20141202, 20:12  #1 
Nov 2007
Halifax, Nova Scotia
70_{8} Posts 
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 distributedmemory parallelism, as well as numerous improvements resulting in constantfactor 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 (FRQNT). The results were doublechecked 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 recalculated 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 
20141203, 14:04  #2 
"Ben"
Feb 2007
3,617 Posts 
Congratulations, an impressive calculation! Can you say how long it took on the various clusters?

20141203, 14:18  #3 
Nov 2007
Halifax, Nova Scotia
2^{3}×7 Posts 
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* friendsIhaven'tmetyet learn my capabilities.

20141203, 19:44  #4 
Jan 2008
France
1000101111_{2} Posts 
Great achievement, can't wait to see more information

20141204, 01:46  #5 
Aug 2006
5979_{10} Posts 

20141204, 02:13  #6 
Nov 2007
Halifax, Nova Scotia
2^{3}×7 Posts 
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.unibonn.de/people/j...alyticPiX.html 
20141204, 04:55  #7  
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
7221_{10} Posts 
Quote:


20141204, 22:45  #8 
Jan 2005
Minsk, Belarus
620_{8} Posts 
http://www.primefan.ru/stuff/primes/table.html is updated :)

20141205, 10:21  #9 
Einyen
Dec 2003
Denmark
6357_{8} Posts 
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 20141205 at 10:21 
20141205, 10:37  #10  
Jan 2008
France
13×43 Posts 
Quote:
Quote:


20141206, 00:07  #11 
Nov 2007
Halifax, Nova Scotia
2^{3}·7 Posts 
I am happy to announce an additional power of two:
pi(2^82) = 86631124695994360074872 
Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
New confirmed pi(10^27),... pi(10^29) prime counting function records  kwalisch  Computer Science & Computational Number Theory  40  20220312 18:57 
Counting Goldbach Prime Pairs Up To...  Steve One  Miscellaneous Math  8  20180306 19:20 
Prime counting function  Steve One  Miscellaneous Math  20  20180303 22:44 
Fourier Series for Prime Number Counting Functions  SteveC  Analysis & Analytic Number Theory  10  20161014 21:48 
Legendre's prime counting function  pbewig  Information & Answers  0  20110714 00:47 