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I'd like to ask how to fastly calculate the Bell number $B_n$ modulo a prime power, where $n$ is around one million.

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I'll answer my question by myself. This paper ("Congruences for Bell and Tangent Numbers" by Ira Gessel) proposed a favorable solution.

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    $\begingroup$ This paper does not answer the posed question. It may help to compute Bell numbers modulo a factorial of a small number, but not modulo a (large) prime power. $\endgroup$ Sep 20, 2018 at 4:42
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    $\begingroup$ This is just a link, that can be made unavailable any time, please include important parts in the answer itself. $\endgroup$
    – Sil
    Sep 20, 2018 at 6:09
  • $\begingroup$ I've updated the title and author information. $\endgroup$
    – Hang Wu
    Sep 20, 2018 at 10:17

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