Harristotelian Logic

So I went ahead and thought more carefully about my latest surrogate factoring equations, and realized why they couldn't work well.

I figured out that to get solutions you needed with

f_1 f_2 = k_2 z^2

and

g_1 g_2 = k_2 z^2 + T

for f_1 + f_2 = g_1 + g_2,

with all the variables rationals, and I chased down that angle and determined that you need to pick k_2 z^2 perfectly, and the odds are slim. So those equations won't work well.

I did notice something weird though, as if you have, with some variable changes for simplicity,

f_1 f_2 = k

and

g_1 g_2 = k + M

with f_1 + f_2 = g_1 + g_2,

it's trival to show that requires that

(2f_1 g_1 - (f_1^2 + k))^2 = (f_1^2 - (k + 2M))^2 - 4M(k + M)

so I thought maybe you could let k be your target, and search for an M that will work to give a rational f_1, but maybe I did some of the algebra wrong as I tried a simple example:

k = 15, M=1

and that gives f_1 = 9, as a possible, which is close, but it'd be nicer if it were f_1 = 3.

In any event, it looks like another failure for the surrogate factoring idea.

The equations could factor, but only if you picked k_2 z^2 precisely and the odds of getting them right are long--unless you already have T factored, so, another failure.