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?
# posted by James Harris @ 22:18 
