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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+ε 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.
MSC:
11B68Bernoulli and Euler numbers and polynomials
11Y60Evaluation of constants
Software:
PARI/GP; Sage