Prime counting function records

[D. B. Staple]

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

[bsquared]

Congratulations, an impressive calculation! Can you say how long it took on the various clusters?

[D. B. Staple]

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.

[ldesnogu]

Great achievement, can't wait to see more information

[CRGreathouse]

Congrats!

Originally Posted by D. B. Staple

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.

[D. B. Staple]

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

[Dubslow]

Originally Posted by D. B. Staple

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.

[XYYXF]

http://www.primefan.ru/stuff/primes/table.html is updated :-)

[ATH]

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?

[ldesnogu]

Originally Posted by ATH

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

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?

[D. B. Staple]

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