## JSH: Pure, pure math

I have been working feverishly to find something HUGE that will push things forward with my latest result, but I'm starting to think it's pure, pure math, as in being such a raw result that it's hard to find anything practical with it.

The result, of course, as I like talking about it so much so you probably have seen it if you read my threads, is with natural numbers, n_1, n_2, C, and k, where

C = n_1 + n_2

and k is a difference of factors of 2C, it must be true that

(8C + k^2) mod n_1

is a square modulor n_1, and

(8C + k^2) mod n_2 is a square modulo n_2,

where I have written that differently in the past, as it also means that

(8n_2 + k^2) mod n_1 is a square modulo n_1, and vice versa,

which follows, of course, fomr C = n_1 + n_2.

That is a pure, pure math result relating the factorization of twice a natural to the quadratic residues of ALL natural numbers, except 1, of course, less than that natural number.

In a way it's a bizarre result, as why in the hell is the addition of two numbers related to the factorization of twice the sum in any way at all?

Well, it may seem strange, but that's the mathematics. The mathematical logic of it is perfect but for creatures who think of addition as trivial, it's odd to find out that there is all this mathematical machinery relating all the freaking natural numbers below a given natural to the difference of factors of twice that natural.

Oh, goody, here's another pure, pure result in there, as why twice the natural?

Maybe it has something to do with primeness as if you have a prime, then with just that prime the only possible difference is p-1, looking at absolute value. But with twice it, you have p-2 and 2p-1, so maybe the math needs at least two possibilities for eveything to work right.

I'd like to think this result might have some practicality in determining primeness, but I think I'm reaching again. Freaking pure, pure math results.

I need something a lot less pure, and a lot mroe practical to break the stupid impasse with dumb mathematicians all over the world ignoring my research.

Maybe in a few years this one will get less pure, and more powerful, but until then, I guess the search for a back breaker against the mathematical community, will continue.

I will find it. And I will be madder than ever when I do.