Re: Gist of surrogate factoring theorem
From: Proginoskes (proginoskes_at_email.msn.com)
Date: 04/10/05
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Date: 9 Apr 2005 22:38:00 -0700
jstevh@msn.com wrote:
> N. Silver wrote:
> > Rick Decker wrote:
> [...]
> Not really, as in factoring, for years a difference of
> squares has been important, while the surrogate factoring
> theorem gives you an *infinite* number of solutions to
> a difference of squares with your target.
Unfortunately, most of them are useless to solving the real factoring
problem.
> That is just given, as you take factors of Tj^2, and as
> you can take rational factors, you have an infinite
> supply, and for *each* set of factors f_1 and f_2 such
> that
>
> f_1 f_2 = Tj^2
>
> you get factors g_1 and g_2 such that
>
> g_1 g_2 = TM^2
>
> where M is your target.
I can do the same thing, by letting g_1 = M f_1 / j and
g_2 = M f_2 / j, because if f_1 f_2 = Tj^2, then
g_1 g_2 = (M f_1 / j) * (M f_2 / j) = M^2 / j^2 * f_1 f_2
= M^2 / j^2 * T j^2 = T M^2.
The existence of a correspondence is not enough to say anything about
factoring; you must find the right one and then take the step that you
seem not to have taken.
> So the theorem as it is gives you what has never before
> been seen, guaranteed an infinite supply of potentially
> non-trivial solutions to a difference of squares.
"Potentially non-trivial"? That's a good one.
> > > Fix that and you might have something.
> >
> > If you can prove something, you might have a proof.
>
> Words. I present a theorem,
Once again, the theorem just says that if you have some equations (like
yx^2 - Ax + M^2 = 0 and yz^2 - Az + M^2 = T, which you seem to have
pulled out of thin air), then some other equations are true. Period.
> and posters go through a lot of social
> effort to hide it as that is the real math world.
>
> It's about fluff and stuff.
"Fluff and stuff" can hardly describe the cold, hard facts that people
like "Nora Baron" use to show that your results are wrong, and why.
> Think about it. Mathematicians TELL people the factoring problem is
> hard because they can't figure it out.
Exactly; if no one can on Earth can do a particular problem, it's
called hard. That's a definition. Unless, of course, it can be shown
that the task is impossible (such as squaring the circle or trisecting
the angle).
> Mostly mathematicians or people
> trained by mathematicians or using tools used by
> mathematicians work to factor numbers.
What other tools should be used for factoring numbers, macrame?
> The world takes on this system believing it secure.
>
> But my collegework was training to be a physicist as my
> B.Sc. is in physics, so I was trained to challenge and
> not to follow lock-step.
And it also explains your SERIOUS mathematical deficiency. The
mathematics requirement for physics majors is algebra, calculus,
differential equations, and maybe linear algebra. If you took courses
which were taught from an application point of view, you never did see
any proofs.
For instance, you may know the Fundamental Theorem of Calculus (the
integral from a to b of F'(x) is F(a) - F(b)), but you don't know WHY
it's true (by defining G(x) to be the integral from a to x of F'(x),
then using the definition of the derivative, the mean value theorem,
the squeeze theorem, etc, to show that G'(x) = F'(x)). It's something
that you accepted as being true, or maybe which was forced down your
throat.
This is not something that is your fault; the physics community is full
of people with this view of mathematics.
Well, here's the hard, cold truth: To work with factoring integers, you
need a background in mathematics and computer science. You need to take
a course in proofs, a number theory course, a course about algorithm
design, and a course in running time of algorithms. ("Course" can mean
a literal class at the college level, or self-study with someone to
make sure you are getting the concepts right.)
The effort you're putting towards mathematics is roughly the same as
the effort that a person learning about Aristotle's physics puts
towards physics. Yes, they are learning something, but it's old, and
incorrect, and much has been learned since then. Your situtation
related to mathematics is the same.
There's another aspect of the mathematics community, and that is what
exactly constitutes proof. The idea of mathematical proof is roughly
that of "beyond the shadow of a doubt" in law. You have to show how,
starting with agreed-upon definitions (what's an integer? what's a
factor?), why some other statement is also true. There can be no
"gaps"; you cannot assume anything (unless, in the statement of the
theorem, you've said that you will be assuming it), no matter how
likely it may be.
This leads to the phrase "Put up or shut up"; if you can't prove
something beyond the shadow of a doubt, you have no right saying that
you can.
Strictly speaking, it is impossible to prove that any theory of physics
is absolutely correct. The law F = m a may just happen to be true all
the times that people have tested it, and the next time someone tries,
it may be false. (There is no way to prove that it MUST be true.)
However, in learning physics career, you learned it and considered it
absolute.
Your assumption about F = m a is at the same level as that of
mathematicians learning (believing) that factoring is hard. It's been
tested a lot, and it's been found to be true every time.
> I was trained for the discipline of physics.
>
> Most people from the physics world don't bother with
> coming over to work on "pure math" problems, but I
> found myself focusing on them for many reasons.
And once again, there's a good analogy here. If someone in the
mathematics community were to take up an interest in quantum computing
(which does happen), then they would be expected to learn about the
field before doing research. Otherwise, what they are doing may have no
connection with the real world, they may be duplicating things that are
already known, they will be trying approaches which are known to lead
nowhere, etc.
Would you believe the results of someone who said that they thought
Newton's second law was actually F = m^2 a? No, because it's been
disproven before.
> The physics world trains you to solve problems.
So does the mathematics world. So does the computer science world.
However, the mathematics community's definition of having solved a
problem is more strict than the physic community's.
> It does not train you to accept that a problem is hard
> because people say so, or because no one else has found
> a simple solution.
Granted, a gifted high school student may solve the problem tomorrow,
but once a problem has been "open" (unsolved) for a long time, the
probability of it having a short proof goes down, and the longer and
more work it takes to solve it.
> I am a product of the physics world.
>
> Math people are trained to believe that what can be
> simply done has already simply been done, so they are
> trained to look for complex solutions that build on
> ***what's already known*** and not to look for simple
> solutions, in "hard" areas, where they are told what is hard.
What mathematics builds on is what is collectively accepted to be true.
Certainly, you could re-define geometry and try to solve problems with
this new system, but you may run across problems like the whole
universe collapsing down into a single point, or proving that 2 + 2 =
5. In those situations, mathematics says that you have to throw out
some assumption you've made because the consequences are horrendous.
(If you can prove that 2 + 2 = 5, then you can -- logically -- prove
that everything is true.)
> Math people are a herd.
Physicists are a herd. Chemists are a herd. Catholics are a herd.
What's your point?
> So even easy problems are effectively hard because they convince
> themselves they are hard,
The definition of "hard" is a relative, social, thing. To an advanced
extra-terrestrial race, the Four Color Theorem is easy and
straightforward. To us, because as human beings we don't have the
capacity to comprehend the proof all at once, we need to build and
program computers to help out convincing ourselves that any given proof
of the 4CT actually does consider all cases.
> and with most math people following along,
> you get a situation like what is facing the world today, where math
> people convinced themselves something was hard, and then convinced
> the world---or most of it.
>
> They didn't convince me, so I went looking, and I found the
> surrogate factoring theorem.
So how does the SF Theorem make factoring an integer into INTEGERS
easy? If you want to have a mathematical result (and publish the paper
in a mathematical journal), you have to play by their rules:
(1) Devise an algorithm (procedure) which allegedly factors integers.
(2) Prove it works, no matter what number you're given.
(3) Prove that it doesn't take too long to execute.
You have given no algorithm, the only thing that is related to the SF
Theorem is algebraic manipulation, and you only say that it works "50%
of the time"; you have not given any defendable justification for that
remark.
The closest thing to a good reason you've given is that when you write
M (a number with exactly two prime factors p_1, p_2) as p * q, where p
and q are rational numbers, then there are two cases: Either p_1 and
p_2 are split up, or they appear in the numerator of one of p and q
(and nowhere else in p or q).
Since each case has an infinite number of possibilities, the
probabilities may be different, and also I showed a third way for p * q
to be M.
I hope you read this and think about it a lot before you respond.
--- Christopher Heckman
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