Harristotelian Logic

Factoring may be simpler than previously believed as consider the following generalized factoring idea:

With T the target composite:

x^2 - y^2 = 0 mod T

is how far researchers previously went, and that area is well-worked showing it difficult to factor T as T increases in size, but my research shows it to just be a primitive case of a more general solution found by using two additional variables, S and k, where

S - 2xk = 0 mod T

which allows you to now use quadratic methods as usual as you easily then have

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

where n is some non-zero integer, and notice, importantly, these generalized factoring equations default to the well-known ones with S=k=0.

But with S and k non-zero they indicate a factorization of S + k^2 + nTas the route to factoring T itself, as the general factoring method.

With n nonzero, thorough analysis of when the ideas shown here lead to a non-trivial factorization of a composite T do not show the normal rules, like indications that the size of T matters. I've just done a bit of analysis in this area and as of yet have found no indication that these ideas cannot be made practical, though I haven't done it myself, only having done initial theory.

But consider, all that I did in actuality was find a more generalized set of factoring equations, which include those typically used in previously known approaches when S=k=0.