Monday, January 14, 2008
JSH: Was a simple math test
So I was kind of tricky in terms of not explaining in very simple terms what I was doing though it was obvious if you bothered to look.
All I did mathematically was constrain the standard difference of squares:
z^2 = y^2 + nT
with one more equation:
x^2 = y^2 + pr_1
where p is an odd prime, and you have all integers.
That's it. I discovered it was easy to do that and limit some possibilities so that you are working through a smaller set than you are with the classical unconstrained equation.
The only loss in terms of prime factors you can use for T is 3. And that's no loss for any practical factoring.
With the constraints came rules on which integers could work for z and y, and specifically came equations for z mod p, and y mod p.
Now that's the mathematics. It's not complicated and that is the absolute.
The practical question though is, can that be used to factor?
Well, you get more information than you do without knowing there exists the additional constraint.
If more information is useless for factoring then you have a logical dilemma.
The equations do not remove any information that was there before, so they just give you more information in addition to what was known before.
The logical issue then is, how can more information be useless in mathematics?
Now from a philosophical perspective, it's kind of a fun thing to consider that maybe in mathematics more information IS crap, even when you constrain a system in such a way that you pull more information out of it than you had before, except that's nonsensical.
The system was ALWAYS constrained, but mathematicians only knew one piece!
They worked with just z^2 = y^2 + nT because they didn't know any better, but behind the scenes the other constraining equation was still impacting behavior.
All I did was remove the curtain.
All I did mathematically was constrain the standard difference of squares:
z^2 = y^2 + nT
with one more equation:
x^2 = y^2 + pr_1
where p is an odd prime, and you have all integers.
That's it. I discovered it was easy to do that and limit some possibilities so that you are working through a smaller set than you are with the classical unconstrained equation.
The only loss in terms of prime factors you can use for T is 3. And that's no loss for any practical factoring.
With the constraints came rules on which integers could work for z and y, and specifically came equations for z mod p, and y mod p.
Now that's the mathematics. It's not complicated and that is the absolute.
The practical question though is, can that be used to factor?
Well, you get more information than you do without knowing there exists the additional constraint.
If more information is useless for factoring then you have a logical dilemma.
The equations do not remove any information that was there before, so they just give you more information in addition to what was known before.
The logical issue then is, how can more information be useless in mathematics?
Now from a philosophical perspective, it's kind of a fun thing to consider that maybe in mathematics more information IS crap, even when you constrain a system in such a way that you pull more information out of it than you had before, except that's nonsensical.
The system was ALWAYS constrained, but mathematicians only knew one piece!
They worked with just z^2 = y^2 + nT because they didn't know any better, but behind the scenes the other constraining equation was still impacting behavior.
All I did was remove the curtain.
# posted by James Harris @ 22:11 
