Harristotelian Logic

Playing around with various approaches to the factoring problem I noticed that if I did something as simple as consider T = 9 mod 11 and T = 2 mod 13, I would find that the minimum positive number for T that would work would be 119, which is a number I like to use in my examples and it occurred to me that the primes were forcing something.

So I started thinking about what information prime numbers could give about factors as if the two primes were forcing T to be 119 or greater, then they were also forcing the factors to be certain values.

So I expanded out a factorization with primes:

(f_1 + c_1*p_1)(f_2 + c_2*p_1) = T = (r_1 + k_1*p_1)

and

(g_1 + d_1*p_2)(g_2 + d_2*p_2) = T = (r_2 + k_2*p_2)

And I multiplied out and solved for k_1 and k_2 respectively and again pondered, what if you GUESSED using f_1*f_2 = T mod p_1 and g_1*g_2 = T mod p_2, so you'd know the f's and the g's?

And I realized that then you could use

(f_1 + c_1*p_1) = (g_1 + d_1*p_2)

and

(f_2 + c_2*p_1) = (g_2 + d_2*p_2)

and reduce to a solution for d_1.

d_1 = (f_1 - g_1)*p_2^{-1} mod p_1

and then you also need d_2, so

d_2 = (f_2 - g_2)*p_2^{-1} mod p_1.

So suddenly I had it!!! Information from the intersection of the primes.

The two prime numbers were now telling me something about a key variable in the expanded factorization!!!

And now you can go back to

(g_1 + d_1*p_2)(g_2 + d_2*p_2) = T = (r_2 + k_2*p_2)

multiply out, and divide by p_2, using m_2 = (g_1*g_2/p_2), and you can get to

k_2 = d_1*g_2 + d_2*g_1 + d_1*d_2*p_2 + m_2

so now you just substitute modulo p_1 and compare what you get on the right side with k_2 mod p_1 as you know what k_2 is, since

k_2 = floor(T/p_2).

Now then, does this work?

Well, is there an answer for d_1 mod p_1 and d_2 mod p_2?

Of course, yes. If I have the correct residues for the f's and g's then will that answer be given by

d_1 = (f_1 - g_1)*p_2^{-1} mod p_1

and

d_2 = (f_2 - g_2)*p_2^{-1} mod p_1?

The answer is, of course, yes.

Now perturb them. Shift f_1 or g_1 and will you change d_1 and d_2?

YES!!!

So then, this method will work to tell you when you have them correctly so the answer to the key question of factoring is available: what is f_1 mod p_1 and f_2 mod p_2?

As you answer that question prime by prime you can factor the target, as a minimum positive value for f_1 is forced, like T = 9 mod 11 and T = 2 mod 13 forced a minimum positive value for T.

Now that is easy math. Trivial algebra.

Hating mathematics is about hating truths that you don't like, and I know that feeling so I can understand why so many of you despise mathematics, even if you claim to be mathematicians.

You despise it because you do not control it.

That's why your society came up with "delicate proofs" and "logical contradictions" as you found a human solution to an inhuman discipline.

The math does not care. What's true is true even if you can't make a living in the field telling the truth.

So most of you learned to lie as otherwise, you'd have to make your living some other way, and if society let you, why not live the fantasy?

Why not just pretend? Why not act? Actors are heroes in this society right?

Like Melanie Griffith. Or Brad Pitt. Or Morgan Freeman. Or Claire Forlani.

Actors make big bucks, and get accolades so why not just act with mathematics too?

So if they could, why not you?