#7539: primes.p0.spkg with "prime_sieve.c" functionality
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Reporter: GeorgSWeber | Owner: rohana
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.3.1
Component: number theory | Keywords:
Work_issues: | Author: rohana, GeorgSWeber
Upstream: N/A | Reviewer:
Merged: |
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Comment(by kevin.stueve):
Sorry. Right now you have the values:
[[BR]]
x pi(x)[[BR]]
18440 00000 00000 00000 : 425 50425 77541 37607[[BR]]
18446 74407 37095 51616 : 425 65628 40352 17743 [[BR]]
18450 00000 00000 00000 : 425 72967 93423 72554[[BR]]
from http://www.primefan.ru/stuff/primes/table.html and
http://www.ieeta.pt/~tos/bib/5.4.pdf.
You could sieve part of this interval for more values. You could contact
one of the authors of one of the variants of combinatorial method asking
for other values (Such as Gourdon, TOS, etc).
Another possibility is using a O(x^1/2^ log(x)) algorithm that computes
the parity (even or odd) of the prime counting function, for a limited
amount of verification of values of the prime counting function. This
algorithm is described by Terence Tao and others at the polymath project
at
http://michaelnielsen.org/polymath1/index.php?title=Prime_counting_function
and by Henri Lifchitz at
http://web.archive.org/web/20070325064748/http://ourworld.compuserve.com/homepages/hlifchitz/Henri/us/ParPius.htm
[[BR]]
Gourdon, an author of a variant of the combinatorial prime_pi method links
to this page at
http://numbers.computation.free.fr/Constants/Primes/countingPrimes.html,
however Gourdon's link to Henri Lifchitz's page is dead and you have to
use the wayback machine. At Lifchitz's page is his paper on calculating
the parity of prime_pi and a program that works up to (unfortunately only)
1e19, and seems to be a little slow.
[[BR]]
Gourdon has a pix program released at
http://numbers.computation.free.fr/Constants/Primes/Pix/contributepixtable.html
that he used for his distributed pix project. I'm not sure what range is
allowed, how long computation takes, or if the code is open source. But
it ''might'' work for you. There is also a table of 7702/9001 values of
pi(x) computed in the range 1000e14 -> 10000e14 available from Gourdon's
pix project at
http://numbers.computation.free.fr/Constants/Primes/Pix/resultstable.php.
Not the range you were asking for, but still worth noting. Take this data
and program with a grain of salt- although Gourdon is surely a better
programmer and mathematician than myself, he had to halt his pix program
when he found his code was off by one in the global checks for
prime_pi(10^23^)
[[BR]]
TOS will probably release more values of prime_pi up to 4e18 after his
Golbach conjecture verification project is complete, but since these
aren't in the range you need, I would recommend contacting Gourdon or TOS
directly asking for values for verification.
[[BR]]
Kevin Stueve
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/7539#comment:15>
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