mersenneforum.org  

Go Back   mersenneforum.org > Math Stuff > Computer Science & Computational Number Theory

Reply
 
Thread Tools
Old 2014-12-02, 20:12   #1
D. B. Staple
 
D. B. Staple's Avatar
 
Nov 2007
Halifax, Nova Scotia

23×7 Posts
Default Prime counting function records

I am pleased to announce the following computations of the prime counting function:
pi(10^26) = 1699246750872437141327603
pi(2^81) = 43860397052947409356492

These values were computed using an enhanced version of the combinatorial method originally due to Meissel. Starting from the version of the algorithm published by T. Oliveira e Silva, I incorporated modifications permitting shared- and distributed-memory parallelism, as well as numerous improvements resulting in constant-factor reductions in time and space.

Calculations were performed on the Guillimin, Briarée, and Colosse supercomputers from McGill University, Université de Montréal, and Laval Université, managed by Calcul Québec and Compute Canada. The operation of these supercomputers is funded by the Canada Foundation for Innovation (CFI), NanoQuébec, RMGA and the Fonds de recherche du Québec - Nature et technologies (FRQ-NT).

The results were double-checked by running independent calculations on separate clusters with different numerical parameters. Specifically, pi(10^26) was computed on Guillimin and Briarée, while pi(2^81) was computed on Guillimin and Colosse. Furthermore, I have re-calculated pi(10^n) and pi(2^m) for all previously known values of n and m.

I would like to extend my heartfelt gratitude to Karl Dilcher for guiding my education in elementary, algebraic, and computational number theory, and for his insightful comments and suggestions regarding these calculations and the underlying algorithm.




Douglas B. Staple, Dr. rer. nat.

Department of Mathematics and Statistics
Dalhousie University
Halifax NS
Canada
D. B. Staple is offline   Reply With Quote
Old 2014-12-03, 14:04   #2
bsquared
 
bsquared's Avatar
 
"Ben"
Feb 2007

63428 Posts
Default

Congratulations, an impressive calculation! Can you say how long it took on the various clusters?
bsquared is offline   Reply With Quote
Old 2014-12-03, 14:18   #3
D. B. Staple
 
D. B. Staple's Avatar
 
Nov 2007
Halifax, Nova Scotia

23×7 Posts
Default

Thank you! I will eventually release technical details, including a scaling analysis. For the moment, though, I am going to sit on that information, lest my competitors *ahem* friends-I-haven't-met-yet learn my capabilities.
D. B. Staple is offline   Reply With Quote
Old 2014-12-03, 19:44   #4
ldesnogu
 
ldesnogu's Avatar
 
Jan 2008
France

52810 Posts
Default

Great achievement, can't wait to see more information
ldesnogu is offline   Reply With Quote
Old 2014-12-04, 01:46   #5
CRGreathouse
 
CRGreathouse's Avatar
 
Aug 2006

172A16 Posts
Default

Congrats!

Quote:
Originally Posted by D. B. Staple View Post
pi(2^81) was computed on Guillimin and Colosse. Furthermore, I have re-calculated pi(10^n) and pi(2^m) for all previously known values of n and m.
Did you compute pi(2^78), pi(2^79), and pi(2^80)? Those aren't known, at least to me.
CRGreathouse is offline   Reply With Quote
Old 2014-12-04, 02:13   #6
D. B. Staple
 
D. B. Staple's Avatar
 
Nov 2007
Halifax, Nova Scotia

3816 Posts
Default

Yes, I calculated pi(2^m) for all m<=81, and pi(10^n) for all n<=26. For pi(2^78), pi(2^79) and pi(2^80), I found the same values as were previously calculated by J. Franke, T. Kleinjung, J. Büthe and A. Jost under the assumption of the Riemann Hypothesis:
http://www.math.uni-bonn.de/people/j...alyticPiX.html
D. B. Staple is offline   Reply With Quote
Old 2014-12-04, 04:55   #7
Dubslow
Basketry That Evening!
 
Dubslow's Avatar
 
"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

3·2,399 Posts
Default

Quote:
Originally Posted by D. B. Staple View Post
For pi(2^78), pi(2^79) and pi(2^80), I found the same values as were previously calculated by J. Franke, T. Kleinjung, J. Büthe and A. Jost under the assumption of the Riemann Hypothesis:
http://www.math.uni-bonn.de/people/j...alyticPiX.html
...well now I have to say I'm firmly in the believer camp lol.
Dubslow is offline   Reply With Quote
Old 2014-12-04, 22:45   #8
XYYXF
 
XYYXF's Avatar
 
Jan 2005
Minsk, Belarus

24·52 Posts
Default

http://www.primefan.ru/stuff/primes/table.html is updated :-)
XYYXF is offline   Reply With Quote
Old 2014-12-05, 10:21   #9
ATH
Einyen
 
ATH's Avatar
 
Dec 2003
Denmark

1011100101112 Posts
Default

Congratulations on the achievement.

I do not know how this calculation is done, so excuse me if this is a stupid question:

Why is the pi(2^m) "only" at 2^81 when pi(10^n) is at 10^26 ~ 2^86.37 ?



The link below has pi(2^77) to pi(2^80) and OEIS has up to pi(2^52): http://oeis.org/A007053/list

Anyone know where pi(2^53) to pi(2^76) can be found?

Last fiddled with by ATH on 2014-12-05 at 10:21
ATH is offline   Reply With Quote
Old 2014-12-05, 10:37   #10
ldesnogu
 
ldesnogu's Avatar
 
Jan 2008
France

10000100002 Posts
Default

Quote:
Originally Posted by ATH View Post
Why is the pi(2^m) "only" at 2^81 when pi(10^n) is at 10^26 ~ 2^86.37 ?
If I had to bet, I'd say 2^81 was chosen just to beat the previous record of 2^80 set by Franke et al

Quote:
The link below has pi(2^77) to pi(2^80) and OEIS has up to pi(2^52): http://oeis.org/A007053/list

Anyone know where pi(2^53) to pi(2^76) can be found?
You can find them on Tomás Oliveira e Silva page. Perhaps someone can add them to OEIS?
ldesnogu is offline   Reply With Quote
Old 2014-12-06, 00:07   #11
D. B. Staple
 
D. B. Staple's Avatar
 
Nov 2007
Halifax, Nova Scotia

23·7 Posts
Default

I am happy to announce an additional power of two:
pi(2^82) = 86631124695994360074872

D. B. Staple is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
New confirmed pi(10^27), pi(10^28) prime counting function records kwalisch Computer Science & Computational Number Theory 32 2020-08-31 16:58
Counting Goldbach Prime Pairs Up To... Steve One Miscellaneous Math 8 2018-03-06 19:20
Prime counting function Steve One Miscellaneous Math 20 2018-03-03 22:44
Fourier Series for Prime Number Counting Functions SteveC Analysis & Analytic Number Theory 10 2016-10-14 21:48
Legendre's prime counting function pbewig Information & Answers 0 2011-07-14 00:47

All times are UTC. The time now is 02:24.

Sun Oct 25 02:24:43 UTC 2020 up 44 days, 23:35, 0 users, load averages: 2.53, 2.08, 1.88

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.