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.

But I have proven problems in this area before with other simple proofs, as you can see in posts here and on my blog.

The problem with bad math though is that it pays some people's bills. And until there is a big enough shock to make it harder for them to keep going with bad math ideas than go with correct ones, this situation will continue.