## SFT: Experimentation starts

Well I finally couldn't resist the impulse to check out the SFT, and hey, the equations actually do work!

It's always nice to see that at least you got the equations right.

In any event, I did verify though that as you use *integer* factors to get your surrogate, the factoring percentage drops as the size of the number increases.

So one of my assumptions was wrong, as I felt that it wouldn't care and factor about 50% of the time without regard to size, which was not the case--with integer factors of the surrogate.

However, in checking that result, I also looked at what I call z in the generalized SFT and found that it dropped in size relative to the number I call x.

So there seems to be a definite movement in one direction when only integers are used for the surrogate, and that direction is to smaller z's and a smaller factoring percentage.

However, the theorem works over all rationals, and focusing on integers is a human choice, where it looks like I can narrow down mathematical reasons for it mattering, which is what I'll probably move towards as I experiment.

There are some simple reasons why the SFT cannot in general care about what factor it gives you of your target--in rationals--but also, it does seem to know integers, and behave differently when integers are involved.

Further experimentation and theorizing should reveal why.

I'll include the generalized SFT for reference.

Generalized Surrogate Factoring Theorem:

Given non-zero integers A and B, let

f_1 f_2 = A^2 (A^2 - B^2)

then

f_1 = (-(z - 2A^2)+ sqrt((z - 2A^2)^2 - 4A^2(A^2 - B^2)))/2

and

f_2 = (-(z - 2A^2) - sqrt((z - 2A^2)^2 - 4A^2(A^2 - B^2)))/2

and

z = x(x +/- sqrt((x - 2B^2)^2 + 4B^2 (A^2 - B^2)))/(2x - 2A^2)

and x is given by

x = +/- (g_1 - g_2) + 2B^2

where

g_1 g_2 = B^2(A^2 - B^2).