Difference of squares ,SF Theorem

jstevh_at_msn.com
Date: 04/10/05 

Date: 9 Apr 2005 16:16:34 -0700

So I've focused on the surrogate factoring theorem for a couple of
important reasons:

1. A theorem is an absolute, once the proof is gone over, there's no
room for rational doubt.

2. The theorem shows how to calculate a difference of squares.

That second is the important point that posters routinely avoid, as
they must, as beyond the issue of practicality, there is the
mathematical reality that no other method known just gives you a
difference of squares, potentially non-trivial, like the SF Theorem
does.

Surrogate Factoring Theorem:

Given M, a target natural number to be factored, and j, an integer
chosen such that j^2>M^2, a rational factor b_2 of M is given by

b_2 f_1 = (-(Az - 2M^2)+/- sqrt((Az - 2M^2)^2 - 4TM^2))/2

where T = M^2 - j^2, and f_1 is a rational factor of T, and where Az is
given by

Az = Ax(Ax +/- sqrt((Ax - 2j^2)^2 + 4Tj^2))/(2Ax - 2M^2)

where Ax is then given by

Ax = +/- (k_1 + k_2) + 2j^2

where k_1 k_2 = -Tj^2, and k_1 and k_2 are rationals.

So if you factor Tj^2, where you pick j and T = M^2 - j^2, where M is
your target to be factored, then you can use those factors to get a
difference of squares seen in

b_2 f_1 = (-(Az - 2M^2)+/- sqrt((Az - 2M^2)^2 - 4TM^2))/2

and yes, the natural inclination is to factor into integers, but you
can factor into rationals, so you have an *infinite* number of
difference of squares that the theorem just gives you.

The Number Field Sieve and the QS work so hard just go get a few to try
and factor, and the SF Theorem gives you an infinite number.

That is important enough to explain in detail as let

Az = n/d

where n and d are coprime nonzero integers, so I'm just saying, assume
it's a fraction, and now you can look at

sqrt((Az - 2M^2)^2 - 4TM^2) = sqrt((n/d - 2M^2)^2 - 4TM^2)

and you have simple enough that

sqrt((n - 2dM^2)^2 - 4Td^2 M^2)

must be an integer, and you have it given to you simply by solving for
Az and finding n and d coprime nonzero integers.

The only area where rational argument can still take place then is in
noting that d can keep varying so maybe the numbers continually shift
so that you always just factor T and d, while M refuses to be factored.

Now that's a rational discussion, where I can explain why it doesn't
look like that can happen, but still, there is no other set of
relations known in mathematics that give you the difference of squares
that aren't set to only give you trivial factors.

Like yeah, you can easily write expressions that are difference of
squares--if you only want trivial factors--bu the SF Theorem just
doesn't tell you one way or the other, and posters aren't claiming that
it NEVER gives an answer.

Regardless, how functions behave can be determined by mathematical
means, and the mathematics just does not show any preference for
trivial or non-trivial factors.

It's like, mathematically, it's a moot point, which indicates that it
will factor in a way that appears to be truly random, giving you
non-trivial factors about 50% of the time.

But regardless, since there is nothing in mathematics like the SF
Theorem, it is important in its own right, so I can just keep talking
about it without bothering to test it, in the practical arena.

Part of my point is that math people including mathematicians lie about
mathematics, and that "pure math" is an area where they lie the most,
and the most easily.

If they didn't lie, then there wouldn't be such denial over such
obvious points.

But they DO lie and they lie quite boldly.

If you don't believe me, read up on factoring, find out about the
importance of a difference of squares, and learn.

I'm not really just some "crank" or "troll" or "crackpot" but a real
researcher with major results which affect a lot of mathematicians who
have chosen to lie about those results rather than accept what is
mathematically true.

I'm faced with social issues from people who are adept at lying, who
clearly feel motivated to lie, and now have even more reason to lie as
the situation is far huger than it ever was before, as the consequences
of those lies can affect quite a few people, including many who have no
interest in mathematics.

James Harris

Relevant Pages

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