John,
please consider that my Numbers.jl package is something I did as my first
trial
in working with Julia -- and left it behind after a few days. It is
certainly
not in a state nor has the intention to be comparable in speed or
completeness
to other such systems.
As I said in another thread, when number theoretic computations are
involved,
PARI/GP is another system to look at closely, not only Mathematica or
Maple.
For instance, returning the next prime for really large input values takes
only
milliseconds ("See Ma, no tables!"):
gp> nextprime(10^100)
time = 9 ms.
%1 = 100000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000267
There is no prime counting function. Probably as there is no direct need
for
this functionality for mathematicians, as you have already suspected. So
the
following trick counts primes:
gp> n = 0;
gp> forprime (p = 1841378967856, 1850000000000, n++);
time = 1min, 49,432 ms.
gp> n
%4 = 305232179
Of course, a prime counting function based on tables cannot be beaten. That
is
why I'm quite glad you have provided this library for Julia.
If someone is thinking about a more extended number theory module for
Julia, I
would be glad to join and be of help.
