Wednesday, March 07, 2007
Factoring has two legs
The theoretical side of my latest research is to show one and only one more remaining fundamental factoring congruence, showing you could say, that factoring has two legs:
With T the target integer to be factored
x^2 = y^2 mod T
and
k^2 = 2xk mod T
and, of course, the first has been known for centuries and a highly refined development of it is the modern Number Field Sieve, while the second is my addition, for what I call surrogate factoring.
The second congruence—the second fundamental leg of factoring—represents the mathematics that brings in what I christened surrogate factoring, where you can factor your target T, by solving for the two congruences, with a chosen non-zero integer k, using explicit equations.
There is no debate about whether or not it can be done, but a general consensus I've gathered that it is not important, and research using the second leg cannot be made practical, seems to be the opinion of the mathematical community.
But it is the only other congruence that extends factoring, and I'm sure some would try to debate that, as I make a theoretical point, so I leave it as an open challenge—knowing I've often lost these!!!—for someone to show that it is not.
My own research into this second leg, the mathematics that gives you surrogate factoring, indicates that it is not picky, as there must exist solutions for any integer k, so there are solutions for the difference of squares, and that the methods that gives you solutions cannot be shown to be picky either, so that hey, it could work well enough with the Number Field Sieve to greatly extend the range of
numbers that can be factored.
That is, bringing the two factoring legs together allows you to go much further.
But, who knows. At this point it's early research, as while I've tossed the idea of surrogate factoring around for years, I just realized how simple the surrogate factoring piece actually is—back around August of last year.
For mathematical research that's like, just found, so it's rather new, but kind of cool from what I've tested with factoring numbers, and from what I've seen with small numbers—the easier for me to test—the results follow theory.
Those who don't understand how I use Usenet need to know that I tend to post on Usenet to announce, while previously I posted as well to work out theory, and often to grouse about non-acceptance of my mathematical research.
This post is primarily a continuation of me announcing.
I have factoring research. Here it is in terms of the basics. I say, hey, other people should care, but if they don't, what then?
I don't know. But if it is important, somehow, someway, then, of course, I figure that will come out eventually.
With T the target integer to be factored
x^2 = y^2 mod T
and
k^2 = 2xk mod T
and, of course, the first has been known for centuries and a highly refined development of it is the modern Number Field Sieve, while the second is my addition, for what I call surrogate factoring.
The second congruence—the second fundamental leg of factoring—represents the mathematics that brings in what I christened surrogate factoring, where you can factor your target T, by solving for the two congruences, with a chosen non-zero integer k, using explicit equations.
There is no debate about whether or not it can be done, but a general consensus I've gathered that it is not important, and research using the second leg cannot be made practical, seems to be the opinion of the mathematical community.
But it is the only other congruence that extends factoring, and I'm sure some would try to debate that, as I make a theoretical point, so I leave it as an open challenge—knowing I've often lost these!!!—for someone to show that it is not.
My own research into this second leg, the mathematics that gives you surrogate factoring, indicates that it is not picky, as there must exist solutions for any integer k, so there are solutions for the difference of squares, and that the methods that gives you solutions cannot be shown to be picky either, so that hey, it could work well enough with the Number Field Sieve to greatly extend the range of
numbers that can be factored.
That is, bringing the two factoring legs together allows you to go much further.
But, who knows. At this point it's early research, as while I've tossed the idea of surrogate factoring around for years, I just realized how simple the surrogate factoring piece actually is—back around August of last year.
For mathematical research that's like, just found, so it's rather new, but kind of cool from what I've tested with factoring numbers, and from what I've seen with small numbers—the easier for me to test—the results follow theory.
Those who don't understand how I use Usenet need to know that I tend to post on Usenet to announce, while previously I posted as well to work out theory, and often to grouse about non-acceptance of my mathematical research.
This post is primarily a continuation of me announcing.
I have factoring research. Here it is in terms of the basics. I say, hey, other people should care, but if they don't, what then?
I don't know. But if it is important, somehow, someway, then, of course, I figure that will come out eventually.
# posted by James Harris @ 21:27 
