# Is there a binary spigot algorithm for log(23) or log(89)?

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
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
@kuch, because 2047 = 23 * 89... I noticed it and thought of this old question. :) – Dan Brumleve Oct 25 '11 at 8:09
Can a fast series for log(23) be obtained by accelerating a slow one? math.stackexchange.com/questions/1605204/… – Jaume Oliver Lafont Jan 12 at 22:44

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.