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A multimodular algorithm for computing Bernoulli numbers. (English) Zbl 1215.11016
Summary: We describe an algorithm for computing Bernoulli numbers. Using a parallel implementation, we have computed ${B}_{k}$ for $k={10}^{8}$, a new record. Our method is to compute ${B}_{k}$ modulo $p$ for many small primes $p$ and then reconstruct ${B}_{k}$ via the Chinese Remainder Theorem. The asymptotic time complexity is $O\left({k}^{2}{log}^{2+\epsilon }k\right)$, matching that of existing algorithms that exploit the relationship between ${B}_{k}$ and the Riemann zeta function. Our implementation is significantly faster than several existing implementations of the zeta-function method.
##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials 11Y60 Evaluation of constants
PARI/GP; Sage