Harristotelian Logic

I really do not know if this latest idea of mine works or not, but I can give advice on testing it out. I myself am doing little to no testing as I'm just not that motivated, as I figure it it's worth something, someone in the world will figure that out.

My idea was to use

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

because the square roots cannot have a single solution, and if you do solve that equation out by multiplying it out, squaring to get rid of sqrt(xy), and completing the square you do find—according to Tim Peters—that you have a factorization dependent on other than just S.

So the point is that even if you have S, there is some other number, I'll call unknown that is also being factored.

Now Tim Peters posted some work where he said he used math software following my instructions to get to that key expression where you have completed the squares, twice.

If those equations give what I think they give, then the number multiplied by S^2 at that point is the number that is being shadow factored, so if you change signs, you get factors of that number.

If that is the case, then this idea can be used on the factoring problem.

If you use the equations that I've put up recently—pulled from what Tim Peters posted--and find that changing signs does NOT give you that number, does not give you unknown, then something interesting is going on:

1. Either he didn't do his work right.

Or

2. The algebra does not depend on the factorization at all, so there's some random thing going on.

Hey, maybe it is random. Maybe there is no rhyme or reason to the number that is shadow factored by a shift in signs with

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

and at this interesting area, mathematics makes no sense, the algebra is not consistent, and there is no way to determine the other number being factored, so it's just a mess.

Maybe I found a way to show that at its heart, mathematics as human beings understand it, is fatally flawed, as in being inconsistent, where things can happen for NO REASON.

I like to think that there is order in mathematics, a beauty and purpose that means that everything has a reason, but then again, I am often wrong.

Maybe, there is no mathematical way to determine the second factorization—the shadow factorization—and posters who are still working to dismiss this research are right.

But think about what they prove:

They prove that in this area I have shown where mathematics fails.

I have found a place where the numbers can't be controlled. The algebra is just a random mess.

And the shadow factorization is uncontrollable, unknowable, and not capable of being controlled by rational thought.

Neat! Fascinating either way.

I may have found a door into chaos.

I suggest to you instead that the shadow factorization CAN be controlled, even if so far no one has figured out how to do it, and that people who push the idea that mathematics is inconsistent—are wrong.

If it is consistent, then there is a way to know what the second factorization will be.

If that's possible then the k's can be set so that it is your target composites.

If so, this approach can be used against the factoring problem.