Friday, August 31, 2007
JSH: Surrogate factoring, periodic behavior
Having completed better analysis on surrogate factoring I found the equations that explain a periodic behavior at least one person has noted in posts, where for a given k and n, if you find a prime factor p of your target T with that n, then you will find other solutions by adding multiples of p to n.
Two of the equations determining that behavior are
Cw = n + (k + 2xr_1*p_1)( k + 2xr_2*p_2) - ((k + 2xr_1*p_1)( k + 2xr_2*p_2) - 2k^2)/T
and
w = k + 2xr_2*p_2 mod T
where if the second equation is true for a given n, then you will have a solution to the surrogate factoring equations at that n, but that is an only if. There C doesn't matter but is just some non-zero integer, as w just needs to be any factor of the right side—which is an integer I should note as the T must divide through—for which the second condition is met.
That is the primary decision relation that determines if a surrogate factorization can work or not.
Remember the surrogate factorization involves factoring a target composite T by solving
(x+k)^2 = y^2 + 2k^2 + nT
where the primary question has been, how do you pick k and n?
If they are picked correctly then some solution for x and y will also be a solution for
x^2 = y^2 mod p
where p is a prime factor of T.
[A reply to someone who asked why should James' method process have a high probability of working.]
Yes, questions, the mark of true researchers and human beings in general, as human curiosity is such a wonderful thing.
We wonder why and in looking for answers humanity finds new things.
So yeah, like I mentioned in another thread, the question in my mind for some time has been how so many of you seem to lack basic human curiosity.
Does the idea work at all? If not, why not? If so, how?
Learning begins with questions.
Now I have worked for years at answering questions presented by an idea, which was, could you factor one number with another, and I kept at it despite derision and insults from people like you.
You are the jocks of the schoolyard who tease that strange little boy who is so fascinated with his books.
Whether you wanted to be or not, or thought you hated those people growing up, that is your behavior against me and always has been. Maybe you hated them growing up because you wanted to BE them, and given the slightest excuse they are who you became.
You are the cruel jocks picking on the kid you call nothing.
And I am the genius.
Two of the equations determining that behavior are
Cw = n + (k + 2xr_1*p_1)( k + 2xr_2*p_2) - ((k + 2xr_1*p_1)( k + 2xr_2*p_2) - 2k^2)/T
and
w = k + 2xr_2*p_2 mod T
where if the second equation is true for a given n, then you will have a solution to the surrogate factoring equations at that n, but that is an only if. There C doesn't matter but is just some non-zero integer, as w just needs to be any factor of the right side—which is an integer I should note as the T must divide through—for which the second condition is met.
That is the primary decision relation that determines if a surrogate factorization can work or not.
Remember the surrogate factorization involves factoring a target composite T by solving
(x+k)^2 = y^2 + 2k^2 + nT
where the primary question has been, how do you pick k and n?
If they are picked correctly then some solution for x and y will also be a solution for
x^2 = y^2 mod p
where p is a prime factor of T.
[A reply to someone who asked why should James' method process have a high probability of working.]
Yes, questions, the mark of true researchers and human beings in general, as human curiosity is such a wonderful thing.
We wonder why and in looking for answers humanity finds new things.
So yeah, like I mentioned in another thread, the question in my mind for some time has been how so many of you seem to lack basic human curiosity.
Does the idea work at all? If not, why not? If so, how?
Learning begins with questions.
Now I have worked for years at answering questions presented by an idea, which was, could you factor one number with another, and I kept at it despite derision and insults from people like you.
You are the jocks of the schoolyard who tease that strange little boy who is so fascinated with his books.
Whether you wanted to be or not, or thought you hated those people growing up, that is your behavior against me and always has been. Maybe you hated them growing up because you wanted to BE them, and given the slightest excuse they are who you became.
You are the cruel jocks picking on the kid you call nothing.
And I am the genius.
# posted by James Harris @ 10:09 
