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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 $O(k^2 \log^{2+\varepsilon} k)$, 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.
11B68Bernoulli and Euler numbers and polynomials
11Y55Calculation of integer sequences
11Y16Algorithms; complexity (number theory)
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