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Regardless of form, I want to know how I can get some interpretation (closed formula preferred) of $B_n$ that is equivalent to $m$ digits of precision, and I'd like to know the fastest way to do it (in terms of calculations required to get the estimate). Can someone please help?

For your information, you may want to consider Wikipedia's page on Bernoulli numbers. I've been attempting to use the "explicit definition" that they give, but I've been running into problems. I'll keep you informed of the progress that I make...

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    What exactly is your question? Does not http://en.wikipedia.org/wiki/Bernoulli_numbers#Efficient_computation_of_Bernoulli_numbers answer it, with references?2011-11-02
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    @lhf: No, they give formulas for how to get $B_n$ - but they don't tell which method is fastest for a given amount of precision. I want to know the fastest way to get $B_n$ to $m$ digits of precision.2011-11-02
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    Have you seen [Fillebrown's](http://dx.doi.org/10.1016/0196-6774(92)90048-H) and [Harvey's](http://arxiv.org/abs/0807.1347) papers?2011-11-02
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    @J.M.: Thanks for the links! I am looking for an estimate, though, which may be easier and quicker to calculate...2011-11-02

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