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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 $23 \times 89 = 2^{11}-1$ 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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    For reference, the list of primes p<100 where $\log(p)$ has no known spigot algorithm is $p=23, 47, 53, 59, 67, 71, 79, 83, 89, 97$. However, I think $97 = 2^5\times3+1$ might have one.2017-07-04

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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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    +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.)2011-07-30