JSH: Difference of squares

jstevh_at_msn.com
Date: 03/22/05 

Date: 21 Mar 2005 17:44:14 -0800

I think that maybe many of you don't know the significance of my
surrogate factoring methods so easily giving you a difference of
squares with some target M to be factored, so I'll explain again how
factoring is typically done.

Consider

x(x+a) = M

where M is your target, as then you can use the quadratic formula to
solve for x, and get

x = (-a +/- sqrt(a^2 - 4M))/2

and most of the work in factoring has to do with figuring out some a,
such that the square root is an integer, and you factor M.

Now with surrogate factoring though, you get

sqrt((Az + 2M^2)^2 + 4M^2T))

where you have M lined up for a difference of squares, and you just
have Az as the variable, where you get a LOT of solutions for Az just
by factoring T and some other number I call j, which is what has been
done before.

Here, though, importantly, you have rationals, which makes it harder
upfront, but still doesn't explain preferential factoring of M, such
that you only get trivial factors.

The weird thing you see, is that the previous methods I tried worked so
badly.

Clearly *something* was blocking the factorization of M into
non-trivial factors.

If you know anything about factoring, seeing how easily I can get a
difference of squares should scare you more than just a little bit, as
it means that even if I'm wrong now, that finding a way to get this to
work might just be some little thing, as you have the difference of
square--easily.

You see, the other factoring methods, like even the Number Field Sieve,
basically work to get you to a difference of squares.

Surrogate factoring hands it to you from the start.

SOMETHING must have been blocking that difference of squares from
factoring M with my earlier work, and if that is handled, then that's
it.

My own perspective is that you people are NUTS, as if someone figures
this out, then who knows what will happen?

But your confidence in the face of such basic mathematics--in believing
it must be wrong I guess because I talk about it--is foolhardy at best.

James Harris