Harristotelian Logic

In desperation over the math community's ability to ignore my research I have for some years been searching with little success to find some answer to the factoring problem, thinking that would be research mathematicians couldn't get away with ignoring.

I've been at it more than ever lately with a lot of frustration and failure, but I had this simple idea that just kind of came to me Saturday, but I am afraid to check it.

It represents hope for me, and checking it and finding it wrong, would mean I was still desperate, still searching with very long odds for a way of climbing out of my Hell.

But maybe it's right? If so, there's no way it can be ignored for long, right?

Here it is.

I started by just out of the blue writing down the following equation:

x^2 - y^2 = S - 2*x*k

You can solve for x and y in terms of the other variables easily as that is just

x^2 + 2*x*k + k^2 = y^2 + S + k^2

so

x+k = sqrt(y^2 + S + k^2)

and finding y is just a matter of factoring (S+k^2)/4.

But get this, now I just let

x^2 - y^2 = 0 mod T

then

S - 2*x_res*k = 0 mod T

where I put in x_res as I don't know x, so I'm using its residue.

So x_res = x mod T, and I can solve for k, assuming 2, S and x are coprime to T, with

k = S*(2*xres)^{-1} mod T

where (2*x_res)^{-1} is the modular inverse of (2*x_res) mod T.

So just pick some x_res and S, and find k, which now allows calculation of y, and x.

If that idea is solid then they can't ignore it, right? The mathematicians can't fight something that has monetary value, right?