From: mpd@netcom.com (Mike Duvos)
Date: Sat, 6 Jan 1996 13:52:36 +0800
To: cypherpunks@toad.com
Subject: Re: Pi Stuff
In-Reply-To: <m0tYMQF-0008ykC@pacifier.com>
Message-ID: <199601060513.VAA12559@netcom21.netcom.com>
MIME-Version: 1.0
Content-Type: text/plain


jim bell <jimbell@pacifier.com> writes:

 > While I'm not an expert at this, I think you're
 > misrepresented the Chudnovsky result.  They formulated an
 > equation that allowed "you" to continue the calculation
 > past "N" digits as long as you had the result that far.

That property would be possessed by any self-correcting iteration
which converged in a neighborhood of Pi.  It would not be
necessary to repeat ones earlier calculations at increased
precision in order to determine Pi to additional digits.  One
could just use the previous calculations as a starting point and
continue to iterate, doing the new calculations to extended
precision.

I believe the Chudnovskys proved a much stronger result than
this, although precisely what it was escapes me at the moment.

[Please hum the theme to "Final Jeopardy" while I look up
 Chudnovsky's formula]

Good - it's in the sci.math FAQ.

Set k_1 = 545140134
    k_2 = 13591409
    k_3 = 640320
    k_4 = 100100025
    k_5 = 327843840
    k_6 = 53360;

Then pi = (k_6 sqrt(k_3))/(S), where

S = sum_(n = 0)^oo (-1)^n ((6n)!(k_2 +nk_1))/(n!^3(3n)!(8k_4k_5)^n)

This converges linearly at about 14 digits a term, and carries
forward a sufficiently small amount of state that one can iterate
into the billions of digits without the CPU requirements becoming
painful.  So it basically functions as a digit generator for Pi,
which, when appropriately initialized, will work on any part of
the number and emit the appropriate output.  The denominator
simplifies in a special way which keeps the computation localized
to a small neighborhood of the place where the new digits are
appearing.

 > As far as I know, they DID NOT generate any formula for the
 > generation of isolated digits of pi, the more recent news.

I guess you're right about it not having the specific form of a
function which takes "i" as input and emits the "ith" bit.
Nonetheless, the discovery of this particular formula and the
way in which its computational requirements expand tastefully
with increasing numbers of digits hints strongly at the existence
of the aforementioned closed solution.

--
     Mike Duvos         $    PGP 2.6 Public Key available     $
     mpd@netcom.com     $    via Finger.                      $