Harristotelian Logic

One of my most powerful simple ideas was to use identities against a hard math problem, as it is such a simple idea—once you consider it—to use identities and subtracting out an equation to be analyzed so that you use the algebraic residue.

For instance, consider the maybe seemingly trivial identity:

x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2

where since residues are what's important you can move to

x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2)

and now manipulate the equations in various ways, where because it is an identity, x, y, z, and v can be ANY numbers that you choose. So like you can say the ring is algebraic integers, and any algebraic integer will be ok, for x, y, z or v, because you're just manipulating an identity.

I call

x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2)

a tautological space, as it is a space in that you have variables, and in this case 4 variables so I call it a 4-dimensional space, and it's tautological in that it's always true, as you're just using an identity, where I figured out a way to use tautologies in mathematics to do detailed analysis.

For instance, consider x^3 + y^3 = z^3, and you can subtract that out of your tautological space, do some algebra and find that

(v^3+1)z^6 - 3x^2y^2(vz^2) - 2x^3y^3 = (a_1 z^2 + b_1 xy)(a_2 z^2 + b_2 xy)(a_3 z^2 + b_3 xy)

where the a's are roots of

a^3 -3va^2 + v^3+1 = 0

and for the b's you have b_1*b_2*b_3 = -2.

Lot of complexity there that just exploded out at you, right? But what I did was just go from

x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2)

to

x^2 + y^2 = -vz^2 (mod x^2 + y^2 + vz^2)

and cubed, and I took x^3 + y^3 = z^3 and squared and subtracted, to get the algebraic residue

(v^3+1)z^6 - 3x^2y^2(vz^2) - 2x^3y^3 = 0 (mod x^2 + y^2 + vz^2)

so it's not really so complicated but it is powerful, as remember, I subtracted from an identity, so in analyzing the residue I'm actually analyzing x^3 + y^3 = z^3.

But crucially v is a free variable, so I can make it whatever I wish, which is the handle given to me by this analysis technique. Now you can proceed to prove that x, y and z cannot be integers in a straightforward way that is easily extended to p odd prime.

And that is a quick introduction to tautological spaces and algebraic residues as I extended concepts began by Gauss.

Algebraic residues are the natural next step from his research where it just took a hundred years or so for the progression but that is the way mathematics actually works, as new techniques cannot be discovered until humanity is ready for them.