On Tue, May 6, 2008 at 10:57 AM, David Harvey <[EMAIL PROTECTED]> wrote:
>
>
>
> On May 6, 2008, at 1:18 PM, William Stein wrote:
>
> >
> > On Tue, May 6, 2008 at 10:15 AM, <[EMAIL PROTECTED]> wrote:
> >>
> >> William has mentioned some congruence tests that we can perform
> >> -- I'd like to make sure that I got the right answer before we pat
> >> ourselves on the back too much.
> >>
> >>
> >
> > David Harvey's congruence tests would be pretty good. Just choose
> > *any* prime p > 10^7 + 10
> > say and compute B_{10^7+4} modulo it using David Harvey's function;
> >
> > sage: p = next_prime(10^7+10)
> > sage: time bernoulli_mod_p_single(p, 10^7+4)
> > CPU times: user 0.49 s, sys: 0.00 s, total: 0.49 s
> > Wall time: 0.51
> > 8087583
> >
> > Pretty cool, eh?
>
> ..... and the natural question is, could you then reconstruct the
> actual bernoulli number, by computing it modulo sufficiently many
> primes? Well, I did the estimates a few days ago, and it turns out
> that the asymptotic behaviour of this proposed algorithm is pretty
> much *the same* as the one using the zeta function (i.e. the one Pari
> uses); they are both around n^2 log^2(n), if I'm not mistaken.
> Unfortunately, I did some further tests, and even if I sped up
> bernoulli_mod_p_single() by the largest factor I could conceive of,
> overall it would still be maybe 50x slower than Pari. If anyone can
> find an extra constant factor of 50x in this algorithm, I'd love to
> hear about it.
>
I think maybe yours parallelizes much more easily than
Pari's, so just use 1000 computers :-).
-- William
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