Harristotelian Logic

Yesterday I made a post with some doodlings—I was actually just playing around--which had some simple dumb errors, easily corrected, but the quadratics easily disprove standard teachings in several areas.

To give you some grasp of the contempt some people have for the truth look over that thread to see the easy conclusions ignored.

So I will explain again, with the corrections.

Consider

x^2 - 5x + 2 = 0

and

y^2 - 7y + 2 = 0

and solve each where I'll not use +/- and just use plusses:

x = (5 + sqrt(17))/2 and y = (7 + sqrt(41))/2.

Now I can subtract

x^2 - 5x + 2 = 0

from

y^2 - 7y + 2 = 0

to get

y^2 - x^2 = 7y - 5x

and group a bit, and divide by y - x to get

y + x = 2y/(y-x) + 5

and y-x must have 2 as a factor if y and x are not coprime to each other because of y+x on the left hand side, but because of the 5 on the right hand side, y-x must divide out ALL factors of 2 in 2y that are in common with x and y.

The gist of it then is that any factors x has in common with 2 it MUST have in common with y, or it must be completely coprime to y, which breaks standard ideas from Galois theory, as it means factors of 2 distribute the same way in both

x^2 - 5x + 2 = 0

and

y^2 - 7y + 2 = 0.

Moving on and making the substitutions then I have

(7 + sqrt(41))/2 + (5 + sqrt(17))/2 = 2(5 + sqrt(17))/(2 + sqrt(41) - sqrt(17)) + 5

which proves that 2 + sqrt(41) - sqrt(17) must be a factor of 2.