Wednesday, January 23, 2008
JSH: Brilliant little approach
I'm waiting to see if I'm validated by others recognizing value in this research approach, like by acceptance by the math journal which has the paper, or for de-validation, by someone being able to find something actually wrong with it, which is why I keep checking math newsgroups.
But there is no doubt about it being a brilliant little approach.
What I did was to just add one more congruence to the traditional difference of squares so that instead of just having x^2 = y^2 mod T, where T is the target composite (mathematicians often use "N"), I use two congruences:
x^2 = y^2 mod p
and
z^2 = y^2 mod T.
And I find I get these nifty little relations that give f_1 mod p and f_2 mod p, where f_1*f_2 = nT.
If modern mathematicians were an ounce of what they present themselves as then such a simple but remarkably powerful approach would get an objective consideration, or if it's not new, someone could cite a source where that approach has been done:
f_1 = ak mod p
and
f_2 = a^{-1}(1 + a^2)k mod p
where
k^2 = (a^2+1)^{-1}(nT) mod p
and T is the target composite with integer factors f_1 and f_2, such that
f_1*f_2 = nT, and with any prime p, those relations will be true 75% of the
time.
That's just mathematics. Those equations don't have opinions. They don't have feelings. And they don't give a damn what you think.
They have always existed and will always exist long after you are quite dead, and forgotten.
Whether they have value or not in this small space on some little planet full of people who can say one thing and be another is a different issue all about how small many of you actually are, despite your desire to be THOUGHT of as great.
While I would rather be great.
But there is no doubt about it being a brilliant little approach.
What I did was to just add one more congruence to the traditional difference of squares so that instead of just having x^2 = y^2 mod T, where T is the target composite (mathematicians often use "N"), I use two congruences:
x^2 = y^2 mod p
and
z^2 = y^2 mod T.
And I find I get these nifty little relations that give f_1 mod p and f_2 mod p, where f_1*f_2 = nT.
If modern mathematicians were an ounce of what they present themselves as then such a simple but remarkably powerful approach would get an objective consideration, or if it's not new, someone could cite a source where that approach has been done:
f_1 = ak mod p
and
f_2 = a^{-1}(1 + a^2)k mod p
where
k^2 = (a^2+1)^{-1}(nT) mod p
and T is the target composite with integer factors f_1 and f_2, such that
f_1*f_2 = nT, and with any prime p, those relations will be true 75% of the
time.
That's just mathematics. Those equations don't have opinions. They don't have feelings. And they don't give a damn what you think.
They have always existed and will always exist long after you are quite dead, and forgotten.
Whether they have value or not in this small space on some little planet full of people who can say one thing and be another is a different issue all about how small many of you actually are, despite your desire to be THOUGHT of as great.
While I would rather be great.
# posted by James Harris @ 01:13 
