Harristotelian Logic

Stepping just slightly away from equation familiar from a congruence of squares gives a generalized factoring method which shows a means to balance the difficulty of factoring a number against computational resources.

Beginning with

x^2 = y^2 + aT and k^2 = 2xk + bT

where T is the target composite to be factored, you can easily solve to find

(x+k)^2 = y^2 + 2k^2 + (a-b)T

so that it is clear that 'a' and 'b' need not be determined as you need a-b, which is chosen to be some non-zero integer as is k.

Trivially, if you introduce integer factors f_1 and f_2, you can have

4f_1*f_2 = 2k^2 + (a-b)T

then y = f_1 - f_2 and

x = f_1 + f_2 - k

so by factoring 2k^2 + (a-b)T you can potentially non-trivially factor your target composite T.

Experiments with this method show that the larger 2k^2 + (a-b)T is, the more likely you are to factor the target T by factoring it, but that means it can be roughly as hard to factor as your original composite!

But there are a couple of saving graces, the most important being that the method focuses on the smaller primes first—pulling them out of the composite preferentially.

And, for composites 2k^2 + (a-b)T very close in factoring difficulty to your target composite T, the method works approximately 67% of the time.

So it is a method that allows you to adjust depending on what factoring probability you can handle with your computational power.