Harristotelian Logic

A remarkable but simple result throws a wrench in old ideas about numbers, with yet another demonstration by me that those ideas DO NOT work.

Begin simply with

x^2 - ax + n = 0

and

y^2 - by + n = 0

and subtract the first from the second and simplify a bit to get

y^2 - x^2 = by - ax

so

y^2 - x^2 = (b-a)y + ay - ax

and now I easily I have

(y + x - a)(y - x) = (b-a)y.

But now let's go to work,

let a = sqrt(5), b=sqrt(13), and n=-3, then I have as a solution

y = (sqrt(13) + sqrt(13 + 12))/2 = (sqrt(13) + 5)/2

and

x = (sqrt(5) + sqrt(5 + 12))/2 = (sqrt(5) + sqrt(17))/2

so

((sqrt(13) + 5)/2 + (sqrt(5) + sqrt(17))/2 - sqrt(5))((sqrt(13) + 5)/2 - (sqrt(5) + sqrt(17))/2) = (sqrt(13) - sqrt(5))((sqrt(13) + 5)/2)

which is

(sqrt(13) - sqrt(5) + 5 + sqrt(17))(sqrt(13) - sqrt(5) + 5 - sqrt(17))/2 = (sqrt(13) - sqrt(5))((sqrt(13) + 5)

which proves that sqrt(13) - sqrt(5) and sqrt(13) + 5 must share the same factors in common with 2.

And easily enough key ideas in the standard teaching of Galois Theory go out the window.

Now consider more carefully why computers are not used to check arguments claimed to be proofs as it's not because the math is actually brilliant, as much of it is actually wrong.

As I show yet again in a small space with easy algebra.

The problem with denial is that people deny for a reason—to preserve what they have.

And math people who use the flawed ideas get something from them, more than they feel they get from the truth, so they just lie, and lie on so many subjects from computer checking, to prime numbers, to factoring without regard for the consequences as all they want to do is—hold on to what they have.