Быстрое вычисление чисел Бернулли

Abstract

Bernoulli numbers are often found in mathematical analysis, number theory, combinatorics, and other areas of mathematics. In some monographs on number theory there are separate chapters devoted only to Bernoulli numbers and their properties. Algorithms for calculating Bernoulli numbers are built into all popular mathematical packages: Mathematica, Matlab, Magma, Pari GP, etc. In this paper, we propose an algorithm for calculating Bernoulli numbers, which is faster than the known ones. The essence of our approach is to improve the Harvey multimodular method due to making sparse of large calculated sums, when we express the sums in half the interval in terms of sums in the intervals of length 1/12 or even 1/15 of the length of the summation interval. It is proved in the paper that such a reduction in the summation intervals can be achieved for the vast majority of prime numbers. At the same time, inconvenient prime numbers (and there are them no more than 0.01% for large values of n) can be simply excluded from the list those by modulo of which the current Bernoulli number is calculated. The algorithm of fast calculation proposed by us in the article (acceleration is more than three times as compared with the Harvey algorithm) of Bernoulli numbers modulo prime numbers can also be successfully used to find irregular prime numbers, irregular pairs (p, n), and also when calculating Iwasawa invariants. When calculating the irregular pairs (p, n)\xa0and Iwasawa invariants, the algorithm presented in our work is significantly (more than 10 times) more efficient.