primecount - Man Page
count prime numbers
Synopsis
primecount x [options]
Description
Count the number of primes less than or equal to x (<= 10^31) using fast implementations of the combinatorial prime counting function algorithms. By default primecount counts primes using Xavier Gourdon’s algorithm which has a runtime complexity of O(x^(2/3) / log^2 x) operations and uses O(x^(2/3) * log^3 x) memory. primecount is multi-threaded, it uses all available CPU cores by default.
Options
- -d, --deleglise-rivat
Count primes using the Deleglise-Rivat algorithm.
- -g, --gourdon
Count primes using Xavier Gourdon’s algorithm (default algorithm).
- -l, --legendre
Count primes using Legendre’s formula.
- --lehmer
Count primes using Lehmer’s formula.
- --lmo
Count primes using the Lagarias-Miller-Odlyzko algorithm.
- -m, --meissel
Count primes using Meissel’s formula.
- --Li
Approximate pi(x) using the Eulerian logarithmic integral: Li(x), with Li(x) = li(x) - li(2).
- --Li-inverse
Approximate the nth prime using the inverse Eulerian logarithmic integral: Li^-1(x).
- -n, --nth-prime
Calculate the nth prime.
- -p, --primesieve
Count primes using the sieve of Eratosthenes.
- --phi X A
phi(x, a) counts the numbers <= x that are not divisible by any of the first a primes.
- -R, --RiemannR
Approximate pi(x) using the Riemann R function: R(x).
- --RiemannR-inverse
Approximate the nth prime using the inverse Riemann R function: R^-1(x).
- -s, --status[=NUM]
Show the computation progress e.g. 1%, 2%, 3%, ... Show NUM digits after the decimal point: --status=1 prints 99.9%.
- --test
Run various correctness tests and exit.
- --time
Print the time elapsed in seconds.
- -t, --threads=NUM
Set the number of threads, 1 <= NUM <= CPU cores. By default primecount uses all available CPU cores.
- -v, --version
Print version and license information.
- -h, --help
Print this help menu.
Advanced Options for the Deleglise-Rivat Algorithm
- --P2
Compute the 2nd partial sieve function.
- --S1
Compute the ordinary leaves.
- --S2-trivial
Compute the trivial special leaves.
- --S2-easy
Compute the easy special leaves.
- --S2-hard
Compute the hard special leaves.
Tuning factor
The alpha tuning factor mainly balances the computation of the S2_easy and S2_hard formulas. By increasing alpha the runtime of the S2_hard formula will usually decrease but the runtime of the S2_easy formula will increase. For large pi(x) computations with x >= 10^25 you can usually achieve a significant speedup by increasing alpha.
The alpha tuning factor is also very useful for verifying pi(x) computations. You compute pi(x) twice but for the second computation you use a slightly different alpha factor. If the results of both pi(x) computations match then pi(x) has been verified successfully.
- -a, --alpha=NUM
Set the alpha tuning factor: y = x^(1/3) * alpha, 1 <= alpha <= x^(1/6).
Advanced Options for Xavier Gourdon’s Algorithm
- --AC
Compute the A + C formulas.
- --B
Compute the B formula.
- --D
Compute the D formula.
- --Phi0
Compute the Phi0 formula.
- --Sigma
Compute the 7 Sigma formulas.
Tuning factors
The alpha_y and alpha_z tuning factors mainly balance the computation of the A, B, C and D formulas. When alpha_y is decreased but alpha_z is increased then the runtime of the B formula will increase but the runtime of the A, C and D formulas will decrease. For large pi(x) computations with x >= 10^25 you can usually achieve a significant speedup by decreasing alpha_y and increasing alpha_z. For convenience when you increase alpha_z using --alpha-z=NUM then alpha_y is automatically decreased.
Both the alpha_y and alpha_z tuning factors are also very useful for verifying pi(x) computations. You compute pi(x) twice but for the second computation you use a slightly different alpha_y or alpha_z factor. If the results of both pi(x) computations match then pi(x) has been verified successfully.
- --alpha-y=NUM
Set the alpha_y tuning factor: y = x^(1/3) * alpha_y, 1 <= alpha_y <= x^(1/6).
- --alpha-z=NUM
Set the alpha_z tuning factor: z = y * alpha_z, 1 <= alpha_z <= x^(1/6).
Examples
primecount 1000
Count the primes <= 1000.
primecount 1e17 --status
Count the primes <= 10^17 and print status information.
primecount 1e15 --threads 1 --time
Count the primes <= 10^15 using a single thread and print the time elapsed.
Homepage
Author
Kim Walisch <kim.walisch@gmail.com>