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. $