#5130: [with patch; not ready for review] create a prime_pi function that
doesn't
just compute len(prime_range(n))
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Reporter: was | Owner: was
Type: defect | Status: new
Priority: major | Milestone: sage-3.3
Component: number theory | Keywords:
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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.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5130>
Sage <http://sagemath.org/>
Sage - Open Source Mathematical Software: Building the Car Instead of
Reinventing the Wheel
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