Hey, I do not make what is mathematically true, true.

It was always true.

Some of you are simply not equipped to handle a world where what is believed to be true is not connected to your own self-image.

### Monday, June 26, 2006

## SF: Tandem factorization, factoring problem solved?

I have the theory, and while I've not worked out the equations in detail, I feel confident enough to say, the factoring problem is done, as a hard problem.

Remember I finally came up with

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

which means that

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

notice that to be integers, sqrt(xy) must be a factor of S-T, and the point of this method is to look for integer solutions to all the variables.

Here's why.

It's trivial to show that with S = f_1*f_2 and T = g_1 * g_2

2*sqrt(x)*k_1 = f_1 + g_1

2*sqrt(y)*k_2 = f_1 - g_1

2*sqrt(x)*k_3 = f_2 + g_2

2*sqrt(y)*k_4 = f_2 - g_2

and what's remarkable here is that with over a century of methods focusing on just one number to factor, where you tend to end up with maybe g_1 + g_2 or g_1 - g_2, when T is your target, here by simply using TWO NUMBERS, in a tandem factorization, the algebra just tosses up a route to factoring both, trivially.

Why is it trivial?

Because when you use squares for x and y that work, then you have 4 variables left with two equations, and rather trivial Diophantine methods for solving those out to get the finite set of integer solutions.

More specifically, you focus on k_1 and k_2 or k_3 and k_4, solving for them using your two equations, as then you get a ratio with the remaining two variables.

e.g.

k_1 = F_1(k_3, k_4)/G_1(k_3, k_4)

k_2 = F_2(k_3, k_4)/G_2(k_3, k_4)

where the F's and G's are functions of the remaining k's.

>From that equation it is trivial to solve out for the finite set of integer values that will work.

So the factoring problem is simply solved by a bit of lateral thinking:

Don't just factor one number, but factor two at a time.

The algebra for the tandem factorization gives up solutions without forcing you to search through as large a space as focusing on one.

It's a counter-intuitive thing that factoring two numbers at a time would break the problem.

But that's how easy problems become hard:

To get to the solution, you have to think outside the box, and do the unexpected.

Factoring problem is done. Deny it if you wish. Shouldn't take long for the news to travel though.

In a remarkable display of why politicians don't make good mathematicians, Tim Peters solved out for the trivial x=y=1 case, and posted. I replied in that thread but thought I'd reply here as well, demonstrating the rest of the method.

<Quote>

> Here, I'll hand them to you for the x=y=1 case:

> k_1 = (k_3*(S+T) - k_4*(S-T)) / d

> k_2 = (k_3*(S-T) - k_4*(S+T)) / d

> where:

> d = 2*(k_3 + k_4)*(k_3 - k_4)

> Do you

> those efficiently? Didn't think so, and bluffing isn't good enough.

</Quote>

Here's how you do it.

k_3 + k_4 = h_1, so

k_3 + k_4 = 0 mod h_1

k_3*(S+T) - k_4*(S-T) = 0 mod h_1

and you solve out for k_3 or k_4 by, for instance, multiplying the first equation by (S+T) and subtracting.

Here you see why using x=y=1 is a trivial case as notice then h_1 is a factor of S.

Next you use k_3 - k_4 = h_2, so

k_3 - k_4 = 0 mod h_2

k_3*(S+T) - k_4*(S-T) = 0 mod h_2

and you have that h_2 is a factor of T.

Now use another case besides x=y=1, and watch it work.

Oh yeah, once you get h_1 and h_2, of course, then you have k_3 and k_4.

As, for instance,

k_3 = (h_1 + h_2)/2

and now all anyone has to do is use something other than x=y=1.

If I'm wrong, post it! See if again you find that h_1 or h_2 has to be a factor of T.

But the problem with current math society is that is a social society.

You people lack comprehension of what mathematics is.

To you it's what you believe to be true.

But I say, mathematics is what's proven to be true.

The mathematical community today has a lot of power to convince, and from what I've seen, it is religious in its ability to fight against mathematical proof.

So this may still take a while.

Notice for those of you who thought you were doing something important with your involvement in the mathematical community, if you were fighting me, you were wrong.

You were fighting against civilization itself.

People like me have fought for the truth against people like you for millennia.

Remember I finally came up with

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

which means that

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

notice that to be integers, sqrt(xy) must be a factor of S-T, and the point of this method is to look for integer solutions to all the variables.

Here's why.

It's trivial to show that with S = f_1*f_2 and T = g_1 * g_2

2*sqrt(x)*k_1 = f_1 + g_1

2*sqrt(y)*k_2 = f_1 - g_1

2*sqrt(x)*k_3 = f_2 + g_2

2*sqrt(y)*k_4 = f_2 - g_2

and what's remarkable here is that with over a century of methods focusing on just one number to factor, where you tend to end up with maybe g_1 + g_2 or g_1 - g_2, when T is your target, here by simply using TWO NUMBERS, in a tandem factorization, the algebra just tosses up a route to factoring both, trivially.

Why is it trivial?

Because when you use squares for x and y that work, then you have 4 variables left with two equations, and rather trivial Diophantine methods for solving those out to get the finite set of integer solutions.

More specifically, you focus on k_1 and k_2 or k_3 and k_4, solving for them using your two equations, as then you get a ratio with the remaining two variables.

e.g.

k_1 = F_1(k_3, k_4)/G_1(k_3, k_4)

k_2 = F_2(k_3, k_4)/G_2(k_3, k_4)

where the F's and G's are functions of the remaining k's.

>From that equation it is trivial to solve out for the finite set of integer values that will work.

So the factoring problem is simply solved by a bit of lateral thinking:

Don't just factor one number, but factor two at a time.

The algebra for the tandem factorization gives up solutions without forcing you to search through as large a space as focusing on one.

It's a counter-intuitive thing that factoring two numbers at a time would break the problem.

But that's how easy problems become hard:

To get to the solution, you have to think outside the box, and do the unexpected.

Factoring problem is done. Deny it if you wish. Shouldn't take long for the news to travel though.

<Quote>

> Here, I'll hand them to you for the x=y=1 case:

> k_1 = (k_3*(S+T) - k_4*(S-T)) / d

> k_2 = (k_3*(S-T) - k_4*(S+T)) / d

> where:

> d = 2*(k_3 + k_4)*(k_3 - k_4)

> Do you

__truly__have any idea how to go about finding integer solutions to

> those efficiently? Didn't think so, and bluffing isn't good enough.

</Quote>

Here's how you do it.

k_3 + k_4 = h_1, so

k_3 + k_4 = 0 mod h_1

k_3*(S+T) - k_4*(S-T) = 0 mod h_1

and you solve out for k_3 or k_4 by, for instance, multiplying the first equation by (S+T) and subtracting.

Here you see why using x=y=1 is a trivial case as notice then h_1 is a factor of S.

Next you use k_3 - k_4 = h_2, so

k_3 - k_4 = 0 mod h_2

k_3*(S+T) - k_4*(S-T) = 0 mod h_2

and you have that h_2 is a factor of T.

Now use another case besides x=y=1, and watch it work.

Oh yeah, once you get h_1 and h_2, of course, then you have k_3 and k_4.

As, for instance,

k_3 = (h_1 + h_2)/2

and now all anyone has to do is use something other than x=y=1.

If I'm wrong, post it! See if again you find that h_1 or h_2 has to be a factor of T.

But the problem with current math society is that is a social society.

You people lack comprehension of what mathematics is.

To you it's what you believe to be true.

But I say, mathematics is what's proven to be true.

The mathematical community today has a lot of power to convince, and from what I've seen, it is religious in its ability to fight against mathematical proof.

So this may still take a while.

Notice for those of you who thought you were doing something important with your involvement in the mathematical community, if you were fighting me, you were wrong.

You were fighting against civilization itself.

People like me have fought for the truth against people like you for millennia.

## Facts speak, math people just lie

I am not a happy person about this, as I have all kinds of facts that back me up, but I live in a world where mathematicians have built up a reputation, when it was past mathematicians who did the work, and a small number of applied mathematicians who still actually do provably correct and valuable work.

But most mathematicians today focus on what they call "pure math".

It is distinguished by being impractical and uncheckable except by other mathematicians.

There is no real world test of the work.

And I have been able to prove errors, show them to mathematicians, and watch them just walk away.

I had a paper published in a formally peer reviewed mathematical journal, a small electronic one, but hey, it was a legitimate math journal. I say was as the journal died.

A little while after publishing my paper readers of sci.math heard about it, some of them emailed the journal convincing the editors my paper was wrong, and they yanked it:

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

That link is to a site mirror, which is the last one still standing as there used to be somewhere around 10, but the rest dropped when the main site dropped, when the journal just shut down.

I posted a bit earlier about a letter I wrote to TIME magazine in the run-up to war with Iraq, and talked about how the editors changed what I said, toning it down in places, and removing some of the analysis, which isn't a big surprise, for those who know something about how the world works.

My own degree is in physics as I have a B.Sc. from Vanderbilt University.

I have an open source project called Class Viewer and you can find it easily enough by doing a google search on "Class Viewer" as it comes up number one.

That's just a slice.

Oh, one other thing, I wrote the first prime counting function article for the Wikipedia, because I was frustrated by mathematicians lying about my prime number research, and that writing is now in the history of the page:

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

I have been searching for over three years for a simple solution to the factoring problem as the means of proving that mathematicians do lie.

I think most of you though will not care about facts that don't matter to you because you have your beliefs, and I know that calling yourself a skeptic is nothing like actually being one.

But this time, reality can cost you money.

The system is at fault when it comes to mathematicians as the world gave a bunch of people a system where they needed each other above anyone else, and could not be checked except by each other, so they learned to value the opinion of each other—above all else.

They protect each other. Their lies about mathematical research put paychecks in each other's pockets. They know who they need and depend on.

And it's not you, if you're not a mathematician.

The mathematicians are victims in a sense of a world that should have known better.

Today computers offer the means to have even "pure" results that have no practical value, checked.

Computers can check mathematical arguments claimed to be proofs, when the work is done to build the expert systems.

That work is way behind now, and I suggest to you it is behind because mathematicians are invested in their current system, so they block, halting research in computer science if they can that would lead to proof checking by computer.

And they will sit on my factoring research until there is no way for them to sit.

And the pressure may be your accounts broken into, or your company's, or someone you know who gets victimized in some way, until that happens well above the level that people can blame anything other than the actual system protecting the Internet.

Sit back. It could take a while. I know from experience.

I have major mathematical research going back over three years.

These people can sit quietly, and wait. They'll wait for you or someone like you to hurt, to feel pain, and cry out loud enough, for the world to pay attention.

So we wait with them.

[A reply to someone who wrote that he would never have guessed that James is not happy about all this.]

Who would be?

Think about it, you take the time to go out and figure out some major new thing—a major accomplishment—and then you find that math professors ignore your research for political reasons, wouldn't you be upset?

So your reward for all the effort and pain is more pain, as society sits by while some people abuse you and call you a crackpot?

And it's so stupid too that mathematicians get away with just having their WORD trusted in these 'pure math' areas, when they push those areas where you just trust, over the important ones where the damn mathematics has to actually work, as proven by being useful in the real world, like to build better atomic weaponry.

A little dark humor there, but if my research were about nuclear weapons there wouldn't be any debate, none of this "crackpot" nonsense, and people would be fallng all over themselves to make sure I wasn't some very angry person, looking for ways to fight an academic world gone bad.

Consider that people.

If my research were about building better bombs there would not be this nonsense.

But because it's not, you can sit back and think it ok for some major discoverer to get punished for success, and what I say here is just ranting to you, comfortable where you might be.

And no computers able to check.

Think about it. If computer checking were around I could just put my resarch through it and all these people who fight my research would be undone, just like that.

Pure intellectual pursuits have this little problem that if people want to ignore you, they can.

But even with that, why do you people trust mathematicians in these "pure" areas, so that research like that of Andrew Wiles is just looked at by some people who claim it's correct, and you have no other way to check?

How can so many people around the world be so trusting given what we know about how readily even large groups of people—yes, math professors are human and can lie as a group—can just get things wrong?

Why do mathematicians get this blind trust in this day and age?

The problem is the possibility—which rational adults will acknowledge—that given a lack of outside checking, mathematicians can as a group miss errors in mathematical arguments that are just looked over by people, as people can make mistakes.

I think that many who recognize that mathematics is such an important area where we could not have the technology that we do without it, assume that there is some kind of other checking that goes on when they see in the news that this or that mathematician has a major discovery.

But the mathematics needed to get a working computer, or to make planes, or build a laser is different from the mathematics where some person just writes something up, for some other people to look at, and they all call it "pure math".

We as a society simply cannot just hope—hope that human nature is somehow purified in mathematics so that people can be trusted.

Did any of you know that Nobel did not create a math prize, so the people who run the Nobel prizes recently created one they call the Abel prize?

Any of you know how much that prize is?

Yet people can win with work that has no applicability in the real world, with research that has only been checked by human eyes, when their colleagues—other mathematicians—are the ones doing the checking.

And I'm not just griping here where there is no solution.

I firmly believe that computers can be programmed to check mathematical arguments.

It is up to us as a world community to push the development.

And it is past time to act like we are intelligent adults who value our mathematical world.

Can any of you rationally deny the importance of this field?

Then why are we leaving it to possibly rot?

It's up to us to force the changes—expert systems backing up human beings, looking over these arguments that can net people millions of dollars.

But most mathematicians today focus on what they call "pure math".

It is distinguished by being impractical and uncheckable except by other mathematicians.

There is no real world test of the work.

And I have been able to prove errors, show them to mathematicians, and watch them just walk away.

I had a paper published in a formally peer reviewed mathematical journal, a small electronic one, but hey, it was a legitimate math journal. I say was as the journal died.

A little while after publishing my paper readers of sci.math heard about it, some of them emailed the journal convincing the editors my paper was wrong, and they yanked it:

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

That link is to a site mirror, which is the last one still standing as there used to be somewhere around 10, but the rest dropped when the main site dropped, when the journal just shut down.

I posted a bit earlier about a letter I wrote to TIME magazine in the run-up to war with Iraq, and talked about how the editors changed what I said, toning it down in places, and removing some of the analysis, which isn't a big surprise, for those who know something about how the world works.

My own degree is in physics as I have a B.Sc. from Vanderbilt University.

I have an open source project called Class Viewer and you can find it easily enough by doing a google search on "Class Viewer" as it comes up number one.

That's just a slice.

Oh, one other thing, I wrote the first prime counting function article for the Wikipedia, because I was frustrated by mathematicians lying about my prime number research, and that writing is now in the history of the page:

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

I have been searching for over three years for a simple solution to the factoring problem as the means of proving that mathematicians do lie.

I think most of you though will not care about facts that don't matter to you because you have your beliefs, and I know that calling yourself a skeptic is nothing like actually being one.

But this time, reality can cost you money.

The system is at fault when it comes to mathematicians as the world gave a bunch of people a system where they needed each other above anyone else, and could not be checked except by each other, so they learned to value the opinion of each other—above all else.

They protect each other. Their lies about mathematical research put paychecks in each other's pockets. They know who they need and depend on.

And it's not you, if you're not a mathematician.

The mathematicians are victims in a sense of a world that should have known better.

Today computers offer the means to have even "pure" results that have no practical value, checked.

Computers can check mathematical arguments claimed to be proofs, when the work is done to build the expert systems.

That work is way behind now, and I suggest to you it is behind because mathematicians are invested in their current system, so they block, halting research in computer science if they can that would lead to proof checking by computer.

And they will sit on my factoring research until there is no way for them to sit.

And the pressure may be your accounts broken into, or your company's, or someone you know who gets victimized in some way, until that happens well above the level that people can blame anything other than the actual system protecting the Internet.

Sit back. It could take a while. I know from experience.

I have major mathematical research going back over three years.

These people can sit quietly, and wait. They'll wait for you or someone like you to hurt, to feel pain, and cry out loud enough, for the world to pay attention.

So we wait with them.

[A reply to someone who wrote that he would never have guessed that James is not happy about all this.]

Who would be?

Think about it, you take the time to go out and figure out some major new thing—a major accomplishment—and then you find that math professors ignore your research for political reasons, wouldn't you be upset?

So your reward for all the effort and pain is more pain, as society sits by while some people abuse you and call you a crackpot?

And it's so stupid too that mathematicians get away with just having their WORD trusted in these 'pure math' areas, when they push those areas where you just trust, over the important ones where the damn mathematics has to actually work, as proven by being useful in the real world, like to build better atomic weaponry.

A little dark humor there, but if my research were about nuclear weapons there wouldn't be any debate, none of this "crackpot" nonsense, and people would be fallng all over themselves to make sure I wasn't some very angry person, looking for ways to fight an academic world gone bad.

Consider that people.

If my research were about building better bombs there would not be this nonsense.

But because it's not, you can sit back and think it ok for some major discoverer to get punished for success, and what I say here is just ranting to you, comfortable where you might be.

And no computers able to check.

Think about it. If computer checking were around I could just put my resarch through it and all these people who fight my research would be undone, just like that.

Pure intellectual pursuits have this little problem that if people want to ignore you, they can.

But even with that, why do you people trust mathematicians in these "pure" areas, so that research like that of Andrew Wiles is just looked at by some people who claim it's correct, and you have no other way to check?

How can so many people around the world be so trusting given what we know about how readily even large groups of people—yes, math professors are human and can lie as a group—can just get things wrong?

Why do mathematicians get this blind trust in this day and age?

The problem is the possibility—which rational adults will acknowledge—that given a lack of outside checking, mathematicians can as a group miss errors in mathematical arguments that are just looked over by people, as people can make mistakes.

I think that many who recognize that mathematics is such an important area where we could not have the technology that we do without it, assume that there is some kind of other checking that goes on when they see in the news that this or that mathematician has a major discovery.

But the mathematics needed to get a working computer, or to make planes, or build a laser is different from the mathematics where some person just writes something up, for some other people to look at, and they all call it "pure math".

We as a society simply cannot just hope—hope that human nature is somehow purified in mathematics so that people can be trusted.

Did any of you know that Nobel did not create a math prize, so the people who run the Nobel prizes recently created one they call the Abel prize?

Any of you know how much that prize is?

Yet people can win with work that has no applicability in the real world, with research that has only been checked by human eyes, when their colleagues—other mathematicians—are the ones doing the checking.

And I'm not just griping here where there is no solution.

I firmly believe that computers can be programmed to check mathematical arguments.

It is up to us as a world community to push the development.

And it is past time to act like we are intelligent adults who value our mathematical world.

Can any of you rationally deny the importance of this field?

Then why are we leaving it to possibly rot?

It's up to us to force the changes—expert systems backing up human beings, looking over these arguments that can net people millions of dollars.

### Sunday, June 25, 2006

## Still a puzzle to me, my crackpot label

Not surprisingly to me as I've learned just how deep corruption runs in the modern math community, people who are a danger to that community, in terms of outing them, are either ignored—if that is possible—or they are labeled with titles like crank or crackpot.

That's not a surprise.

What has surprised me is that it doesn't matter what your real world accomplishments are, or how intelligent you've proven yourself to be, or how rational you sound, like in posts like this one, the label seems to be all-powerful.

I have yet to see anyone beat it.

Like consider Dr. Halton Arp in another field, where he is a distinguished researcher who was an assistant to Hubble himself, and he is well-known for very valid and accepted research—but he diverges from the bulk of astronomers in one key area, and they call him a crackpot.

I looked over his research in that key area, and it is sound. Yes, he has some other ideas that I think are nonsense that are related, but his primary points are solid.

Or coming back to mathematics consider Anatoly Plotnikov with a published paper that to my understanding, if correct, proves P=NP.

Did you know there is a million dollar prize for proving what he claims to have proven, back in 1998?

I have seen no refutations of his work. The Clay Institute can ignore it though because by their rules, what they will accept must have been published in a major journal.

Democracy.

It's built-in to the system.

Believe that Dr. Arp is just a nut? Think Plotnikov is just some deluded mathematician who thinks he solved P=NP?

What if Arp is right and Plotnikov was right, back in 1998?

I AM a skeptic. I see a lot of things that people say that I know are crap, or I should say, I firmly believe they are crap. But I did once pride myself on my own ability to convince people.

I do so no longer.

This group thing over research results is too powerful, even for me.

I can talk about my research, and even get people who seem convinced, but they do nothing.

Did you know I have an open source project?

It is a tool for Java developers that is a quick reference for information about classes.

It's used all over the world.

But some nobody on sci.math with no credentials or known accomplishments can reply to me calling me an idiot, talking about how my mathematics is crap, and that person is going to get more mileage than I will, no matter how carefully and thoroughly I explain.

What's wrong with this world?

Why do the people who lie and pretend to be something keep winning over the people who can actually accomplish things and have demonstrated it over and over again?

Andrew Wiles from what I've managed to deduce, has few, if any actual mathematical accomplishments.

So in terms of real mathematical ability, he is probably very low.

But clearly he has become master of a game that is about style over substance so right now he is internationally famous as supposedly one of the greatest mathematicians of all time.

But reality has a way of balancing things out.

I read about global warming and I get a sense of what's coming, when this easy life that we've had for so long begins to fade, and as intelligence becomes ever more crucial, lots of people will die.

And they will die ignorantly, unbelieving, calling out to people they trust, or to God.

Begging for rain, or for the rain to stop. Scrounging for food, even eating each other, all the time holding on to stupid and wrong beliefs that will not save them.

And as those millions of people die, this time, and these times, when people who should have been listened to, were not, won't even register in their minds.

But I plan on being prepared and surviving, thinking back later, to people like you, who had fatally wrong beliefs.

You bet on the wrong horse, so you died.

That's an epitaph I've already planned as my general one, for so many graves.

That's not a surprise.

What has surprised me is that it doesn't matter what your real world accomplishments are, or how intelligent you've proven yourself to be, or how rational you sound, like in posts like this one, the label seems to be all-powerful.

I have yet to see anyone beat it.

Like consider Dr. Halton Arp in another field, where he is a distinguished researcher who was an assistant to Hubble himself, and he is well-known for very valid and accepted research—but he diverges from the bulk of astronomers in one key area, and they call him a crackpot.

I looked over his research in that key area, and it is sound. Yes, he has some other ideas that I think are nonsense that are related, but his primary points are solid.

Or coming back to mathematics consider Anatoly Plotnikov with a published paper that to my understanding, if correct, proves P=NP.

Did you know there is a million dollar prize for proving what he claims to have proven, back in 1998?

I have seen no refutations of his work. The Clay Institute can ignore it though because by their rules, what they will accept must have been published in a major journal.

Democracy.

It's built-in to the system.

Believe that Dr. Arp is just a nut? Think Plotnikov is just some deluded mathematician who thinks he solved P=NP?

What if Arp is right and Plotnikov was right, back in 1998?

I AM a skeptic. I see a lot of things that people say that I know are crap, or I should say, I firmly believe they are crap. But I did once pride myself on my own ability to convince people.

I do so no longer.

This group thing over research results is too powerful, even for me.

I can talk about my research, and even get people who seem convinced, but they do nothing.

Did you know I have an open source project?

It is a tool for Java developers that is a quick reference for information about classes.

It's used all over the world.

But some nobody on sci.math with no credentials or known accomplishments can reply to me calling me an idiot, talking about how my mathematics is crap, and that person is going to get more mileage than I will, no matter how carefully and thoroughly I explain.

What's wrong with this world?

Why do the people who lie and pretend to be something keep winning over the people who can actually accomplish things and have demonstrated it over and over again?

Andrew Wiles from what I've managed to deduce, has few, if any actual mathematical accomplishments.

So in terms of real mathematical ability, he is probably very low.

But clearly he has become master of a game that is about style over substance so right now he is internationally famous as supposedly one of the greatest mathematicians of all time.

But reality has a way of balancing things out.

I read about global warming and I get a sense of what's coming, when this easy life that we've had for so long begins to fade, and as intelligence becomes ever more crucial, lots of people will die.

And they will die ignorantly, unbelieving, calling out to people they trust, or to God.

Begging for rain, or for the rain to stop. Scrounging for food, even eating each other, all the time holding on to stupid and wrong beliefs that will not save them.

And as those millions of people die, this time, and these times, when people who should have been listened to, were not, won't even register in their minds.

But I plan on being prepared and surviving, thinking back later, to people like you, who had fatally wrong beliefs.

You bet on the wrong horse, so you died.

That's an epitaph I've already planned as my general one, for so many graves.

## intelligence failures, Iraq, RSA

I read with interest yet another news story about the intelligence failures in the US leading up to war with Iraq:

"New details on WMD 'fabricator' emerge"

"Warnings that Iraqi was lying about bioweapons ignored, ex-CIA aide says"

http://www.msnbc.msn.com/id/13493736/from/RS.4/

Now I remember that period around Colin Powell's speech, and I remember thinking that it didn't sound credible to me, just on what was already publicly known at that time.

But lots of people believed it.

Now I have some math ideas that I say may simply solve what's called the factoring problem.

I just go ahead and put up the math as well.

Trouble for some of you gullible people who think you are skeptics is that the people you rely on for their experties are completely shown to be fauds by my research and their reaction to it.

For years they have painted me as a crackpot and ignored my mathematical findings.

I see no reason to suspect they will not do so here.

Frustrated with this situation I have been pursuing a solution to the factoring problem for over three years now, with a lot of cases where I was wrong, but was more certain than I am here.

If the math community were doing its job then there wouldn't be what I see as political speeches—little different from Powell at the UN—from both sides.

I have my little political speeches. Math people replying to me have theirs.

And you can trust that I am wrong, like I've been wrong before, but if I'm right, then you may lose your life savings, along with a lot of other people, or worse.

One thing for sure, if I am right, then this already is a massive intelligence failure as SOMEONE in the world's cryptographic community should have stepped up by now, so I say that hopefully this idea is wrong again.

But I can put it up so easily:

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

now multiply both out, which gives

S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and

T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and now subtract one from the other for one result, and add one to the other for another:

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

so you can now go and find all the variables. Like if S=15, and T is your public key used to secure some Internet transaction.

Then xy is the square of a factor of S-T. So you pick x and y using that criteria.

You divide sqrt(xy) from both sides giving your first equation:

(S - T)/sqrt(xy) = 2*(k_2*k_3 + k_1*k_4)

and now you plug in your x and y into the second equation, which leaves you with 4 variables, the k's.

Well, you pick two to solve for, like k_1 and k_2, or k_3 and k_4, and after solving for them, you pick integers for the others such that all are integers, which is the vaguest part of this method at this point—as I haven't solved out the equations yet.

If you think that is the hole here, you may be right, but it's not hard to do. I just have not done it in a post.

But what if it IS easy, and this method works, and the math community is full of frauds as I say?

Then your life may change irrevocably in a few days, when you had the information in front of you, couod have acted, even thought you were a skeptic, but did nothing, just like so many people in the run-up to war with Iraq.

But I did something back then as I sent an email to TIME magazine, and they published a version of it in a November 2002:

http://www.time.com/time/archive/preview/0,10987,1003625,00.html

You can't see the letter there without paying up, so I'll go ahead and post what TIME printed, and then tell you what I sent them:

"Your report on the weapons that the U.S. could use in a war with Iraq [WORLD, Oct. 21] noted that Iraq's best tactic would be to deploy weapons of mass destruction. While Saddam Hussein used chemical weapons against Iran, today his troops would have trouble getting close enough to deliver them. So what would be a possible Iraqi gambit? If the U.S. began military operations to soften up Iraq, Saddam would quickly ask the U.N. to send in weapons inspectors. He would then show the inspection team he doesn't have any weapons of mass destruction. There would be an international outcry to lift the sanctions and force the U.S. to pay reparations for any damage done. The U.S. needs the inspectors to go in before we attack. JAMES HARRIS Atlanta"

That's what they printed in a heavily edited piece.

What I sent was that Saddam Hussein had probably figured out from his war with Iran that chemical weapons were not decisive, and that it was quite possible he did not have any because he realized they were useless.

I did note that part of that was worrying about getting close enough to use them against our troops.

So I outlined what I thought at that time was a possible way Iraq could out-fox us, which was to wait till war started, get some people in to show they didn't have NBC weapons, and try to use that politically.

I even mentioned that Bush might be accused of war crimes, in that part where I talked about an international outcry, and the US being forced to pay war reparations.

Now I am just some guy who has ideas, many of them wrong, and I read the news like lots of people, but how many were motivated to send warnings, and got published?

At that time many people in this "land of the free, and home of the brave" were terrified to talk out against going to war, and if they didn't know what might happen to them if they did, they had examples like the Dixie Chicks to show them exactly what could happen.

(Thankfully some people did try anyway, and marched and did other protests but, of course, were not listened to either.)

People do stupid things. In groups lots of people do stupid things.

You can sit back, not check, and suppose I'm wrong, and tomorrow you may get blown up in some terrorist plot where terrorists used information that people who are supposed to protect us, are ignoring.

So? Life will go on. The planet will still keep spinning.

But you will be dead. And who really will care? Maybe some family if they survive? You might get a news blurb?

But does any of it really matter? Does it really matter how many of you reading this today may be dead in the next few months?

More and more I'm beginning to think it does not because people make their choices, and the consequences from one perspective are then, inevitable.

[A reply to someone who spotted a hole in James' method.]

That doesn't sound like a hole to me.

It sounds like you're making—a political post.

The politics here are brutal. I say the math community is full of frauds who deliberately ignore valid mathematical methods, like my new factoring idea.

They ARE frauds, so you see replies that have no mathematical value.

And yes, I have failed many times at finding a solution to the factoring problem, after claiming success.

I may be wrong here.

But I'm an actual researcher—versus being a politician—in a nightmare situation where I've found out that mathematicians routinely lie about mathematics, so that I've pursued a problem with major financial implications to prove that beyond any doubt.

Previous attempts have failed. I don't think this one does.

But the answer to me is not to put my failed history, or to make a political ad as if this were an American presidential campaign, instead of a question of whether or not a particular method works.

The answer is to objectively go over the mathematics shown.

If you see that and it shows I'm wrong, then fine.

But when you don't see that understand that's because the math community is not what it claims to be, so it can't behave as expected here.

Since they are not really mathematicians, members of that community can't objectively shoot down my mathematical results.

Which is why they rely so much on ridicule, politics, and playing to your naivette.

Think back to that letter I wrote to TIME. Some of you should be a little surprised at how heavily they edited. And why exactly did they throw out the part about Bush being accused of war crimes?

Welcome to the real world people. Your refusal to accept the truth can get you killed here.

That's the math community I know.

These people are frauds. They relied on a system where they could not

be easily checked and came up with a security method that is fatally

flawed.

They are cons so when caught what do they do?

What do ANY cons do?

They just play the same game that much harder.

These people will let other people die to protect their game, like any other cons.

[Another reply.]

And that is math society.

Those of you who bothered to check may be surprised to find how often people you think have been refuted mathematically, like me, were instead simply called names, and their work was simply called false.

They are lazy because people believe them.

Mathematicians don't have to argue. They just pronounce other people to be wrong.

Notice this poster also chose to take out information that shows you I'm not just your ordinary person mouthing off, as I can talk about a version of an email that I sent to TIME magazine before the Iraq war where I sound like I am a better analyst then the entire US intelligence community—using published information.

TIME magazine heavily edited my letter taking out some controversial things, and trashing the analysis where I considered the possibility that having found chemical weapons to be non-decisive against Iran, Saddam Hussein had no reason to hoard them.

And they took out the part where I cautioned against Bush being accused of war crimes.

Why?

Any of you have a clue? Why would a major magazine take out controversial pieces when this country was taking such a major step as to go to war when the consequences for screwing up could be so huge?

They thought that Bush and company wouldn't screw up, that's why.

They figured someway, somehow those powerful people in that important office would figure out a way to make it work out ok, no matter what.

I think they have watched too many TV shows and movies where the "hero" always wins.

And they decided Bush was the hero because he is the American president.

I say, you people do the same thing—maybe having watched too many TV shows—and you think the mathematicians are the heroes and can't lose.

I say they are the cons, and they are losing right now.

These discussions are a sideshow. The real action may be happening on your computer right now while you don't know it.

Who knows who is peering into your world, right now.

"New details on WMD 'fabricator' emerge"

"Warnings that Iraqi was lying about bioweapons ignored, ex-CIA aide says"

http://www.msnbc.msn.com/id/13493736/from/RS.4/

Now I remember that period around Colin Powell's speech, and I remember thinking that it didn't sound credible to me, just on what was already publicly known at that time.

But lots of people believed it.

Now I have some math ideas that I say may simply solve what's called the factoring problem.

I just go ahead and put up the math as well.

Trouble for some of you gullible people who think you are skeptics is that the people you rely on for their experties are completely shown to be fauds by my research and their reaction to it.

For years they have painted me as a crackpot and ignored my mathematical findings.

I see no reason to suspect they will not do so here.

Frustrated with this situation I have been pursuing a solution to the factoring problem for over three years now, with a lot of cases where I was wrong, but was more certain than I am here.

If the math community were doing its job then there wouldn't be what I see as political speeches—little different from Powell at the UN—from both sides.

I have my little political speeches. Math people replying to me have theirs.

And you can trust that I am wrong, like I've been wrong before, but if I'm right, then you may lose your life savings, along with a lot of other people, or worse.

One thing for sure, if I am right, then this already is a massive intelligence failure as SOMEONE in the world's cryptographic community should have stepped up by now, so I say that hopefully this idea is wrong again.

But I can put it up so easily:

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

now multiply both out, which gives

S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and

T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and now subtract one from the other for one result, and add one to the other for another:

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

so you can now go and find all the variables. Like if S=15, and T is your public key used to secure some Internet transaction.

Then xy is the square of a factor of S-T. So you pick x and y using that criteria.

You divide sqrt(xy) from both sides giving your first equation:

(S - T)/sqrt(xy) = 2*(k_2*k_3 + k_1*k_4)

and now you plug in your x and y into the second equation, which leaves you with 4 variables, the k's.

Well, you pick two to solve for, like k_1 and k_2, or k_3 and k_4, and after solving for them, you pick integers for the others such that all are integers, which is the vaguest part of this method at this point—as I haven't solved out the equations yet.

If you think that is the hole here, you may be right, but it's not hard to do. I just have not done it in a post.

But what if it IS easy, and this method works, and the math community is full of frauds as I say?

Then your life may change irrevocably in a few days, when you had the information in front of you, couod have acted, even thought you were a skeptic, but did nothing, just like so many people in the run-up to war with Iraq.

But I did something back then as I sent an email to TIME magazine, and they published a version of it in a November 2002:

http://www.time.com/time/archive/preview/0,10987,1003625,00.html

You can't see the letter there without paying up, so I'll go ahead and post what TIME printed, and then tell you what I sent them:

"Your report on the weapons that the U.S. could use in a war with Iraq [WORLD, Oct. 21] noted that Iraq's best tactic would be to deploy weapons of mass destruction. While Saddam Hussein used chemical weapons against Iran, today his troops would have trouble getting close enough to deliver them. So what would be a possible Iraqi gambit? If the U.S. began military operations to soften up Iraq, Saddam would quickly ask the U.N. to send in weapons inspectors. He would then show the inspection team he doesn't have any weapons of mass destruction. There would be an international outcry to lift the sanctions and force the U.S. to pay reparations for any damage done. The U.S. needs the inspectors to go in before we attack. JAMES HARRIS Atlanta"

That's what they printed in a heavily edited piece.

What I sent was that Saddam Hussein had probably figured out from his war with Iran that chemical weapons were not decisive, and that it was quite possible he did not have any because he realized they were useless.

I did note that part of that was worrying about getting close enough to use them against our troops.

So I outlined what I thought at that time was a possible way Iraq could out-fox us, which was to wait till war started, get some people in to show they didn't have NBC weapons, and try to use that politically.

I even mentioned that Bush might be accused of war crimes, in that part where I talked about an international outcry, and the US being forced to pay war reparations.

Now I am just some guy who has ideas, many of them wrong, and I read the news like lots of people, but how many were motivated to send warnings, and got published?

At that time many people in this "land of the free, and home of the brave" were terrified to talk out against going to war, and if they didn't know what might happen to them if they did, they had examples like the Dixie Chicks to show them exactly what could happen.

(Thankfully some people did try anyway, and marched and did other protests but, of course, were not listened to either.)

People do stupid things. In groups lots of people do stupid things.

You can sit back, not check, and suppose I'm wrong, and tomorrow you may get blown up in some terrorist plot where terrorists used information that people who are supposed to protect us, are ignoring.

So? Life will go on. The planet will still keep spinning.

But you will be dead. And who really will care? Maybe some family if they survive? You might get a news blurb?

But does any of it really matter? Does it really matter how many of you reading this today may be dead in the next few months?

More and more I'm beginning to think it does not because people make their choices, and the consequences from one perspective are then, inevitable.

[A reply to someone who spotted a hole in James' method.]

That doesn't sound like a hole to me.

It sounds like you're making—a political post.

The politics here are brutal. I say the math community is full of frauds who deliberately ignore valid mathematical methods, like my new factoring idea.

They ARE frauds, so you see replies that have no mathematical value.

And yes, I have failed many times at finding a solution to the factoring problem, after claiming success.

I may be wrong here.

But I'm an actual researcher—versus being a politician—in a nightmare situation where I've found out that mathematicians routinely lie about mathematics, so that I've pursued a problem with major financial implications to prove that beyond any doubt.

Previous attempts have failed. I don't think this one does.

But the answer to me is not to put my failed history, or to make a political ad as if this were an American presidential campaign, instead of a question of whether or not a particular method works.

The answer is to objectively go over the mathematics shown.

If you see that and it shows I'm wrong, then fine.

But when you don't see that understand that's because the math community is not what it claims to be, so it can't behave as expected here.

Since they are not really mathematicians, members of that community can't objectively shoot down my mathematical results.

Which is why they rely so much on ridicule, politics, and playing to your naivette.

Think back to that letter I wrote to TIME. Some of you should be a little surprised at how heavily they edited. And why exactly did they throw out the part about Bush being accused of war crimes?

Welcome to the real world people. Your refusal to accept the truth can get you killed here.

That's the math community I know.

These people are frauds. They relied on a system where they could not

be easily checked and came up with a security method that is fatally

flawed.

They are cons so when caught what do they do?

What do ANY cons do?

They just play the same game that much harder.

These people will let other people die to protect their game, like any other cons.

[Another reply.]

And that is math society.

Those of you who bothered to check may be surprised to find how often people you think have been refuted mathematically, like me, were instead simply called names, and their work was simply called false.

They are lazy because people believe them.

Mathematicians don't have to argue. They just pronounce other people to be wrong.

Notice this poster also chose to take out information that shows you I'm not just your ordinary person mouthing off, as I can talk about a version of an email that I sent to TIME magazine before the Iraq war where I sound like I am a better analyst then the entire US intelligence community—using published information.

TIME magazine heavily edited my letter taking out some controversial things, and trashing the analysis where I considered the possibility that having found chemical weapons to be non-decisive against Iran, Saddam Hussein had no reason to hoard them.

And they took out the part where I cautioned against Bush being accused of war crimes.

Why?

Any of you have a clue? Why would a major magazine take out controversial pieces when this country was taking such a major step as to go to war when the consequences for screwing up could be so huge?

They thought that Bush and company wouldn't screw up, that's why.

They figured someway, somehow those powerful people in that important office would figure out a way to make it work out ok, no matter what.

I think they have watched too many TV shows and movies where the "hero" always wins.

And they decided Bush was the hero because he is the American president.

I say, you people do the same thing—maybe having watched too many TV shows—and you think the mathematicians are the heroes and can't lose.

I say they are the cons, and they are losing right now.

These discussions are a sideshow. The real action may be happening on your computer right now while you don't know it.

Who knows who is peering into your world, right now.

### Saturday, June 24, 2006

## JSH: Matter of order

One pure math remarkable thing about the expressions

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

is that they are the SAME expression, in terms of values, as what's to the right side of the equals in both cases has two values because of the square roots, but the order is what's different.

So if you work out all the variables, remarkably, you don't know why one will give what, so taking the negative of sqrt(y) in one case may give you S, but with another set of numbers it may give you T.

So there is this single expression, where you can reverse its order to show its two values, and then go forward with your analysis, which is what I finally figured out yesterday, allowing for the solution to how to pick all the variables, which I've posted in other threads.

It looks like two equations, but it's the same equation.

I can kind of understand why mathematicians wanted to wish away this kind of complexity and declare the square root function to be single-valued, but, where's the pursuit of truth in that?

Yes, mathematics can be difficult and hard to understand, but arbitrary simplification is, well, it's just stupid.

And I know I just proclaimed some people many of you think to be great to be stupid, but they were.

They thought to put human preference over mathematical logic.

And to me that is just stupid.

And consider how remarkable real mathematics is!!!

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

Seemingly two equations, but mathematically they are just one, with two faces, and that's just a way to show both faces at once, by shifting the order in which each shows.

It is a two-faced equation. A bit of the absolute that cannot be changed to be something it is not, no matter how many people think math is a democracy.

After all, soon enough—far less than even a million years—you will

all be dead, and humanity won't even be a memory.

But those equations will still have the same properties, beyond time mathematics is.

Mathematics is from the absolute.

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

is that they are the SAME expression, in terms of values, as what's to the right side of the equals in both cases has two values because of the square roots, but the order is what's different.

So if you work out all the variables, remarkably, you don't know why one will give what, so taking the negative of sqrt(y) in one case may give you S, but with another set of numbers it may give you T.

So there is this single expression, where you can reverse its order to show its two values, and then go forward with your analysis, which is what I finally figured out yesterday, allowing for the solution to how to pick all the variables, which I've posted in other threads.

It looks like two equations, but it's the same equation.

I can kind of understand why mathematicians wanted to wish away this kind of complexity and declare the square root function to be single-valued, but, where's the pursuit of truth in that?

Yes, mathematics can be difficult and hard to understand, but arbitrary simplification is, well, it's just stupid.

And I know I just proclaimed some people many of you think to be great to be stupid, but they were.

They thought to put human preference over mathematical logic.

And to me that is just stupid.

And consider how remarkable real mathematics is!!!

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

Seemingly two equations, but mathematically they are just one, with two faces, and that's just a way to show both faces at once, by shifting the order in which each shows.

It is a two-faced equation. A bit of the absolute that cannot be changed to be something it is not, no matter how many people think math is a democracy.

After all, soon enough—far less than even a million years—you will

all be dead, and humanity won't even be a memory.

But those equations will still have the same properties, beyond time mathematics is.

Mathematics is from the absolute.

## Lying from the math community

If you read my previous post then you know that my position is that mathematicians act as a community as cons, lying about mathematics.

Consider that today in our great modern age where computers dominate so many areas of our lives, they are not setup to check most mathematical arguments that people in the math community claim are proofs i.e. absolutely true.

Without computers to check we rely on the word of people who are part of the same cmmunity, with shared interests.

Millions of dollars in research money flow based on their word.

And I suggest to you that they lie.

Unfortunately such a huge charge against such an influential community is hard to prove, but I may now have the proof in the factoring problem.

Mathematicians proclaimed factoring to be difficult, and built a security system around that position.

I suggest to you they screwed up—because they are frauds, and not real mathematicians.

The nice thing about such a charge is that you do not have to believe me, but the unfortunate thing is that the proof may be your poverty.

If you have stocks, and think yourself wealthy now, but in a few days are broken and penniless, then there will be no question of proof.

So why make a post like this one?

I don't know, maybe I'm myself simply amazed that such a situation is possible so I find myself chatting up about it.

They ARE frauds, and have lied with claims of major proofs, while ignoring major research, and not just mine.

Think you know the math community?

Do a web search on Anatoly Plotnikov, a mathematician who may have proven before the rise of RSA that it was inherently insecure by proving something called P=NP.

The evidence is out there, and it's not just my say-so.

These people may have ignored his research to promote their community, to promote the idea that they could provide value, when what they provided was security based on ignorance, and blind trust.

Now, open your eyes.

Consider that today in our great modern age where computers dominate so many areas of our lives, they are not setup to check most mathematical arguments that people in the math community claim are proofs i.e. absolutely true.

Without computers to check we rely on the word of people who are part of the same cmmunity, with shared interests.

Millions of dollars in research money flow based on their word.

And I suggest to you that they lie.

Unfortunately such a huge charge against such an influential community is hard to prove, but I may now have the proof in the factoring problem.

Mathematicians proclaimed factoring to be difficult, and built a security system around that position.

I suggest to you they screwed up—because they are frauds, and not real mathematicians.

The nice thing about such a charge is that you do not have to believe me, but the unfortunate thing is that the proof may be your poverty.

If you have stocks, and think yourself wealthy now, but in a few days are broken and penniless, then there will be no question of proof.

So why make a post like this one?

I don't know, maybe I'm myself simply amazed that such a situation is possible so I find myself chatting up about it.

They ARE frauds, and have lied with claims of major proofs, while ignoring major research, and not just mine.

Think you know the math community?

Do a web search on Anatoly Plotnikov, a mathematician who may have proven before the rise of RSA that it was inherently insecure by proving something called P=NP.

The evidence is out there, and it's not just my say-so.

These people may have ignored his research to promote their community, to promote the idea that they could provide value, when what they provided was security based on ignorance, and blind trust.

Now, open your eyes.

## New way to factor?

I fear that mathematicians as a community are capable of lying to the public on a huge scale, and it's hard to show that as they protect each other, so I go to another mathematically sophisticated community to present what looks to me like a way to factor which invalidates the current Internet security system.

You can easily check the math yourself, and then consider the charge that the math community is willfully ignoring this information—leaving you and many others vulnerable—because they are cons finally caught in a dangerous game, where they presented factoring as a hard problem, when the solution is what you can see here.

Let

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

now multiply both out, which gives

S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and

T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and now subtract one from the other for one result, and add one to the other for another:

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

and if you have an S number and a target composite T to factor, you have two equations with 6 other unknowns with which to determine values that will work.

It seems to me that letting xy be a square of a factor of S-T, might be useful, which can give you x and y as squares, then you have to pick two of the k's, and the final two are determined by the two equations.

AT the end of that bit of work you are mathematically guaranteed—by remarkably basic algebra—to have the factorization

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

of your target composite.

And there is nothing in the algebra that says that if T is your public key, this method will not work!

It's basic algebra.

If I am right, then mathematicians will do—nothing.

If they acknowledge my research they are outed as frauds as I am a vocal critic of that community with other major research which they have ignored.

But them doing nothing—if you do nothing as well—is just leaving the door open for someone to do something with the research.

It's your choice.

Do something, or wait, in denial of your own mathematical ability, and see what happens.

You can easily check the math yourself, and then consider the charge that the math community is willfully ignoring this information—leaving you and many others vulnerable—because they are cons finally caught in a dangerous game, where they presented factoring as a hard problem, when the solution is what you can see here.

Let

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

now multiply both out, which gives

S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and

T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and now subtract one from the other for one result, and add one to the other for another:

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

and if you have an S number and a target composite T to factor, you have two equations with 6 other unknowns with which to determine values that will work.

It seems to me that letting xy be a square of a factor of S-T, might be useful, which can give you x and y as squares, then you have to pick two of the k's, and the final two are determined by the two equations.

AT the end of that bit of work you are mathematically guaranteed—by remarkably basic algebra—to have the factorization

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

of your target composite.

And there is nothing in the algebra that says that if T is your public key, this method will not work!

It's basic algebra.

If I am right, then mathematicians will do—nothing.

If they acknowledge my research they are outed as frauds as I am a vocal critic of that community with other major research which they have ignored.

But them doing nothing—if you do nothing as well—is just leaving the door open for someone to do something with the research.

It's your choice.

Do something, or wait, in denial of your own mathematical ability, and see what happens.

## SF: Progress, double factorization proven

For years I've wondered how one might use the factorization of one number to factor another, and I introduced the term surrogate factoring some years ago as a title for the process.

And for years I have completely failed at finding anything that can be made practical from the idea in terms of a factoring solution, but finally had what may be the key breakthrough by considering an expression which cannot mathematically represent a single factorization.

Remarkably that is achieved very easily by using square roots—where immediately I run into a convention issue as mathematicians long ago decided that the square root function having two values was inconvenient, so they defined it away!

They said that a function can have only one value, so most mathematicians look at square root functions as single-valued, while engineers and scientists use it as two-valued, but defer to the convention that when discussed as a function, it is single-valued.

This mindless human preference—contradiction with mathematical reality for human arrogance—possibly lead mathematicians to ignore a very obvious way to factor, which I will show here.

Let

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

now multiply both out, which gives

S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and

T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and now subtract one from the other for one result, and add one to the other for another:

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

and if you have an S number and a target composite T to factor, you have two equations with 6 other unknowns with which to determine values that will work.

It seems to me that letting xy be a square of a factor of S-T, might be useful, which can give you x and y as squares, then you have to pick two of the k's, and the final two are determined by the two equations.

AT the end of that bit of work you are mathematically guaranteed—by remarkably basic algebra—to have the factorization

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

of your taget composite.

And there is nothing in the algebra that says that if T is your public key, this method will not work!

But is there anything in there that says it will?

I don't know for sure, but I'm afraid it can be made to work.

And I have years of exprience with the reality of the math community to understand that they are quite capable of ignoring this research, even if it does.

And they are ignoring it now to my knowledge.

My position is that most mathematicians are not really researchers but are people who learned use of math-ese, to LOOK like they were doing important research, and learned that they could claim things were true which are not because people trusted them, and no outside source was there to check.

Note that computers are NOT used to check most claims of mathematical proof in the area of "pure math" so these people can safely rely on the word of each other, in a field that somehow has escaped major cases of reported fraud!!!

I suggest to you that the reality is that the field is instead dominated by fraud, and without objective checking, like by computers, people in the math field can simply claim to have proofs, have other mathematicians back them up, and not have proofs, in a world that can't tell the truth.

And being frauds, with it outed that factoring is not a hard problem, and with the technique revealed that shows they screwed up with the security system for the Internet and wireless communication, what can they do?

Think about it. What would you do if you were a math con and the story got out?

But it's not out, is it? I'm just a "crackpot" mouthing off on Usenet, on the fringe, and being frauds that math community's members can feel safe because they're not real mathematicians.

They figure if I'm just out there on the fringe, maybe no one will notice that the security system of the Internet—does not work.

But there are probably going to be people who will reply with nothing of real mathematical interest as unfortunately the math community is corrupted.

How did this happen?

Well think people! Mathematicians who work in "pure math" areas only need to convince other mathematicians that what they have is correct, as who else is in a position to check?

If their research has no real world value, like in building more fuel efficient engines for cars, how do we really know if it actually is correct?

Knowing they only have each other to convince the society has become a social one, where politics rule, and like politicians everywhere, they lie.

To them lying is just part of being a modern mathematician. It's how you play the game, get funding, get a professorship.

The people who don't lie, like politicians who don't, don't get ahead.

But they picked a practical area with factoring, and here politics don't work.

Their social world ran up against a real world problem, and failed.

[A reply to someone who examined James' method and asked whether he made mistakes.]

I don't know. And I don't care.

You are a minor player in an area where if I'm right, far more intelligent people with a lot more expertise are already working on these equations, so you are probably way, way, way behind, and your input is irrelevant.

If they don't work, who cares? Seems to me that you seem to think they do work, but inefficiently, which means either you made a mistake, or better researchers CAN make these work, and, well, then it's the worst case scenario.

For years I've wondered how one might use the factorization of one number to factor another, and I introduced the term surrogate factoring some years ago as a title for the process.

And for years I have completely failed at finding anything that can be made practical from the idea in terms of a factoring solution, but finally had what may be the key breakthrough by considering an expression which cannot mathematically represent a single factorization.

Remarkably that is achieved very easily by using square roots—where immediately I run into a convention issue as mathematicians long ago decided that the square root function having two values was inconvenient, so they defined it away!

They said that a function can have only one value, so most mathematicians look at square root functions as single-valued, while engineers and scientists use it as two-valued, but defer to the convention that when discussed as a function, it is single-valued.

This mindless human preference—contradiction with mathematical reality for human arrogance—possibly lead mathematicians to ignore a very obvious way to factor, which I will show here.

Let

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

now multiply both out, which gives

S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and

T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and now subtract one from the other for one result, and add one to the other for another:

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

and if you have an S number and a target composite T to factor, you have two equations with 6 other unknowns with which to determine values that will work.

It seems to me that letting xy be a square of a factor of S-T, might be useful, which can give you x and y as squares, then you have to pick two of the k's, and the final two are determined by the two equations.

AT the end of that bit of work you are mathematically guaranteed—by remarkably basic algebra—to have the factorization

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

of your taget composite.

And there is nothing in the algebra that says that if T is your public key, this method will not work!

But is there anything in there that says it will?

I don't know for sure, but I'm afraid it can be made to work.

And I have years of exprience with the reality of the math community to understand that they are quite capable of ignoring this research, even if it does.

And they are ignoring it now to my knowledge.

My position is that most mathematicians are not really researchers but are people who learned use of math-ese, to LOOK like they were doing important research, and learned that they could claim things were true which are not because people trusted them, and no outside source was there to check.

Note that computers are NOT used to check most claims of mathematical proof in the area of "pure math" so these people can safely rely on the word of each other, in a field that somehow has escaped major cases of reported fraud!!!

I suggest to you that the reality is that the field is instead dominated by fraud, and without objective checking, like by computers, people in the math field can simply claim to have proofs, have other mathematicians back them up, and not have proofs, in a world that can't tell the truth.

And being frauds, with it outed that factoring is not a hard problem, and with the technique revealed that shows they screwed up with the security system for the Internet and wireless communication, what can they do?

Think about it. What would you do if you were a math con and the story got out?

But it's not out, is it? I'm just a "crackpot" mouthing off on Usenet, on the fringe, and being frauds that math community's members can feel safe because they're not real mathematicians.

They figure if I'm just out there on the fringe, maybe no one will notice that the security system of the Internet—does not work.

But there are probably going to be people who will reply with nothing of real mathematical interest as unfortunately the math community is corrupted.

How did this happen?

Well think people! Mathematicians who work in "pure math" areas only need to convince other mathematicians that what they have is correct, as who else is in a position to check?

If their research has no real world value, like in building more fuel efficient engines for cars, how do we really know if it actually is correct?

Knowing they only have each other to convince the society has become a social one, where politics rule, and like politicians everywhere, they lie.

To them lying is just part of being a modern mathematician. It's how you play the game, get funding, get a professorship.

The people who don't lie, like politicians who don't, don't get ahead.

But they picked a practical area with factoring, and here politics don't work.

Their social world ran up against a real world problem, and failed.

[A reply to someone who said that the whole problem is about finding factors

If you look for integer k's, then there are a FINITE number of possibles.

There is no way to limit the possibilities with other methods.

So, like with an RSA key, someone with these ideas can absolutely guarantee factoring the key if they loop through all possible integer k's, but there is no such guarantee possible in a reasonable space, with any other known method.

Besides, you people talk as if finding new factoring methods was an everyday thing.

Ok, yes, for me, sure I can find new factoring methods easily, and up until now, none have been practical.

So maybe you've forgotten that in the real world, there are maybe 4 or 5 factoring methods that have been come up with by people besides me in the history of humanity.

[A reply to someone who wanted to know what qualifies James to describe the other poster as a minor player in a field where James is not respected at all.]

You really think this is about respect?

It's about the mathematics.

There is only ONE way those equations can't be important.

That one way is if you need to know the factorization of T ahead of time to get that finite set of integer values.

If you don't, then that's it. It's over. RSA is finished. The company will soon go bankrupt and the entire Internet will be changed.

Companies will die, while new ones may be born.

Billions of dollars and then trillions of dollars will shift around the world.

Something on this scale is so huge you cannot begin to comprehend it, and yes, you people are less than bit players on a stage where presidents and other heads of state may soon get phone calls telling them of the news.

All you people are good enough for is to show this to be wrong, but it's not, so you have no real role here.

I'm mostly just posting now to hear myself talk. None of you are actually important at this point.

You have no impact. No ability to greatly affect things one way or another.

You are just flotsam dragged along with the current in something that is huger than anything civilization has ever faced before.

The lives of millions in the balance? Yup. So yeah, you're a nobody on that scale.

None of you on sci.math actually matter any more.

Your role is done.

And for years I have completely failed at finding anything that can be made practical from the idea in terms of a factoring solution, but finally had what may be the key breakthrough by considering an expression which cannot mathematically represent a single factorization.

Remarkably that is achieved very easily by using square roots—where immediately I run into a convention issue as mathematicians long ago decided that the square root function having two values was inconvenient, so they defined it away!

They said that a function can have only one value, so most mathematicians look at square root functions as single-valued, while engineers and scientists use it as two-valued, but defer to the convention that when discussed as a function, it is single-valued.

This mindless human preference—contradiction with mathematical reality for human arrogance—possibly lead mathematicians to ignore a very obvious way to factor, which I will show here.

Let

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

now multiply both out, which gives

S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and

T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and now subtract one from the other for one result, and add one to the other for another:

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

and if you have an S number and a target composite T to factor, you have two equations with 6 other unknowns with which to determine values that will work.

It seems to me that letting xy be a square of a factor of S-T, might be useful, which can give you x and y as squares, then you have to pick two of the k's, and the final two are determined by the two equations.

AT the end of that bit of work you are mathematically guaranteed—by remarkably basic algebra—to have the factorization

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

of your taget composite.

And there is nothing in the algebra that says that if T is your public key, this method will not work!

But is there anything in there that says it will?

I don't know for sure, but I'm afraid it can be made to work.

And I have years of exprience with the reality of the math community to understand that they are quite capable of ignoring this research, even if it does.

And they are ignoring it now to my knowledge.

My position is that most mathematicians are not really researchers but are people who learned use of math-ese, to LOOK like they were doing important research, and learned that they could claim things were true which are not because people trusted them, and no outside source was there to check.

Note that computers are NOT used to check most claims of mathematical proof in the area of "pure math" so these people can safely rely on the word of each other, in a field that somehow has escaped major cases of reported fraud!!!

I suggest to you that the reality is that the field is instead dominated by fraud, and without objective checking, like by computers, people in the math field can simply claim to have proofs, have other mathematicians back them up, and not have proofs, in a world that can't tell the truth.

And being frauds, with it outed that factoring is not a hard problem, and with the technique revealed that shows they screwed up with the security system for the Internet and wireless communication, what can they do?

Think about it. What would you do if you were a math con and the story got out?

But it's not out, is it? I'm just a "crackpot" mouthing off on Usenet, on the fringe, and being frauds that math community's members can feel safe because they're not real mathematicians.

They figure if I'm just out there on the fringe, maybe no one will notice that the security system of the Internet—does not work.

But there are probably going to be people who will reply with nothing of real mathematical interest as unfortunately the math community is corrupted.

How did this happen?

Well think people! Mathematicians who work in "pure math" areas only need to convince other mathematicians that what they have is correct, as who else is in a position to check?

If their research has no real world value, like in building more fuel efficient engines for cars, how do we really know if it actually is correct?

Knowing they only have each other to convince the society has become a social one, where politics rule, and like politicians everywhere, they lie.

To them lying is just part of being a modern mathematician. It's how you play the game, get funding, get a professorship.

The people who don't lie, like politicians who don't, don't get ahead.

But they picked a practical area with factoring, and here politics don't work.

Their social world ran up against a real world problem, and failed.

[A reply to someone who examined James' method and asked whether he made mistakes.]

I don't know. And I don't care.

You are a minor player in an area where if I'm right, far more intelligent people with a lot more expertise are already working on these equations, so you are probably way, way, way behind, and your input is irrelevant.

If they don't work, who cares? Seems to me that you seem to think they do work, but inefficiently, which means either you made a mistake, or better researchers CAN make these work, and, well, then it's the worst case scenario.

For years I've wondered how one might use the factorization of one number to factor another, and I introduced the term surrogate factoring some years ago as a title for the process.

And for years I have completely failed at finding anything that can be made practical from the idea in terms of a factoring solution, but finally had what may be the key breakthrough by considering an expression which cannot mathematically represent a single factorization.

Remarkably that is achieved very easily by using square roots—where immediately I run into a convention issue as mathematicians long ago decided that the square root function having two values was inconvenient, so they defined it away!

They said that a function can have only one value, so most mathematicians look at square root functions as single-valued, while engineers and scientists use it as two-valued, but defer to the convention that when discussed as a function, it is single-valued.

This mindless human preference—contradiction with mathematical reality for human arrogance—possibly lead mathematicians to ignore a very obvious way to factor, which I will show here.

Let

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

now multiply both out, which gives

S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and

T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and now subtract one from the other for one result, and add one to the other for another:

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

and if you have an S number and a target composite T to factor, you have two equations with 6 other unknowns with which to determine values that will work.

It seems to me that letting xy be a square of a factor of S-T, might be useful, which can give you x and y as squares, then you have to pick two of the k's, and the final two are determined by the two equations.

AT the end of that bit of work you are mathematically guaranteed—by remarkably basic algebra—to have the factorization

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

of your taget composite.

And there is nothing in the algebra that says that if T is your public key, this method will not work!

But is there anything in there that says it will?

I don't know for sure, but I'm afraid it can be made to work.

And I have years of exprience with the reality of the math community to understand that they are quite capable of ignoring this research, even if it does.

And they are ignoring it now to my knowledge.

My position is that most mathematicians are not really researchers but are people who learned use of math-ese, to LOOK like they were doing important research, and learned that they could claim things were true which are not because people trusted them, and no outside source was there to check.

Note that computers are NOT used to check most claims of mathematical proof in the area of "pure math" so these people can safely rely on the word of each other, in a field that somehow has escaped major cases of reported fraud!!!

I suggest to you that the reality is that the field is instead dominated by fraud, and without objective checking, like by computers, people in the math field can simply claim to have proofs, have other mathematicians back them up, and not have proofs, in a world that can't tell the truth.

And being frauds, with it outed that factoring is not a hard problem, and with the technique revealed that shows they screwed up with the security system for the Internet and wireless communication, what can they do?

Think about it. What would you do if you were a math con and the story got out?

But it's not out, is it? I'm just a "crackpot" mouthing off on Usenet, on the fringe, and being frauds that math community's members can feel safe because they're not real mathematicians.

They figure if I'm just out there on the fringe, maybe no one will notice that the security system of the Internet—does not work.

But there are probably going to be people who will reply with nothing of real mathematical interest as unfortunately the math community is corrupted.

How did this happen?

Well think people! Mathematicians who work in "pure math" areas only need to convince other mathematicians that what they have is correct, as who else is in a position to check?

If their research has no real world value, like in building more fuel efficient engines for cars, how do we really know if it actually is correct?

Knowing they only have each other to convince the society has become a social one, where politics rule, and like politicians everywhere, they lie.

To them lying is just part of being a modern mathematician. It's how you play the game, get funding, get a professorship.

The people who don't lie, like politicians who don't, don't get ahead.

But they picked a practical area with factoring, and here politics don't work.

Their social world ran up against a real world problem, and failed.

[A reply to someone who said that the whole problem is about finding factors

**efficiently**.]If you look for integer k's, then there are a FINITE number of possibles.

There is no way to limit the possibilities with other methods.

So, like with an RSA key, someone with these ideas can absolutely guarantee factoring the key if they loop through all possible integer k's, but there is no such guarantee possible in a reasonable space, with any other known method.

Besides, you people talk as if finding new factoring methods was an everyday thing.

Ok, yes, for me, sure I can find new factoring methods easily, and up until now, none have been practical.

So maybe you've forgotten that in the real world, there are maybe 4 or 5 factoring methods that have been come up with by people besides me in the history of humanity.

[A reply to someone who wanted to know what qualifies James to describe the other poster as a minor player in a field where James is not respected at all.]

You really think this is about respect?

It's about the mathematics.

There is only ONE way those equations can't be important.

That one way is if you need to know the factorization of T ahead of time to get that finite set of integer values.

If you don't, then that's it. It's over. RSA is finished. The company will soon go bankrupt and the entire Internet will be changed.

Companies will die, while new ones may be born.

Billions of dollars and then trillions of dollars will shift around the world.

Something on this scale is so huge you cannot begin to comprehend it, and yes, you people are less than bit players on a stage where presidents and other heads of state may soon get phone calls telling them of the news.

All you people are good enough for is to show this to be wrong, but it's not, so you have no real role here.

I'm mostly just posting now to hear myself talk. None of you are actually important at this point.

You have no impact. No ability to greatly affect things one way or another.

You are just flotsam dragged along with the current in something that is huger than anything civilization has ever faced before.

The lives of millions in the balance? Yup. So yeah, you're a nobody on that scale.

None of you on sci.math actually matter any more.

Your role is done.

### Friday, June 23, 2006

## SF: Full solution?

Arguing with Tim Peters gave me an idea, as in one reply he deleted out my equations to put in something with T, and that got me to thinking.

What if I used

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))?

Multiply them out, subtract one from the other, and I think the correct way to get the k's might be there, but I STILL think there is a way going from just the solution to the equation with S if you actually try to get a solution versus win points in stupid newsgroup arguments.

If this idea works, then people can see how dangerous it can be when political minded people can hold so much leeway on math newsgroups, and lie freely, with no fear of consequences.

I say, if this idea works, make the people who distracted, pay.

Make them pay enough that the next time someone thinks it's fun and harmless to go after someone in a public forum, they think twice, knowing that if they're wrong, the weight of the world can fall on them.

Well, I didn't work everything out, but just came up with this idea.

So let's do so now.

S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and

T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

So

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

and it looks like between the two of them you CAN in fact, define the k's, where it looks like you're want to choose squares for x and y, like maybe dividing off some factors of

S-T?

Then you have two equations with 4 unknowns, so you'd still have to pick two of the k's and then that's it.

So no, I don't know what you're griping about this time Silver as that looks like a route to a solution to me, which completely refutes Tim Peters and Rick Decker, as well as any other people who have been working to shoot down this idea.

It looks completely viable at this point. God help us.

So simple too. It's so easy once you see how it all works out.

What if I used

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))?

Multiply them out, subtract one from the other, and I think the correct way to get the k's might be there, but I STILL think there is a way going from just the solution to the equation with S if you actually try to get a solution versus win points in stupid newsgroup arguments.

If this idea works, then people can see how dangerous it can be when political minded people can hold so much leeway on math newsgroups, and lie freely, with no fear of consequences.

I say, if this idea works, make the people who distracted, pay.

Make them pay enough that the next time someone thinks it's fun and harmless to go after someone in a public forum, they think twice, knowing that if they're wrong, the weight of the world can fall on them.

Well, I didn't work everything out, but just came up with this idea.

So let's do so now.

S = k_1*k_3*x + (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

and

T = k_1*k_3*x - (k_2*k_3 + k_1*k_4)*sqrt(xy) + k_2*k_4*y

So

S - T = 2*(k_2*k_3 + k_1*k_4)*sqrt(xy)

and

S+T = 2*k_1*k_3*x + 2*k_2*k_4*y

and it looks like between the two of them you CAN in fact, define the k's, where it looks like you're want to choose squares for x and y, like maybe dividing off some factors of

S-T?

Then you have two equations with 4 unknowns, so you'd still have to pick two of the k's and then that's it.

So no, I don't know what you're griping about this time Silver as that looks like a route to a solution to me, which completely refutes Tim Peters and Rick Decker, as well as any other people who have been working to shoot down this idea.

It looks completely viable at this point. God help us.

So simple too. It's so easy once you see how it all works out.

## JSH: Why you people are NOT pure

One of the most amazing lies that people who claim to care about mathematics tell is that they are "pure" in that they are fascinated by mathematical questions, care about what's mathematically true, and prize questions that have no practical answer.

But I can give

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and note that there are two factorizations, and your "purity" fails.

Let's play with it a bit, and stick in some numbers.

k_1=1, k_2 =2, k_3= 3, k_4 = 4, and use x=1, y=25

and take the positive of the square root, then

S = (1 + 2(5))(3 + 4(5)) = 253

But the sqrt(25) is 5 or -5, so what if I shift some signs?

(1-10)(3-20) = 153

and that's the shadow factorization.

That second number, where does it come from? How is it determined?

Is it just some random monster? Does 153 just materialize from mystery zone, with no reason, no meaning?

Is this a case where mathematics fails?

Of course not. Those of you with an ounce of mathematical acumen know that there are rules that determine—that force—153 to be the second number in that case.

If those rules can be found and understood, why can't you make some target composite of your chose the shadow to be factored?

Why can't you?

I think many of you hate those equations. It makes you sick to your stomach that the mathematics is there in front of you with an answer you don't like because it could mean so much misery for mathematicians.

What is the shadow? How is the shadow determined?

Why with those numbers that I just gave was it 153 and not 223?

Some of you may guess that I know how this ends.

But my point really is that some people can play in areas they don't understand. Unleash forces they can't comprehend, and pay a cost they can't imagine.

Or they would never have done it.

And I like how you people consider time. A week goes by and you sigh with relief.

Years go by and you party as if you were safe.

You have no comprehension of the length and breadth of history.

No idea of time when billions of years are like a summer night.

[A reply to someone who said that James should see that he's really pondering whether it's difficult to find integers a, b, c, and d such that T = (a + b)*(c + d) - 2*(a*d + b*c).]

That's not necessarily true even if

S = (a+b)*(c+d)

because of the square roots.

And notice though Peters just deleted out what I gave.

The expression I gave was

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

where the entire point is that the square roots are ambiguous with regard to sign, so why not

T = (a+b)*(c+d) + 2*(a*d + b*c)?

My purpose in picking an ambiguous expression was in order to see if you could pick some S unrelated to your target composite T, and then find k's based on your target T such that factors of T are given by the alternate factorization, without having to factor T first.

That's it. It's a brilliant idea—if it can be made to work. Trouble is, I have to deal with people like Tim Peters who are working to distract from the mathematical issues.

Look in this thread, and look at the nincompoops replies.

These people are DEDICATED. They are political animals.

They know their game, and their game is distraction.

But just remember, if these ideas can be made viable, then it may be your Inbox that some hacker walks into, or your wife's pc, or your kids pc, or your bank account, or your company's sensitive files.

And then they won't seem harmless. You will see the people replying to me in this thread to distract for what they are.

Time is the issue here people. While we sit here yammering about stupid stuff, if ANYONE in the world that is a hostile develops these ideas—if they can be made viable as I'm still not sure—then each day is a gift to them.

It's another day they have to penetrate without the world being on alert.

It's another day they get to see how many ways they can kill you or someone else using sensitive information—if these ideas can be made viable—while people think they are safe.

I'm not asking for much here. Just some posters who are actually interested in figuring out if these ideas are worth anything or not.

But I can give

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and note that there are two factorizations, and your "purity" fails.

Let's play with it a bit, and stick in some numbers.

k_1=1, k_2 =2, k_3= 3, k_4 = 4, and use x=1, y=25

and take the positive of the square root, then

S = (1 + 2(5))(3 + 4(5)) = 253

But the sqrt(25) is 5 or -5, so what if I shift some signs?

(1-10)(3-20) = 153

and that's the shadow factorization.

That second number, where does it come from? How is it determined?

Is it just some random monster? Does 153 just materialize from mystery zone, with no reason, no meaning?

Is this a case where mathematics fails?

Of course not. Those of you with an ounce of mathematical acumen know that there are rules that determine—that force—153 to be the second number in that case.

If those rules can be found and understood, why can't you make some target composite of your chose the shadow to be factored?

Why can't you?

I think many of you hate those equations. It makes you sick to your stomach that the mathematics is there in front of you with an answer you don't like because it could mean so much misery for mathematicians.

What is the shadow? How is the shadow determined?

Why with those numbers that I just gave was it 153 and not 223?

Some of you may guess that I know how this ends.

But my point really is that some people can play in areas they don't understand. Unleash forces they can't comprehend, and pay a cost they can't imagine.

Or they would never have done it.

And I like how you people consider time. A week goes by and you sigh with relief.

Years go by and you party as if you were safe.

You have no comprehension of the length and breadth of history.

No idea of time when billions of years are like a summer night.

[A reply to someone who said that James should see that he's really pondering whether it's difficult to find integers a, b, c, and d such that T = (a + b)*(c + d) - 2*(a*d + b*c).]

That's not necessarily true even if

S = (a+b)*(c+d)

because of the square roots.

And notice though Peters just deleted out what I gave.

The expression I gave was

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

where the entire point is that the square roots are ambiguous with regard to sign, so why not

T = (a+b)*(c+d) + 2*(a*d + b*c)?

My purpose in picking an ambiguous expression was in order to see if you could pick some S unrelated to your target composite T, and then find k's based on your target T such that factors of T are given by the alternate factorization, without having to factor T first.

That's it. It's a brilliant idea—if it can be made to work. Trouble is, I have to deal with people like Tim Peters who are working to distract from the mathematical issues.

Look in this thread, and look at the nincompoops replies.

These people are DEDICATED. They are political animals.

They know their game, and their game is distraction.

But just remember, if these ideas can be made viable, then it may be your Inbox that some hacker walks into, or your wife's pc, or your kids pc, or your bank account, or your company's sensitive files.

And then they won't seem harmless. You will see the people replying to me in this thread to distract for what they are.

Time is the issue here people. While we sit here yammering about stupid stuff, if ANYONE in the world that is a hostile develops these ideas—if they can be made viable as I'm still not sure—then each day is a gift to them.

It's another day they have to penetrate without the world being on alert.

It's another day they get to see how many ways they can kill you or someone else using sensitive information—if these ideas can be made viable—while people think they are safe.

I'm not asking for much here. Just some posters who are actually interested in figuring out if these ideas are worth anything or not.

## SF: Simpler test of new idea

I really do not know if this latest idea of mine works or not, but I can give advice on testing it out. I myself am doing little to no testing as I'm just not that motivated, as I figure it it's worth something, someone in the world will figure that out.

My idea was to use

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

because the square roots cannot have a single solution, and if you do solve that equation out by multiplying it out, squaring to get rid of sqrt(xy), and completing the square you do find—according to Tim Peters—that you have a factorization dependent on other than just S.

So the point is that even if you have S, there is some other number, I'll call unknown that is also being factored.

Now Tim Peters posted some work where he said he used math software following my instructions to get to that key expression where you have completed the squares, twice.

If those equations give what I think they give, then the number multiplied by S^2 at that point is the number that is being shadow factored, so if you change signs, you get factors of that number.

If that is the case, then this idea can be used on the factoring problem.

If you use the equations that I've put up recently—pulled from what Tim Peters posted--and find that changing signs does NOT give you that number, does not give you unknown, then something interesting is going on:

1. Either he didn't do his work right.

Or

2. The algebra does not depend on the factorization at all, so there's some random thing going on.

Hey, maybe it is random. Maybe there is no rhyme or reason to the number that is shadow factored by a shift in signs with

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and at this interesting area, mathematics makes no sense, the algebra is not consistent, and there is no way to determine the other number being factored, so it's just a mess.

Maybe I found a way to show that at its heart, mathematics as human beings understand it, is fatally flawed, as in being inconsistent, where things can happen for NO REASON.

I like to think that there is order in mathematics, a beauty and purpose that means that everything has a reason, but then again, I am often wrong.

Maybe, there is no mathematical way to determine the second factorization—the shadow factorization—and posters who are still working to dismiss this research are right.

But think about what they prove:

They prove that in this area I have shown where mathematics fails.

I have found a place where the numbers can't be controlled. The algebra is just a random mess.

And the shadow factorization is uncontrollable, unknowable, and not capable of being controlled by rational thought.

Neat! Fascinating either way.

I may have found a door into chaos.

I suggest to you instead that the shadow factorization CAN be controlled, even if so far no one has figured out how to do it, and that people who push the idea that mathematics is inconsistent—are wrong.

If it is consistent, then there is a way to know what the second factorization will be.

If that's possible then the k's can be set so that it is your target composites.

If so, this approach can be used against the factoring problem.

My idea was to use

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

because the square roots cannot have a single solution, and if you do solve that equation out by multiplying it out, squaring to get rid of sqrt(xy), and completing the square you do find—according to Tim Peters—that you have a factorization dependent on other than just S.

So the point is that even if you have S, there is some other number, I'll call unknown that is also being factored.

Now Tim Peters posted some work where he said he used math software following my instructions to get to that key expression where you have completed the squares, twice.

If those equations give what I think they give, then the number multiplied by S^2 at that point is the number that is being shadow factored, so if you change signs, you get factors of that number.

If that is the case, then this idea can be used on the factoring problem.

If you use the equations that I've put up recently—pulled from what Tim Peters posted--and find that changing signs does NOT give you that number, does not give you unknown, then something interesting is going on:

1. Either he didn't do his work right.

Or

2. The algebra does not depend on the factorization at all, so there's some random thing going on.

Hey, maybe it is random. Maybe there is no rhyme or reason to the number that is shadow factored by a shift in signs with

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and at this interesting area, mathematics makes no sense, the algebra is not consistent, and there is no way to determine the other number being factored, so it's just a mess.

Maybe I found a way to show that at its heart, mathematics as human beings understand it, is fatally flawed, as in being inconsistent, where things can happen for NO REASON.

I like to think that there is order in mathematics, a beauty and purpose that means that everything has a reason, but then again, I am often wrong.

Maybe, there is no mathematical way to determine the second factorization—the shadow factorization—and posters who are still working to dismiss this research are right.

But think about what they prove:

They prove that in this area I have shown where mathematics fails.

I have found a place where the numbers can't be controlled. The algebra is just a random mess.

And the shadow factorization is uncontrollable, unknowable, and not capable of being controlled by rational thought.

Neat! Fascinating either way.

I may have found a door into chaos.

I suggest to you instead that the shadow factorization CAN be controlled, even if so far no one has figured out how to do it, and that people who push the idea that mathematics is inconsistent—are wrong.

If it is consistent, then there is a way to know what the second factorization will be.

If that's possible then the k's can be set so that it is your target composites.

If so, this approach can be used against the factoring problem.

### Wednesday, June 21, 2006

## JSH: Still retired

My last big idea was to use

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

on the factring problem, relying on the reality—the absolute mathematical fact—that the expressions CANNOT have a single factorization.

There is no debate on that point.

It just cannot only factor S.

That's it. I haven't even bothered to step out a solution of the damn thing, relying on the posted work of Tim Peters.

I'm done. The big ideas are for young people. That was my last big idea.

But I can follow through on it and on my other ideas—or help others follow through.

Maybe it's crap. Maybe algebra is some odd little thing that just wishes to protect humanity from the factoring problem being this dinky nothing, so it will fight ot make sure this approach doesn't work.

I don't think so, but I say it like that because some of you may have no clue about what mathematics is, so you probably suppose that because a LOT of people are invested in this factoring thing, no way the math can show them to be wrong.

But you see, the math doesn't care.

People are wrong about all kinds of things. It's just kind of a human thing—being wrong.

If you trust, then you can just jump to the conclusion of posters who leap at replying to me, to see what their opinion is.

I suggest instead you look to see what their math is.

Right now I am sitting in the United States contemplating a war started by the son of a president, a president who was ridiculed for not finishing with Iraq. That president was also ridiculed for saying he wouldn't cut taxes and then doing it later.

Seems to me his son just went to the big areaa his father was criticized on, and decided he'd fix it, in his own mindless way.

That's the real world. People do weird things, and then a LOT of other people support them on it, as that's what people do.

But mathematics IS pure in that no matter what, no matter how many people get together to decide something that is false is true, you can trace out the logical argument, and find a flaw.

And it doesn't matter if it's the son of some dude who wants to make good on what he thinks his father failed on—no matter who dies.

Mathematics IS pure. But you have to know where the purity actually is.

It's not in the value or usefulness that some people put on it, but in the reality of absolute truth.

[A reply to someone who asked whether or not James could factor 6816852827 using his discoveries.]

Probably, but I do not care to try. I have some earlier stuff from WAY back that did not lead to a way to crack RSA, so I dropped it. It could handle such a dinky number, but it is not worth the effort.

Hey, I come up with ideas. My last big idea was that maybe this surrogate factoring thing could work with an expression that could not represent a single factorization:

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

That expression cannot be a single factorization because of the square roots as there is a positive and a negative solution to a square root, for instance, sqrt(4) is 2 or -2, because

(-2)(-2) = 4

It is a simple idea but mathematicians are taught to only take the positive of the square root, routinely throwing away the negative as human beings prefer the positive.

I think that kind of narrow thinking deflected most people from considering such a simple expression as mine.

But do I have the energy to see if it can be made practical?

Nope. So I put it out there and see what other people do with it.

If nothing, fine. If you think it is a useless idea then do what any normal person would do, wander off.

But if it is a useful idea then someone in the world will eventually develop it, and I will not have had to do a thing beyond point out the obvious, to start the ball rolling.

That over $300k that RSA is offering in prize money is an incentive for other people around the world, not for me.

It means nothing to me.

But for some of you out there it can mean your life changes completely.

And that is the carrot that drives things from here on out.

I have no intentions of developing these ideas further, but will just discuss them as I have been doing.

If they work, I have no doubt that some hungry person somewhere in the world will do the developing, and take the money.

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

on the factring problem, relying on the reality—the absolute mathematical fact—that the expressions CANNOT have a single factorization.

There is no debate on that point.

It just cannot only factor S.

That's it. I haven't even bothered to step out a solution of the damn thing, relying on the posted work of Tim Peters.

I'm done. The big ideas are for young people. That was my last big idea.

But I can follow through on it and on my other ideas—or help others follow through.

Maybe it's crap. Maybe algebra is some odd little thing that just wishes to protect humanity from the factoring problem being this dinky nothing, so it will fight ot make sure this approach doesn't work.

I don't think so, but I say it like that because some of you may have no clue about what mathematics is, so you probably suppose that because a LOT of people are invested in this factoring thing, no way the math can show them to be wrong.

But you see, the math doesn't care.

People are wrong about all kinds of things. It's just kind of a human thing—being wrong.

If you trust, then you can just jump to the conclusion of posters who leap at replying to me, to see what their opinion is.

I suggest instead you look to see what their math is.

Right now I am sitting in the United States contemplating a war started by the son of a president, a president who was ridiculed for not finishing with Iraq. That president was also ridiculed for saying he wouldn't cut taxes and then doing it later.

Seems to me his son just went to the big areaa his father was criticized on, and decided he'd fix it, in his own mindless way.

That's the real world. People do weird things, and then a LOT of other people support them on it, as that's what people do.

But mathematics IS pure in that no matter what, no matter how many people get together to decide something that is false is true, you can trace out the logical argument, and find a flaw.

And it doesn't matter if it's the son of some dude who wants to make good on what he thinks his father failed on—no matter who dies.

Mathematics IS pure. But you have to know where the purity actually is.

It's not in the value or usefulness that some people put on it, but in the reality of absolute truth.

[A reply to someone who asked whether or not James could factor 6816852827 using his discoveries.]

Probably, but I do not care to try. I have some earlier stuff from WAY back that did not lead to a way to crack RSA, so I dropped it. It could handle such a dinky number, but it is not worth the effort.

Hey, I come up with ideas. My last big idea was that maybe this surrogate factoring thing could work with an expression that could not represent a single factorization:

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

That expression cannot be a single factorization because of the square roots as there is a positive and a negative solution to a square root, for instance, sqrt(4) is 2 or -2, because

(-2)(-2) = 4

It is a simple idea but mathematicians are taught to only take the positive of the square root, routinely throwing away the negative as human beings prefer the positive.

I think that kind of narrow thinking deflected most people from considering such a simple expression as mine.

But do I have the energy to see if it can be made practical?

Nope. So I put it out there and see what other people do with it.

If nothing, fine. If you think it is a useless idea then do what any normal person would do, wander off.

But if it is a useful idea then someone in the world will eventually develop it, and I will not have had to do a thing beyond point out the obvious, to start the ball rolling.

That over $300k that RSA is offering in prize money is an incentive for other people around the world, not for me.

It means nothing to me.

But for some of you out there it can mean your life changes completely.

And that is the carrot that drives things from here on out.

I have no intentions of developing these ideas further, but will just discuss them as I have been doing.

If they work, I have no doubt that some hungry person somewhere in the world will do the developing, and take the money.

### Tuesday, June 20, 2006

## SF: Simpler factoring idea, but does it work?

After yet another failure with what I call surrogate factoring, where this time I had been doing some basic algebra wrong, I sat down to think about it all for a while, and considered that I was quite reasonably just going in circles, using equations to try and factor that could only give one answer.

So I started thinking about equations that could give two.

It didn't take long till I was concentrating on:

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

Here the square roots mean that expression can't just factor S, which is what I call the surrogate, as something else is being factored as well, but how do I get that something else to be a target composite?

After I posted that equation and started talking about it, a Tim Peters worked out details following my instructions, but unfortunately, he is a dedicated, um, "crank" buster you might call it, who spends his time trying to shoot down my ideas, so when he worked out the equations, and got to something useable, he promptly began throwing up distracting posts meant to show it was useless.

However, oddly enough, his results can be used quite simply, where the first thing is to use some of his equations, to introduce a target composite, which I call T.

You introduce T using

(k_1*k_4 + k_2*k_3) / (k_1*k_4 - k_2*k_3) = T

Multiply both sides by

k_1*k_4 - k_2*k_3

to get

(k_1*k_4 + k_2*k_3) = T (k_1*k_4 - k_2*k_3)

and just subtract the left from the right to get

0 = (T-1)* k_1*k_4 - (T+1)*k_2*k_3

So

(T-1)* k_1*k_4 = (T+1)*k_2*k_3

And you have

(k_1*k_4)/(k_2*k_3) = (T+1)/(T-1)

so k_1 and k_4 are integer factors of T+1, and k_2 and k_3 are integer factors of T-1.

Easy. Just like that you're most of the way to using the equations.

That gives a finite set of possibles for the k's.

For instance, with

So, for instance if T=15, you have

(k_1*k_4)/(k_2*k_3) = 16/14 = 8/7

so k_1*k_4 = 8, and k_2*k_3 = 7

and one possible setup then is

k_1 = 2, k_4 = 4, k_2 = 7, k_3 = 1

so plug those into

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and you need S, and can use any square you like, so it's easiest just to use

x=y=1, and take the positive result of the square roots to get

S = (2 + 7)(1 + 4) = 45

which promptly factors it, but I'll continue, as that's just one possibility, so you have to check for it.

If S were coprime to T, then you now use factors of S, where with

S = g_1*g_2

you have

k_1*sqrt(x) + k_2*sqrt(y) = g_1

k_3*sqrt(x) + k_4*sqrt(y) = g_2

and you just find find squares for x and y that will work to give you g_1 and g_2, and here's where that second solution from the square roots comes in, as with each set of squares for x and y that will work, you just change the sign of one of the square roots, to get the shadow factorization.

That's it. Remarkably simple, as you go for the hidden factorization.

For instance, still using x=y=1, with my simple example, now take the negative of ONE of the square roots:

S = (2 - 7)(1 + 4) = 55

I guess T=15 is too dinky of an example as it just keeps factoring it, no matter what you do, but at least that still shows the basic idea here.

To recap, I looked at an expression

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

that can't just be the factorization of a single number because of the use of square roots. I posted about it and a Tim Peters worked through some analysis following instructions I gave, which gives up the simple equation

(k_1*k_4 + k_2*k_3) / (k_1*k_4 - k_2*k_3) = T

which can be solved to get

(k_1*k_4)/(k_2*k_3) = (T+1)/(T-1)

so you can find the k's based on the factors of T+1 and T-1, to plug back into the original equations and get an S, and then you use the factors of S to find squares that are solutions for x and y, and then you do the remarkable trick of just switching signs one at a time to get to the hidden factorization.

But does it work beyond toy examples like factoring 15?

Unfortunately, as I said, Tim Peters is a hostile when it comes to my research, so he promptly busied himself trying to obscure use of the equations, while claiming that they don't work. You can look at recent threads I've created on sci.crypt and sci.math to see him at work.

Some of you might be shocked by such behavior, as, hey, it's the factoring problem.

But consider for YEARS Peters and people like him have been shadowing my posts working to convince people that my research is useless.

I assure you that he has worked in this way many times.

He is only doing what he has always been doing, so there is nothing new in his behavior.

So the question is an open one. Does this method work?

I can assure you that the math society that has busied itself ignoring my research for years is in no hurry to acknowledge a result of mine like this one—if it does work—as then they would be completely overturned, now wouldn't they?

And what is your security against theirs?

So they wait, seeing if the "crackpot" label will hold, and they wait, to see if no one can tell if these ideas will work or not, and if they do work, they wait until there is a disaster big enough for the world to care about the truth.

And then, finally, they will be able to wait, no more.

Here I said find squares that will work for x and y, but usually I suspect you will find squares of fractions.

Note that it's easy enough to solve for sqrt(x), and sqrt(y), as you get

sqrt(x) =( k_4*g_1 - k_2*g_2)/(k_1*k_4 - k_3*k_2)

and

sqrt(y) = (k_3*g_1 - k_1*g_2)/(k_2*k_3 - k_1*k_4)

and you can see something familiar in the denominator.

So why do you HAVE to use a particular S?

I don't know. It sounds good to me that you do, but I suspect that if you don't, but just pick square roots, you'll get nothing better than random behavior, which seems consistent with what some mathematically naive posters apparently tried from what I read in this thread.

Remember, the people replying to me are invested in me being wrong.

They don't give a damn about whether or not these ideas can be made to work.

They only care about supporting their old position.

Now I don't know. Maybe these are crap ideas, but I, at least, wish to actually find out.

Short story of it is that people like Tim Peters or Dik Winter or Arturo Magidin or David Ullrich couldn't care less about the factoring problem—if I present something that actually is of value.

They only care about supporting their social positions as, of course, what happens if I come up with a useful idea?

Then they are totally invalidated, and all those posts of theirs come back to haunt them.

Don't trust those people. Sure, this idea may be crap, but I can assure you that if it's not, they will not tell you, but instead will do their BEST to hide the fact.

Use one S, and iterate through its factors.

The bonus is that you get to see those people crushed like little bugs, and watch them squirm as they fight you, try to distract from what you present, and do anything they can to see what they can post to hide the truth—if there is any value in this approach.

Otherwise, oh well, it's just another idea among so many others that failed.

But I at least care about what's true, versus being invested in fighting no matter what against some guy.

You will NOT get a cogent answer from regular posters who have spent so much time fighting me.

These people have lied about my other research.

They have no reason not to lie now.

They're in for a penny, in for a pound.

They will lie until they're broken.

Use a single S, and iterate through its factors. Note that in the replies in this thread, that was not done.

But they congratulated each other anyway.

These people are not researchers!!! They are politicians. They don't give a damn about the factoring problem or any other mathematics, if they think it helps me.

They are—anti-civilization.

[A reply to someone who wanted to know in which category James thought he was ("you", "them", "both" or "neither in "you" nor in "them"").]

Who cares about categories? The problem here is just out and out fraud.

Like consider the replies by Tim Peters and earlier by Rick Decker about this new idea of mine.

But now that I've shown both factorizations with

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

you can multiply them together easily enough and notice then you have

S*T = (k_1^2*x - k_2^2*y)*(k_3^2*x - k_4^2*y)

and now you can multiply out and complete the square twice as before, and this time you will have x and y dependent on the factorization of S*T.

To me that says that somehow BOTH S and T are wrapped up in the original

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

where T gets placed by how you pick the k's, though my newer work also indicates that the values of x and y are important, so maybe that's not quite right.

See what I'm doing? I'm asking questions, as I'd like to know the answers.

The people who have succeeded by pretending to care about mathematics are distinguished by their disinterest in such questions.

They do not actually care. Pretending to care gets some of them paychecks, while others may get prestige, or simply like to lie, and get away with it.

They are frauds and cons.

So as I play with these simple expressions the bulk of that math society will do their best to ignore it, and if they are called out on it—like by government agencies—they will probably come up with clever lies, like any cons.

It's how they could ignore my prime counting function, because they don't actually care about prime numbers.

It's how they could ignore my non-polynomial factorization research, as they don't actually give a damn about the fundamental properties of numbers.

It's how they could ignore my short proof of Fermat's Last Theorem.

Because they actually couldn't care less about really solving it.

They only care if people believe one of their own did solve it.

Why is why they ignored the simple explanations for why the work of Andrew Wiles fails by Cum Hoc, Ergo Propter Hoc.

Lies dominate their research.

They live in a political world where they depend on not being checked, which is why they are likely to have undermined computer checking of mathematical arguments.

Which is a strong suspicion on my part given the wonderful progress of computer science in other areas, while there is a dearth of progress when it comes to checking math people in "pure math" areas, giving a continuing license to steal.

They rely on social forces, like ridicule, when confronted with math, or, on distraction, getting people to ignore the mathematical reality to keep their social world in place, even if they endanger the security of the entire world, as that's what cons do.

If they were big thinkers who could be relied on in such matters, then they wouldn't be cons, now would they?

So I started thinking about equations that could give two.

It didn't take long till I was concentrating on:

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

Here the square roots mean that expression can't just factor S, which is what I call the surrogate, as something else is being factored as well, but how do I get that something else to be a target composite?

After I posted that equation and started talking about it, a Tim Peters worked out details following my instructions, but unfortunately, he is a dedicated, um, "crank" buster you might call it, who spends his time trying to shoot down my ideas, so when he worked out the equations, and got to something useable, he promptly began throwing up distracting posts meant to show it was useless.

However, oddly enough, his results can be used quite simply, where the first thing is to use some of his equations, to introduce a target composite, which I call T.

You introduce T using

(k_1*k_4 + k_2*k_3) / (k_1*k_4 - k_2*k_3) = T

Multiply both sides by

k_1*k_4 - k_2*k_3

to get

(k_1*k_4 + k_2*k_3) = T (k_1*k_4 - k_2*k_3)

and just subtract the left from the right to get

0 = (T-1)* k_1*k_4 - (T+1)*k_2*k_3

So

(T-1)* k_1*k_4 = (T+1)*k_2*k_3

And you have

(k_1*k_4)/(k_2*k_3) = (T+1)/(T-1)

so k_1 and k_4 are integer factors of T+1, and k_2 and k_3 are integer factors of T-1.

Easy. Just like that you're most of the way to using the equations.

That gives a finite set of possibles for the k's.

For instance, with

So, for instance if T=15, you have

(k_1*k_4)/(k_2*k_3) = 16/14 = 8/7

so k_1*k_4 = 8, and k_2*k_3 = 7

and one possible setup then is

k_1 = 2, k_4 = 4, k_2 = 7, k_3 = 1

so plug those into

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and you need S, and can use any square you like, so it's easiest just to use

x=y=1, and take the positive result of the square roots to get

S = (2 + 7)(1 + 4) = 45

which promptly factors it, but I'll continue, as that's just one possibility, so you have to check for it.

If S were coprime to T, then you now use factors of S, where with

S = g_1*g_2

you have

k_1*sqrt(x) + k_2*sqrt(y) = g_1

k_3*sqrt(x) + k_4*sqrt(y) = g_2

and you just find find squares for x and y that will work to give you g_1 and g_2, and here's where that second solution from the square roots comes in, as with each set of squares for x and y that will work, you just change the sign of one of the square roots, to get the shadow factorization.

That's it. Remarkably simple, as you go for the hidden factorization.

For instance, still using x=y=1, with my simple example, now take the negative of ONE of the square roots:

S = (2 - 7)(1 + 4) = 55

I guess T=15 is too dinky of an example as it just keeps factoring it, no matter what you do, but at least that still shows the basic idea here.

To recap, I looked at an expression

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

that can't just be the factorization of a single number because of the use of square roots. I posted about it and a Tim Peters worked through some analysis following instructions I gave, which gives up the simple equation

(k_1*k_4 + k_2*k_3) / (k_1*k_4 - k_2*k_3) = T

which can be solved to get

(k_1*k_4)/(k_2*k_3) = (T+1)/(T-1)

so you can find the k's based on the factors of T+1 and T-1, to plug back into the original equations and get an S, and then you use the factors of S to find squares that are solutions for x and y, and then you do the remarkable trick of just switching signs one at a time to get to the hidden factorization.

But does it work beyond toy examples like factoring 15?

Unfortunately, as I said, Tim Peters is a hostile when it comes to my research, so he promptly busied himself trying to obscure use of the equations, while claiming that they don't work. You can look at recent threads I've created on sci.crypt and sci.math to see him at work.

Some of you might be shocked by such behavior, as, hey, it's the factoring problem.

But consider for YEARS Peters and people like him have been shadowing my posts working to convince people that my research is useless.

I assure you that he has worked in this way many times.

He is only doing what he has always been doing, so there is nothing new in his behavior.

So the question is an open one. Does this method work?

I can assure you that the math society that has busied itself ignoring my research for years is in no hurry to acknowledge a result of mine like this one—if it does work—as then they would be completely overturned, now wouldn't they?

And what is your security against theirs?

So they wait, seeing if the "crackpot" label will hold, and they wait, to see if no one can tell if these ideas will work or not, and if they do work, they wait until there is a disaster big enough for the world to care about the truth.

And then, finally, they will be able to wait, no more.

Here I said find squares that will work for x and y, but usually I suspect you will find squares of fractions.

Note that it's easy enough to solve for sqrt(x), and sqrt(y), as you get

sqrt(x) =( k_4*g_1 - k_2*g_2)/(k_1*k_4 - k_3*k_2)

and

sqrt(y) = (k_3*g_1 - k_1*g_2)/(k_2*k_3 - k_1*k_4)

and you can see something familiar in the denominator.

So why do you HAVE to use a particular S?

I don't know. It sounds good to me that you do, but I suspect that if you don't, but just pick square roots, you'll get nothing better than random behavior, which seems consistent with what some mathematically naive posters apparently tried from what I read in this thread.

Remember, the people replying to me are invested in me being wrong.

They don't give a damn about whether or not these ideas can be made to work.

They only care about supporting their old position.

Now I don't know. Maybe these are crap ideas, but I, at least, wish to actually find out.

Short story of it is that people like Tim Peters or Dik Winter or Arturo Magidin or David Ullrich couldn't care less about the factoring problem—if I present something that actually is of value.

They only care about supporting their social positions as, of course, what happens if I come up with a useful idea?

Then they are totally invalidated, and all those posts of theirs come back to haunt them.

Don't trust those people. Sure, this idea may be crap, but I can assure you that if it's not, they will not tell you, but instead will do their BEST to hide the fact.

Use one S, and iterate through its factors.

The bonus is that you get to see those people crushed like little bugs, and watch them squirm as they fight you, try to distract from what you present, and do anything they can to see what they can post to hide the truth—if there is any value in this approach.

Otherwise, oh well, it's just another idea among so many others that failed.

But I at least care about what's true, versus being invested in fighting no matter what against some guy.

You will NOT get a cogent answer from regular posters who have spent so much time fighting me.

These people have lied about my other research.

They have no reason not to lie now.

They're in for a penny, in for a pound.

They will lie until they're broken.

Use a single S, and iterate through its factors. Note that in the replies in this thread, that was not done.

But they congratulated each other anyway.

These people are not researchers!!! They are politicians. They don't give a damn about the factoring problem or any other mathematics, if they think it helps me.

They are—anti-civilization.

[A reply to someone who wanted to know in which category James thought he was ("you", "them", "both" or "neither in "you" nor in "them"").]

Who cares about categories? The problem here is just out and out fraud.

Like consider the replies by Tim Peters and earlier by Rick Decker about this new idea of mine.

But now that I've shown both factorizations with

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and

T = (k_1*sqrt(x) - k_2*sqrt(y))*(k_3*sqrt(x) - k_4*sqrt(y))

you can multiply them together easily enough and notice then you have

S*T = (k_1^2*x - k_2^2*y)*(k_3^2*x - k_4^2*y)

and now you can multiply out and complete the square twice as before, and this time you will have x and y dependent on the factorization of S*T.

To me that says that somehow BOTH S and T are wrapped up in the original

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

where T gets placed by how you pick the k's, though my newer work also indicates that the values of x and y are important, so maybe that's not quite right.

See what I'm doing? I'm asking questions, as I'd like to know the answers.

The people who have succeeded by pretending to care about mathematics are distinguished by their disinterest in such questions.

They do not actually care. Pretending to care gets some of them paychecks, while others may get prestige, or simply like to lie, and get away with it.

They are frauds and cons.

So as I play with these simple expressions the bulk of that math society will do their best to ignore it, and if they are called out on it—like by government agencies—they will probably come up with clever lies, like any cons.

It's how they could ignore my prime counting function, because they don't actually care about prime numbers.

It's how they could ignore my non-polynomial factorization research, as they don't actually give a damn about the fundamental properties of numbers.

It's how they could ignore my short proof of Fermat's Last Theorem.

Because they actually couldn't care less about really solving it.

They only care if people believe one of their own did solve it.

Why is why they ignored the simple explanations for why the work of Andrew Wiles fails by Cum Hoc, Ergo Propter Hoc.

Lies dominate their research.

They live in a political world where they depend on not being checked, which is why they are likely to have undermined computer checking of mathematical arguments.

Which is a strong suspicion on my part given the wonderful progress of computer science in other areas, while there is a dearth of progress when it comes to checking math people in "pure math" areas, giving a continuing license to steal.

They rely on social forces, like ridicule, when confronted with math, or, on distraction, getting people to ignore the mathematical reality to keep their social world in place, even if they endanger the security of the entire world, as that's what cons do.

If they were big thinkers who could be relied on in such matters, then they wouldn't be cons, now would they?

## JSH: Like baseball

Discovery is not easy.

I am telling you from experience that in terms of stress, and having to push yourself, fighting for every inch, never knowing quite which way is up, or down, there is nothing like the search for truth.

Being wrong is easy, knowing when you're right can be hard, but actually being right and knowing it, is the hardest thing of all.

When it comes to a sports analogy that applies, I think the best is baseball.

You lose most of the time, as a hitter in baseball.

Most of the time, you don't get that homerun.

Most of the time, when you go up to that plate, with that anticipation, the adrenaline surging, all those plans for beating that pitcher beating through your mind—you lose.

You go up—then, you sit back down, and you get to think about it, one more time.

But, you have to get up again, sometime later—if you're lucky—and go up, one more time, against another set of pitches, and try again.

Discovery is like baseball.

You lose most of the time. You learn to feel that pain in your gut. To feel that disappointment from yourself and people watching, and know that you will go back up again, one more time—God willing.

It's not about whether or not you hit that ball. But that you got up there, and you tried, God help you, you tried because that's what you do.

I'm at the end of my run. Mathematics is a young man's game, so far, and hopefully, it'll be a young person's game soon enough, as it's past time for a woman to step up, and I am looking for her.

But for now, it's a young man's game and as I'm past 35 I am well past my prime, and my time is over.

But, in my time, when I had my big dreams, I could step up, swing a lot, miss a lot, and keep dreaming, as I knew I didn't have to connect all that many times, as in mathematics, your homeruns, well, they last, forever.

I am retired. The spirit has died. The Muse has left me. I don't feel any more discovery is left in these bones.

But I look back, and no matter what anyone else says, no matter who calls me "crazy" or a "loon" or a "crackpot" or a "crank", I stepped up to the plate, and I hit some homeruns.

Feeling really old now. Well past my prime, and I had just enough energy for one last go at it.

The deeds are done. Now I can just think back and remember the hard days, the painful days, when nothing was right, when there seemed to be nothing but pain and misery, the laughter and the taunting, all that hurt from people I never met and never want to meet, when all I could think of was that nothing, nothing in this world was going to stop me, because there was nothing else I could do, but win.

I am telling you from experience that in terms of stress, and having to push yourself, fighting for every inch, never knowing quite which way is up, or down, there is nothing like the search for truth.

Being wrong is easy, knowing when you're right can be hard, but actually being right and knowing it, is the hardest thing of all.

When it comes to a sports analogy that applies, I think the best is baseball.

You lose most of the time, as a hitter in baseball.

Most of the time, you don't get that homerun.

Most of the time, when you go up to that plate, with that anticipation, the adrenaline surging, all those plans for beating that pitcher beating through your mind—you lose.

You go up—then, you sit back down, and you get to think about it, one more time.

But, you have to get up again, sometime later—if you're lucky—and go up, one more time, against another set of pitches, and try again.

Discovery is like baseball.

You lose most of the time. You learn to feel that pain in your gut. To feel that disappointment from yourself and people watching, and know that you will go back up again, one more time—God willing.

It's not about whether or not you hit that ball. But that you got up there, and you tried, God help you, you tried because that's what you do.

I'm at the end of my run. Mathematics is a young man's game, so far, and hopefully, it'll be a young person's game soon enough, as it's past time for a woman to step up, and I am looking for her.

But for now, it's a young man's game and as I'm past 35 I am well past my prime, and my time is over.

But, in my time, when I had my big dreams, I could step up, swing a lot, miss a lot, and keep dreaming, as I knew I didn't have to connect all that many times, as in mathematics, your homeruns, well, they last, forever.

I am retired. The spirit has died. The Muse has left me. I don't feel any more discovery is left in these bones.

But I look back, and no matter what anyone else says, no matter who calls me "crazy" or a "loon" or a "crackpot" or a "crank", I stepped up to the plate, and I hit some homeruns.

Feeling really old now. Well past my prime, and I had just enough energy for one last go at it.

The deeds are done. Now I can just think back and remember the hard days, the painful days, when nothing was right, when there seemed to be nothing but pain and misery, the laughter and the taunting, all that hurt from people I never met and never want to meet, when all I could think of was that nothing, nothing in this world was going to stop me, because there was nothing else I could do, but win.

### Saturday, June 17, 2006

## JSH: Those damn fools, math is before us

I remember reading stuff about mathematicians creating proofs as if they were writers, and people would ask questions that were stupid to me, like wondering if mathematics was a creation of human beings.

Of course mathematics is not a creation of human beings.

And no one ever writes a proof—a proof is discovered.

Mathematical proofs have been here since before humanity. They will be here long after humanity is gone.

You go out looking for math proofs. Like searching for treasure.

You can't make one.

The stunning impact of human beings deluded about their place in reality can be seen now with the factoring problem where trillions of dollars are going to move as people pay the price for believing in some of you who thought you create mathematics.

Make no mistake.

Every human being on this planet can die to hold on to the belief that human beings create mathematics and it won't change anything.

Once every person is dead, the mathematics will still be here.

The mathematical proofs, like they were before, will still be here.

Of course mathematics is not a creation of human beings.

And no one ever writes a proof—a proof is discovered.

Mathematical proofs have been here since before humanity. They will be here long after humanity is gone.

You go out looking for math proofs. Like searching for treasure.

You can't make one.

The stunning impact of human beings deluded about their place in reality can be seen now with the factoring problem where trillions of dollars are going to move as people pay the price for believing in some of you who thought you create mathematics.

Make no mistake.

Every human being on this planet can die to hold on to the belief that human beings create mathematics and it won't change anything.

Once every person is dead, the mathematics will still be here.

The mathematical proofs, like they were before, will still be here.

### Friday, June 16, 2006

## SF: Sad part, my discoveries against politics

Now I am sitting here amazed as much as anything at yet another major mathematical find.

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

is just one of those glorious kinds of brilliant ideas that just catch your breath. Who'd a thunk it? Use the ambiguity of the square root function to factor?

Well, I did. So all those stupid arguments where I kept pointing out that the square root function returns TWO VALUES helped, as when I was puzzling over my latest failed factoring idea, it occurred to me to use something that couldn't have just one answer, and asking myself how, I figured, hey, use square roots!!!

But now here's the sad part, where math people prove they are not who most people think they are, which isn't a surprise to me, unfortunately, as here we go again.

I have multiple math discoveries. My techniques use the most modern problem solving techniques, and I have proven time and time again that I get results.

And math people have proven time and time again that they lie about my results.

The first time I really had hard information about how bad it was, was when I'd explained my prime counting function—yet again—and in explaining it on the sci.math newsgroup, I realized that they had to get it.

There was just no way that if you looked with an objective eye at current prime number research, and considered what I found, that you could dismiss it, or claim it wasn't new.

But these people were doing just that.

And I learned that math people were liars. Years worked by as I pieced the puzzle together, and I figured out that there were too many people claiming to be mathematicians to be supported by the reality of the difficulty of innovative mathematical research, and that the human solution had been to start lying about coming up with discoveries, in areas where only the word of people was being used, and call it "pure".

Mathematical discovery is difficult not because there aren't lots of things to discover—mathematics is an infinite subject—but because the people who discover things are so rare, and they also tend to annoy other people who can't stand them.

History tends to forget that, as the stories get white-washed, so that people talk less about how hated and feared Isaac Newton was than nice little stories like of a little boy sitting under a tree getting hit in the head with an apple.

There are an infinite number of things in mathematics to discover, but hey, how many people are going to do something like sit down, consider the factoring problem for over three years, and then suddenly think to use square roots to factor?

Or sit down for over two weeks thinking about prime numbers and come up with their own prime counting function from scratch?

See an old Wikpedia article of mine in the history pages where I wrote the first prime counting function article for the Wikipedia:

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

You can see my prime counting function there, and I look at it, and still don't quite know why I figured it out, or why no one else did over the last two thousand years.

I can do things—after puzzling for years—that no one else seems to have been bothered to do, like my definition of mathematical proof seems easy enough to me. So why am I the one to give it?

So, here we are now with the factoring problem, and if math people try to sit on this like with all my other research, THIS TIME, the world will, I think, find it important to learn why, and all those questions I've wanted to ask, will get asked, as all of you are revealed, your lives peered into, and your computers' hard drives combed, for clues about what you were thinking and doing.

The clock is ticking.

I think it is sad, but at least in this case, the longer you wait, the more thoroughly your lives can be checked, as the public will demand to know, why.

And if enough people demand to know why, your lives will be turned upside down, as people comb through them, check everything, looking for answers.

And along with them, finally, I can find out as well, how you people do some of the things you do, like how you've ignored my mathematical research for so long, but kept acting like you like mathematics, or lectured, or wrote papers.

I want to know how you did it. I want to understand you. Know what was going on in your heads, and how you kept going, when I've been out here, looking for something to end this, and if you had a clue of the necessity of this ending.

Did you try to live your life like nothing, not protecting yourself, not doing anything to defend yourself, as if the mathematics could really be blocked, even though you should have known how the story had to end?

If so, how? If so, why?

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

is just one of those glorious kinds of brilliant ideas that just catch your breath. Who'd a thunk it? Use the ambiguity of the square root function to factor?

Well, I did. So all those stupid arguments where I kept pointing out that the square root function returns TWO VALUES helped, as when I was puzzling over my latest failed factoring idea, it occurred to me to use something that couldn't have just one answer, and asking myself how, I figured, hey, use square roots!!!

But now here's the sad part, where math people prove they are not who most people think they are, which isn't a surprise to me, unfortunately, as here we go again.

I have multiple math discoveries. My techniques use the most modern problem solving techniques, and I have proven time and time again that I get results.

And math people have proven time and time again that they lie about my results.

The first time I really had hard information about how bad it was, was when I'd explained my prime counting function—yet again—and in explaining it on the sci.math newsgroup, I realized that they had to get it.

There was just no way that if you looked with an objective eye at current prime number research, and considered what I found, that you could dismiss it, or claim it wasn't new.

But these people were doing just that.

And I learned that math people were liars. Years worked by as I pieced the puzzle together, and I figured out that there were too many people claiming to be mathematicians to be supported by the reality of the difficulty of innovative mathematical research, and that the human solution had been to start lying about coming up with discoveries, in areas where only the word of people was being used, and call it "pure".

Mathematical discovery is difficult not because there aren't lots of things to discover—mathematics is an infinite subject—but because the people who discover things are so rare, and they also tend to annoy other people who can't stand them.

History tends to forget that, as the stories get white-washed, so that people talk less about how hated and feared Isaac Newton was than nice little stories like of a little boy sitting under a tree getting hit in the head with an apple.

There are an infinite number of things in mathematics to discover, but hey, how many people are going to do something like sit down, consider the factoring problem for over three years, and then suddenly think to use square roots to factor?

Or sit down for over two weeks thinking about prime numbers and come up with their own prime counting function from scratch?

See an old Wikpedia article of mine in the history pages where I wrote the first prime counting function article for the Wikipedia:

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

You can see my prime counting function there, and I look at it, and still don't quite know why I figured it out, or why no one else did over the last two thousand years.

I can do things—after puzzling for years—that no one else seems to have been bothered to do, like my definition of mathematical proof seems easy enough to me. So why am I the one to give it?

So, here we are now with the factoring problem, and if math people try to sit on this like with all my other research, THIS TIME, the world will, I think, find it important to learn why, and all those questions I've wanted to ask, will get asked, as all of you are revealed, your lives peered into, and your computers' hard drives combed, for clues about what you were thinking and doing.

The clock is ticking.

I think it is sad, but at least in this case, the longer you wait, the more thoroughly your lives can be checked, as the public will demand to know, why.

And if enough people demand to know why, your lives will be turned upside down, as people comb through them, check everything, looking for answers.

And along with them, finally, I can find out as well, how you people do some of the things you do, like how you've ignored my mathematical research for so long, but kept acting like you like mathematics, or lectured, or wrote papers.

I want to know how you did it. I want to understand you. Know what was going on in your heads, and how you kept going, when I've been out here, looking for something to end this, and if you had a clue of the necessity of this ending.

Did you try to live your life like nothing, not protecting yourself, not doing anything to defend yourself, as if the mathematics could really be blocked, even though you should have known how the story had to end?

If so, how? If so, why?

## JSH: Politics

As I consider this factoring idea using the square root ambiguity I find it amazing some behavior displayed by two sci.math regulars who stepped in with mathematics when they thought they could dismiss my ideas, but quit posting math when I pushed back a bit, also, of course, I reminded them that factoring is a BIG DEAL, so they needed to be careful.

But why not finish out the equations as I described?

I suggest to you that they never cared about what was mathematically correct, but were playing a political game to win political points on the newsgroup.

You people believe in consensus.

To you a mathematical argument is true because everyone agrees it's true.

I think you're stupid for that belief.

People get together and believe all kinds of dumb things.

A LOT of math people can get together and all be blinded one way or another, declare an argument to be a proof when it's not, and go home happy as hell.

That's human nature.

People are limited. Human beings can convince themselves of just about anything, and fight to the death over it, even when it's wrong.

People can fight to the death, over wrong ideas.

And you people think your system of having some people look over some complicated stuff actually can be relied on to work?

Either you're very stupid, and unwilling to learn lessons of history, or you're so naive that, well, I think for many of you, you are NOT stupid, and NOT naive, but know full well that with today's system, a person could have a full career as a mathematician, and never actually have a single proof.

But instead, have a lot of people who signed off on math-ese and complex arguments that were crafted to LOOK like proofs in a system that is about human judgement—not mathematical truth.

But why not finish out the equations as I described?

I suggest to you that they never cared about what was mathematically correct, but were playing a political game to win political points on the newsgroup.

You people believe in consensus.

To you a mathematical argument is true because everyone agrees it's true.

I think you're stupid for that belief.

People get together and believe all kinds of dumb things.

A LOT of math people can get together and all be blinded one way or another, declare an argument to be a proof when it's not, and go home happy as hell.

That's human nature.

People are limited. Human beings can convince themselves of just about anything, and fight to the death over it, even when it's wrong.

People can fight to the death, over wrong ideas.

And you people think your system of having some people look over some complicated stuff actually can be relied on to work?

Either you're very stupid, and unwilling to learn lessons of history, or you're so naive that, well, I think for many of you, you are NOT stupid, and NOT naive, but know full well that with today's system, a person could have a full career as a mathematician, and never actually have a single proof.

But instead, have a lot of people who signed off on math-ese and complex arguments that were crafted to LOOK like proofs in a system that is about human judgement—not mathematical truth.