Harristotelian Logic

Kind of a quicker post to cover the big picture while the previous one was a long one:

Short of it is that I found that you could approach factoring through a two equation system:

x^2 = y^2 + pr_1

z^2 = y^2 + nT

while mathematicians have traditionally used a single equation system:

x^2 = y^2 + nT or often given as x^2 = y^2 mod T.

where in both cases T is the target to factor.

With the two equation system I found mathematical laws represented by a set of congruence relations:

z = (2a)^{-1} (1 + 2a^2)k mod p, k^2 = (a^2+1)^{-1}(nT) mod p

and

y = (1+2a^2)^{-1} z mod p, or y = -(1+2a^2)^{-1} z mod p.

I pointed out in my prior long posting that you can verify for yourself quite a few things by considering a simple example with n=1, T=21, as y = 2 is a solution, as, of course 5^2 = 2^2 + 21, and possible values for pr_1 go out to infinity with the start of the sequence being useful for understanding important facts:

pr_1 = (5)(1) = 5, pr_1 = (6)(2) = 12, pr_1 = (7)(3) = 21,

pr_1 = (8)(4) = 32, pr_1 = (9)(5) = 45,

pr_1 = (10)(6) = 60, and pr_1 = (11)(7) = 77.

You can add 4 to any of those to see you get another square and see why you will have results for any prime p.

The congruence relations are trivially derived. The idea that a two equation system will give you more information than just one is not a complicated one. And the proof of the existence of solutions is easy.

Mathematicians just used one equation to study a problem that is best handled by using two.

In a way it is a profoundly cool thing that there were these underlying mathematical laws all along controlling things that people just didn't know anything about, but the sense of satisfaction is muted by the potential impact of the result.

However, it is the reality now that the information is known.

I HAVE been notifying mathematicians directly by email, by posting on math newsgroups, and I have submitted a paper to a major mathematical journal. But so far to no avail.