Sunday, April 30, 2006
JSH: Latest factoring idea, might work
I have been fascinated by this very simple yet very powerful quadratic residue result of mine, and not surprisingly, turned it against the factoring problem! (It looks like it can be used against Goldbach's, but that's not as sure a bet for me.)
What's weird here is that the freaking solution, is so freaking simple.
Because I found that the difference of factors of a composite are related to the quadratic residues of the numbers that add to give the composite, you can take any composite, find a couple of numbers that add to give it, and get info about its factors by these congruence relationships.
But I puzzled over that for a while not seeing a way to easily factor with that, until it occurred to me to force the difference of factors.
That is, if you have C, where C = p_1 p_2, then the difference of factors, p_1 - p_2, is, of course, enough information to factor C. But, of course, you don't have that difference.
But what if you multiply by ab, so you have ap_1 - bp_2 = 1?
Now you've forced the difference, but how do you find ab?
Well, my result lets you find ab mod n_1, where n_1 is some natural you get to pick!!!
It's so damn obvious that it just seems too simple, even now.
I am looking for someone to shoot down my exposition of the method in my other thread, to point out what's wrong with this idea.
If there is nothing wrong with it, then someone searching could have a very small search space with a large n_1 to factor even a large composite.
If this idea does work, it will be the second method that I've discovered that I think solves the factoring problem, as my previous hyperbolic factoring method, I think, should work, but it's complicated compared to this.
It is elegant in its own way though, and eventually I want to test it to see if it does work.
Guess I could easily test this latest idea as well, but I just can't bring myself to check factoring problem attempted solutions any more. They're just depressing in a way as they represent the need for force.
I mean, if things were the way they are supposed to be, I should have chosen going after Goldbach's Conjecture versus the factoring problem.
But things are messed up with current mathematicians fighting to keep the world from knowing about flawed mathematical ideas that I've proven don't work, so they have to be moved by force, as at this point, they show no signs of ever giving in to what's mathematically correct of their own free will.
What's weird here is that the freaking solution, is so freaking simple.
Because I found that the difference of factors of a composite are related to the quadratic residues of the numbers that add to give the composite, you can take any composite, find a couple of numbers that add to give it, and get info about its factors by these congruence relationships.
But I puzzled over that for a while not seeing a way to easily factor with that, until it occurred to me to force the difference of factors.
That is, if you have C, where C = p_1 p_2, then the difference of factors, p_1 - p_2, is, of course, enough information to factor C. But, of course, you don't have that difference.
But what if you multiply by ab, so you have ap_1 - bp_2 = 1?
Now you've forced the difference, but how do you find ab?
Well, my result lets you find ab mod n_1, where n_1 is some natural you get to pick!!!
It's so damn obvious that it just seems too simple, even now.
I am looking for someone to shoot down my exposition of the method in my other thread, to point out what's wrong with this idea.
If there is nothing wrong with it, then someone searching could have a very small search space with a large n_1 to factor even a large composite.
If this idea does work, it will be the second method that I've discovered that I think solves the factoring problem, as my previous hyperbolic factoring method, I think, should work, but it's complicated compared to this.
It is elegant in its own way though, and eventually I want to test it to see if it does work.
Guess I could easily test this latest idea as well, but I just can't bring myself to check factoring problem attempted solutions any more. They're just depressing in a way as they represent the need for force.
I mean, if things were the way they are supposed to be, I should have chosen going after Goldbach's Conjecture versus the factoring problem.
But things are messed up with current mathematicians fighting to keep the world from knowing about flawed mathematical ideas that I've proven don't work, so they have to be moved by force, as at this point, they show no signs of ever giving in to what's mathematically correct of their own free will.
# posted by James Harris @ 12:07 
