Harristotelian Logic

Here's an explicit solution without the congruences showing how surrogate factoring can work for those of you who lack familiarity or confidence in congruence relationships.

Let

x^2 - y^2 = a*T

where x and y are to naturals to be determined, 'a' is some unknown integer and T is the target to be factored.

Now let

S = 2*x*k + b*T

where S and k are integers that will be chosen and 'b' is some unknown integer.

So adding y^2 to both sides with the first gives

x^2 = y^2 + a*T, and adding the second equation in gives

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

so I can complete the square with

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

so I have with some simplification and a focus on y:

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

so

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

and I only have that a-b not equal 0, so I can burn a degree of freedome and now let

a - b = 1

and solve for x and y.

I think the explicit solution is necessary for those who are uncomfortable with congruences, as the situation now is beyond bizarre.

Oh yeah, it's now also clear why Tim Peters couldn't get this to always work before, as my original approach is like using

a-b = 0

here, but with slightly different equations and can also be made to work, by use of T itself, which locks the equations to your target composite.

Easy math people. No way if you look it over you can miss it, or not understand how big it should be, as I've found a way to get equal quadratic residues modulo a target composite T, which involves factoring some number other than T which I call the surrogate.

Easy math—potentially huge consequences.

The ball is now in your court.

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It occurs to me that some of you may not know how to proceed from that equatio to solve for x and y, so here's how.

If you have S and k, like S=1, k=1, and use, let's say a-b=1 still so you have

y^2 = (x+k)^2 - 1 - 1 - T = (x+k)^2 - 2 - T

then solving for x and y is just a matter of factoring T+2.

I like to show that using factors f_1 and f_2, where

f_1 * f_2 = 4*(T+2), so

x+k = f_1 + f_2

and

y = f_1 - f_2

and you can see trivially that it all balances out.

So how might this method not work?

I'm puzzling over that, and it seems to me that there is nothing forcing 'a' and 'b' to be integers, while a-b is forced to be an integer, so maybe they can divide out T in both cases, which is the only mathematical way this technique could fail besides producing a trivial factorization, as otherwise it will factor T.

I like simple equations as they leave little room for error or confusion about how they work to hide.

Still it's just basic research at this point. I, still, have yet to factor with these ideas, as I work out the theory.

I'm a theory guy. I leave experimentation for the experimentalists.

Now then, why would 'a' and 'b' consistently block out a factorization by dividing off T?

[A reply to someone who said that James had other reasons for not wanting to do experiments.]

Look over the equations, there is only one way for the approach not to work, which is if 'a' and 'b' are fractions that divide off T, despite a-b being chosen to be an integer.

As an interesting sidepoint, there are 7 variables, and 3 equations, where T is the target composite, so you choose it, of course, and you choose S and k, which takes up 6 degrees of freedom, leaving the final one to be taken by looking for integer solutions to the square root, handling all degrees of freedom.

The approach expressed explicitly can be considered very easily from multiple angles, so there is that thing I love—simplicity.

Simple ideas remove the ability of people to cherry-pick just on the difficulty of anyone else understanding the ideas, and in this case, with factoring, they help make my other point that mathematicians routinely break their own rules.

So you can't get mathematicians to show interest in anything that they don't like, no matter what its mathematical value because they're like spoiled children completely used to using arbitrary decisions without regard to real mathematical value, and one of their arbitrary positions is keeping out people they label "crank", "crackpot", or "loons".

They have no true love of their discipline and no true interest in knowledge for its own sake, but care only about what suits their personal needs, where you can just look over my posting here, with a simple factoring idea, and consider their rejection of it.

And they're clearly not very bright as if these ideas DO have value then a lot of people in the world are going to be mad as hornets when they realize that what I said above is true, and mathematicians are not only weird in ways that most people knew about, but they don't actually care about their own field, or the impact of important research on others, as if they're not in the real world with the rest of us, but in
their own little self-created world.