An efficient algorithm for the computation of Bernoulli numbers
| dc.creator | Fee, Greg | |
| dc.creator | Plouffe, Simon | |
| dc.date | 2007-02-11 | |
| dc.date | 2007-02-25 | |
| dc.date.accessioned | 2026-07-07T07:48:22Z | |
| dc.date.available | 2026-07-07T07:48:22Z | |
| dc.description | This article gives a direct formula for the computation of B(n) using the asymptotic formula $$B (n) \approx 2 {\frac {n!}{π^{n}{2}^{n}}}$$ where n is even and $n >> 1$. This is simply based on the fact that $ζ(n)$ is very near 1 when n is large and since $B (n) = 2 {\frac {ζ(n) n!}{π^{n}{2}^{n}}}$ exactly. The formula chosen for the Zeta function is the one with prime numbers from the well-known Euler product for $ζ(n)$. This algorithm is far better than the recurrence formula for the Bernoulli numbers even if each B(n) is computed individually. The author could compute $B (750,000)$ in a few hours. The current record of computation is now (as of Feb. 2007) $B (5,000,000)$ a number of (the numerator) of 27332507 decimal digits is also based on that idea. | |
| dc.identifier | https://arxiv.org/abs/math/0702300 | |
| dc.identifier | http://arxiv.org/abs/math/0702300 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/124472 | |
| dc.subject | Number Theory | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 11B68 | |
| dc.title | An efficient algorithm for the computation of Bernoulli numbers | |
| dc.type | text |