Re: SF: Experimental mathematics

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 03/07/05 

Date: Sun, 06 Mar 2005 18:43:09 -0600

On 6 Mar 2005 15:01:57 -0800, jstevh@msn.com wrote:

>I know that most of you probably don't have a clue about experimental
>mathematics, so I'm going to help myself think a bit by explaining it
>out.
>
>A lot of mathematical research in the past has been done in a
>relatively simple problem space, with a lot of linear thinking
>following a specific and well-defined line of inquiry.
>
>My work though tends to be in more complicated spaces where it's not
>necessarily clear what is the best direction, so I guess.

Oh for heaven's sake. Yeah, that's right, when most mathematicians
prove theorems it's usually pretty obvious what to do - us ordinary
guys don't have any experience with trying one thing and then
another.

You really believe that?

>Technically what I do is called making a hypothesis.

No, technically what you do is called making a wild-assed
guess based on no evidence or maybe one example, announcing
that you've _proved_ that the guess is correct so you don't
need to look at explanations of why it's wrong, then when
you're finally convinced it's wrong making another wild-
assed guess, announcing that you've proved it's correct...

>I make a hypothesis and proceed to test how that plays out
>mathematically.
>
>It's like what physicists do, and I basically operate a lot like a
>theoretical physicist.
>
>Mathematicians have begun to talk a bit about experimental mathematics,
>but most of you aren't taught anything meaningful about the
>experimental process while I have a B.Sc. in physics, so I have been.
>
>Now the problem space I'm dealing with now involves the factoring
>problem, but is focused on the relatively basic quadratics
>
>yx^2 + Ax - M^2 = 0
>
>and
>
>yz^2 + Az - j^2 = 0
>
>which create a somewhat complex space.
>
>That may seem strange, but you may have noticed I've talked about a LOT
>of different approaches,

Uh, no, actually we have noticed that.

>and I haven't exhausted the space. It's just
>that for most of you an experimental approach--versus someone coming up
>with a well-laid out plan for attacking a problem--is just foreign.

No, your problem-solving technique is well known to all of us.
It's called the monkey-at-typewriter approach, in honor of the
fact that a monkey at a typewriter _will_ type Hamlet if you
wait long enough.

It's not a very efficient approach.

>But a problem like the factoring problem requires it, just like
>problems in the real world require the scientific method of making a
>hypothesis and experimenting.
>
>After a couple of months of theorizing and experimenting I've focused
>on a solution found by solving out y, which is
>
>x = z(-Az +/- sqrt((Az - 2M^2)^2 - 4TM^2))/(2j^2 - 2Az)
>
>which I pick as it shows me two variables linked together closely that
>have to have a prime factor of M for some solution, as Az must have an
>integer solution that has a single prime factor of M, which follows
>from that square root.
>
>Well I decided to call that rather complicated part r, as I have a
>ratio between z and x, as I have
>
>x = zr
>
>with
>
>r = (-Az +/- sqrt((Az - 2M^2)^2 - 4TM^2))/(2j^2 - 2Az)
>
>and I made the easy substitution into
>
>yx^2 + Ax - M^2 = 0, to find I could solve for Az, as I had
>
>yr^2 z^2 + rAz - M^2 = 0
>
>with
>
>yz^2 + Az - j^2 = 0
>
>and I initially assumed, because it was easier that r could always be
>an integer, which lead me to conclude I had my answer.
>
>But some experimenting--trying the resultant algorithms and seeing if
>they factored--by another poster showed that did not work, as in fact r
>may be a fraction, so I've been trying to figure out its denominator,
>as I easily found that letting r = n/d, gave me
>
>Az = (-n T/(n-d) + M^2 (n+d)/n)
>
>where for Az to be an integer, n needs to be a factor of M^2, and n-d a
>factor of T. And I know integer solutions for Az must exist, so now
>the question is, how do I get d?
>
>I've been thinking about it all day, trying various approaches as if I
>can determine what d is, then necessarily I can get that perfect
>factoring algorithm.
>
>And that's what I've been doing, muddling along, theorizing and at
>times testing out, trying to figure out how to get a handle on d, where
>I'd LIKE it to be a factor of T, as that's simple, or even of j^2 or
>Tj^2, so I'm trying to prove that's the case, because it's easier.
>
>You learn when you experiment that you try easy first, and then move on
>to hard.
>
>The problem space here is rather robust in terms of possible paths to
>pursue, and I would hate to have to look at just about every
>possibility before solving the problem, or worse, find the problem
>isn't solvable in this space, but that's the process.
>
>You will notice in threads some posters showing their true colors, and
>here is a good example to highlight how I got my "crank" label.
>
>I'd pursue a problem, trying out a bunch of ideas, which would anger
>posters who saw me as cluttering up the newsgroups with posts, and
>they'd harass me, in various ways, like making fun of me or denigrating
>my efforts.
>
>When I'd fail, they'd make a big deal of it, and still do.
>
>Now on the other hand, as I'm experimenting I'll get excited about a
>result, and jump the gun with an announcement, like saying I'd solved
>the factoring problem. But the result looked so nice, and easy...until
>it didn't work.
>
>So you might say it balances out in a dark way, with me guessing and at
>times making big claims, and other people demonstrating their
>viciousness by posting in various nasty ways.
>
>It all makes for an interesting time.
>
>Ok, back to theorizing, as I've had enough of a break.
>
>Now then, what is going on with that damn d...
>
>
>James Harris

************************

David C. Ullrich

Relevant Pages

  • Re: SF: Experimental mathematics
    ... >Mathematicians have begun to talk a bit about experimental mathematics, ... >Now the problem space I'm dealing with now involves the factoring ... >After a couple of months of theorizing and experimenting I've focused ... >integer solution that has a single prime factor of M, ...
    (sci.crypt)
  • SF: Experimental mathematics
    ... Mathematicians have begun to talk a bit about experimental mathematics, ... But a problem like the factoring problem requires it, ... After a couple of months of theorizing and experimenting I've focused ... integer solution that has a single prime factor of M, ...
    (sci.math)
  • SF: Experimental mathematics
    ... Mathematicians have begun to talk a bit about experimental mathematics, ... But a problem like the factoring problem requires it, ... After a couple of months of theorizing and experimenting I've focused ... integer solution that has a single prime factor of M, ...
    (sci.crypt)
  • Re: Magdalenian (year seven)
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  • Re: Magdalenian (year seven)
    ... me in peace, while the challies gather here, ... do the same when it comes to early mathematics. ... So I began experimenting with both ... "A rose is a rose is a rose" (Gertrude ...
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