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Old 2015-09-06, 19:58   #1
kwalisch
 
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Sep 2015

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Default New confirmed pi(10^27) prime counting function record

I am happy to announce that David Baugh and I have computed the number
of primes below 10^27, the result is:

pi(10^27) = 16,352,460,426,841,680,446,427,399

The computation was performed using an unpublished version of my
primecount program with backup functionality. primecount counts primes
using a highly optimized parallel implementation of the
Deleglise-Rivat algorithm (combinatorial method). The computation took
23.03 CPU core years and the peak memory usage was 235 gigabytes.

Half of the computations were run on multiple EC2 spot instances of
type r3.8xlarge (16 CPU cores, Intel Xeon E5-2670 v2). The other half
(the computation of the hard special leaves) was run on David's dual
socket server (36 CPU cores, Intel Xeon E5-2699 v3).

Our result passes the parity check and the result is also very close
to Riemann R(10^27) i.e.:

| R(10^27) - 16,352,460,426,841,680,446,427,399 | < sqrt(10^27) / log(10^27)

We will start a full verification shortly by recalculating pi(10^27) a
second time using different configuration parameters. We expect this
verification to take about 5 months.

We'd like to thank J&N Computer Services for giving a discount on
David's BigRig server and Shabir Ali for granting free access on his
backup server.

Regards,
Kim Walisch

Last fiddled with by kwalisch on 2015-09-06 at 20:11 Reason: Add link to primecount website
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Old 2015-09-06, 20:36   #2
danaj
 
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Old 2015-09-06, 21:04   #3
bsquared
 
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"Ben"
Feb 2007

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Nice work!
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Old 2015-09-07, 12:51   #4
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Great! Congratulations
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Old 2015-09-08, 12:27   #5
D. B. Staple
 
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Nice work, Kim!
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Old 2015-09-08, 13:47   #6
rogue
 
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"Mark"
Apr 2003
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Old 2016-06-01, 00:22   #7
dbaugh
 
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Default result verified

Last December Kim Walisch and I started a verification run of pi(1e27) using different initial conditions. Kim completed the last piece of the verification calculation of pi(1e27) on May 31, 2016 at 23:00 Luxembourg time. Along with the pieces I finished computing on May 9 , we can now say that our original value for pi(1e27) has been verified. The latest version of primecount is even faster than when we first calculated pi(1e27). I am sure Kim will be providing more details.

Best regards,

David Baugh
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Old 2016-06-01, 07:24   #8
kwalisch
 
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Default pi(10^27) verified!

After roughly 5 months of computation David Baugh and myself have verified pi(10^27):

pi(10^27) = 16,352,460,426,841,680,446,427,399

This time the computation took 20.35 CPU core years, this is 11.6% faster than our first computation. The speed up comes from primecount improvements, particularly I have added pre-sieving and wheel factorization to primecount's sieving algorithms.

Below are the details of the verification:

Code:
x = 1000000000000000000000000000
y = 231112254739
pi(y) = 9199337709
P2 = 4743234949871865833944278
S1 = 45739379279637813150
S2_trivial = 42247262851521121201
S2_easy = 4453498620247012088893172
S2_hard = 16642108769824393833206446
S2 = S2_trivial + S2_easy + S2_hard
pi(x) = S1 + S2 + pi(y) - 1 - P2
pi(x) = 16352460426841680446427399
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Old 2016-06-01, 20:02   #9
CRGreathouse
 
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Fantastic! I've updated the OEIS entry https://oeis.org/A006880.
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Old 2016-06-01, 21:07   #10
kwalisch
 
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Sep 2015

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Thanks, I actually created an account on OEIS earlier today in order to submit it. But you have been faster. So I hope I can use my OEIS account in near future for submitting new prime sum records ;-)
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Old 2016-06-01, 21:40   #11
CRGreathouse
 
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Quote:
Originally Posted by kwalisch View Post
Thanks, I actually created an account on OEIS earlier today in order to submit it. But you have been faster. So I hope I can use my OEIS account in near future for submitting new prime sum records ;-)
Welcome to the OEIS! Let me know if you need anything.
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