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The Bailey-Borwein-Plouffe formula yields a binary spigot algorithm for π, and related formulas give the bits of log(2) and those of the logarithms of some other integers. I got stuck (over a year ago, so none of this is fresh in my head) trying to find a spigot algorithm for the bits of log(23), and I guess that the difficulty is because 23 * 89 is a Mersenne number. Is any binary spigot algorithm known for log(23) or log(89) which is just as fast as those for π and log(2)? If not, is there any reason to think that one doesn't exist?

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Have a look at Page 7 of this link It contains a list of primes whose logarithms have a binary BBP formula. Neither 23 nor 89 are on the list. –  kuch nahi Aug 29 '11 at 6:55
    
"it is unknown whether or not all primes have this property". If it is an open problem I will accept an answer to that effect. –  Dan Brumleve Aug 30 '11 at 6:18
    
My reputation at this moment is 2047 and I'm still hoping for an answer to this question. I didn't follow Gerry's advice but someday I may have the chutzpah. –  Dan Brumleve Oct 25 '11 at 7:51
    
I'm wondering why I got notified without @. Anyhow, regarding the above comment, what has your reputation got to do with it? –  kuch nahi Oct 25 '11 at 8:01
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@kuch, because 2047 = 23 * 89... I noticed it and thought of this old question. :) –  Dan Brumleve Oct 25 '11 at 8:09

1 Answer 1

up vote 1 down vote accepted

Dang-Khoa Do, Spigot algorithm and reliable computation of natural logarithm, Reliable Computing 10 (2004) 489-500, gives spigot algorithms for computing various logarithms. I didn't look at it closely enough to tell whether $\log23$ is amenable to Do's methods.

EDIT: Incidentally, OP is quite correct to relate the difficulty to the Mersenne connection. See the top of page 11 of http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/digits.pdf, Bailey, Borwein, and Plouffe, On the rapid computation of various polylogarithmic constants.

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I don't have free access, and I can't find a free copy on Google, so I can't tell either. I did find docserver.carma.newcastle.edu.au/47/1/machin.pdf which says "There is no non-Aurifeuillian binary Machin-type BBP logarithms formula for ln(23) or for ln(89)" but I don't know what this means and I don't think it addresses my question about the existence of an algorithm as fast as that for log(2) or π. –  Dan Brumleve May 31 '11 at 7:05
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@Dan, perhaps you can find contact information for the author and ask for the paper (or just ask the author about the answer to your question). –  Gerry Myerson May 31 '11 at 13:43
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Gerry, I wouldn't feel comfortable doing that just to satisfy my idle curiosity. I don't have anything to offer in return except upvotes and I guess these guys have students who pay tuition. –  Dan Brumleve May 31 '11 at 23:17
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@Dan, authors love to know that someone is actually reading the papers they are writing and is interested, even if only out of idle curiosity, in the same things that interest them. There is absolutely no reason to feel uncomfortable about asking an author for access to a paper. Alternatively, perhaps you have access to a library that will do interlibrary loan for you? –  Gerry Myerson Jun 1 '11 at 0:43
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+1 for Gerry's suggestion. All mathematicians I know are more than happy to send you electronic copies of their papers. Just make sure not to send your request by post using barely legible handwriting with the demand that a physical copy be FedEx'ed to your location. =) (Unless, of course, you include a pre-paid return mailer.) –  Willie Wong Jul 30 '11 at 5:02

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