#5130: [with patch; not ready for review] create a prime_pi function that
doesn't
just compute len(prime_range(n))
---------------------------+------------------------------------------------
Reporter: was | Owner: was
Type: defect | Status: new
Priority: major | Milestone: sage-3.3
Component: number theory | Resolution:
Keywords: |
---------------------------+------------------------------------------------
Old description:
> The goal of this ticket is to make something eventually competitive with
> Mathematica's PrimePi. The first version won't do that, but it is a step
> in the right direction.
>
> Note that evidently the only program we know of with a fast prime_pi is
> Mathematica (pari, maple, etc., -- none of them have anything at all).
> In contract, Mathematica can do {{{PrimePi[10^14]}}} in "about a minute",
> but the algorithm there doesn't allow larger input.
>
> {{{
> In[11]:= Timing[PrimePi[ 10^13 + 10^8 +10^9]]
> Out[11]= {12., 346102281239}
>
> In[13]:= Timing[PrimePi[10^14]]
> Out[13]= {59.53, 3204941750802}
> }}}
>
> The attached code did {{{10^14}}} in just under an hour. The issue is
> that Mathematica implements a different sublinear algorithm.
>
> In Sage, with extra storage usage:
> {{{
> sage: k = 10^13
> sage: time print prime_pi(k,40)
> 346065536839
> Time: CPU 219.44 s, Wall: 219.44 s
> }}}
>
> So for that one mathematica is 18 times faster.
New description:
The goal of this ticket is to make something eventually competitive with
Mathematica's PrimePi. The first version won't do that, but it is a step
in the right direction.
Note that evidently the only program we know of with a fast prime_pi is
Mathematica (pari, maple, etc., -- none of them have anything at all). In
contract, Mathematica can do {{{PrimePi[10^14]}}} in "about a minute", but
the algorithm there doesn't allow larger input.
{{{
In[11]:= Timing[PrimePi[ 10^13 + 10^8 +10^9]]
Out[11]= {12., 346102281239}
In[13]:= Timing[PrimePi[10^14]]
Out[13]= {59.53, 3204941750802}
}}}
The attached code did {{{10^14}}} in just under an hour. The issue is
that Mathematica implements a different sublinear algorithm.
In Sage, with extra storage usage:
{{{
sage: k = 10^13
sage: time print prime_pi(k,40)
346065536839
Time: CPU 219.44 s, Wall: 219.44 s
}}}
So for that one mathematica is 18 times faster.
Another comparison for smaller input, where this is better:
{{{
sage: time prime_pi(10^10,32)
CPU times: user 0.59 s, sys: 0.04 s, total: 0.63 s
Wall time: 0.63 s
455052511L
...
wst...@sage:~$ math
Mathematica 6.0 for Linux x86 (64-bit)
Copyright 1988-2007 Wolfram Research, Inc.
In[1]:= Timing[PrimePi[10^10]]
Out[1]= {0.12, 455052511}
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5130#comment:2>
Sage <http://sagemath.org/>
Sage - Open Source Mathematical Software: Building the Car Instead of
Reinventing the Wheel
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