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primecount

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primecount is a command-line program and C/C++ library that counts the primes below an integer x ≤ 1031 using highly optimized implementations of the combinatorial prime counting algorithms.

primecount includes implementations of all important combinatorial prime counting algorithms known up to this date all of which have been parallelized using OpenMP. primecount contains the first ever open source implementations of the Deleglise-Rivat algorithm and Xavier Gourdon's algorithm (that works). primecount also features a novel load balancer that is shared amongst all implementations and that scales up to hundreds of CPU cores. primecount has already been used to compute several prime counting function world records.

Installation

The primecount command-line program is available in a few package managers. For doing development with libprimecount you may need to install libprimecount-dev or libprimecount-devel.

Windows: winget install primecount
macOS: brew tap kimwalisch/primecount
brew install primecount
Arch Linux: sudo pacman -S primecount
Fedora: sudo dnf install primecount
openSUSE: sudo zypper install primecount

Build instructions

You need to have installed a C++ compiler and CMake. Ideally primecount should be compiled using GCC or Clang as these compilers support both OpenMP (multi-threading library) and 128-bit integers.

cmake .
make -j
sudo make install

Usage examples

# Count the primes below 10^14
primecount 1e14

# Print progress and status information during computation
primecount 1e20 --status

# Count primes using Meissel's algorithm
primecount 2**32 --meissel

# Find the 10^14th prime using 4 threads
primecount 1e14 --nth-prime --threads=4 --time

Command-line options

Usage: primecount x [options]
Count the number of primes less than or equal to x (<= 10^31).

Options:

  -d, --deleglise-rivat  Count primes using the Deleglise-Rivat algorithm
  -g, --gourdon          Count primes using Xavier Gourdon's algorithm.
                         This is the default algorithm.
  -l, --legendre         Count primes using Legendre's formula
      --lehmer           Count primes using Lehmer's formula
      --lmo              Count primes using Lagarias-Miller-Odlyzko
  -m, --meissel          Count primes using Meissel's formula
      --Li               Approximate pi(x) using the logarithmic integral
      --Li-inverse       Approximate the nth prime using 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
      --Ri               Approximate pi(x) using Riemann R
      --Ri-inverse       Approximate the nth prime using Ri^-1(x)
  -s, --status[=NUM]     Show computation progress 1%, 2%, 3%, ...
                         Set digits after decimal point: -s1 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
Advanced options for the Deleglise-Rivat algorithm:

  -a, --alpha=NUM        Set tuning factor: y = x^(1/3) * alpha
      --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

Advanced options for Xavier Gourdon's algorithm:

      --alpha-y=NUM      Set tuning factor: y = x^(1/3) * alpha_y
      --alpha-z=NUM      Set tuning factor: z = y * alpha_z
      --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

Benchmarks

x Prime Count Legendre Meissel Lagarias
Miller
Odlyzko
Deleglise
Rivat
Gourdon
1010 455,052,511 0.01s 0.01s 0.00s 0.01s 0.00s
1011 4,118,054,813 0.01s 0.01s 0.01s 0.01s 0.01s
1012 37,607,912,018 0.03s 0.02s 0.01s 0.01s 0.01s
1013 346,065,536,839 0.09s 0.06s 0.04s 0.02s 0.02s
1014 3,204,941,750,802 0.42s 0.19s 0.10s 0.10s 0.07s
1015 29,844,570,422,669 2.26s 0.92s 0.29s 0.26s 0.18s
1016 279,238,341,033,925 15.50s 5.12s 1.26s 1.04s 0.52s
1017 2,623,557,157,654,233 123.80s 38.70s 5.54s 2.58s 1.57s
1018 24,739,954,287,740,860 1,039.30s 345.33s 25.67s 9.34s 5.35s
1019 234,057,667,276,344,607 NaN NaN NaN 40.54s 21.00s
1020 2,220,819,602,560,918,840 NaN NaN NaN 170.92s 83.23s
1021 21,127,269,486,018,731,928 NaN NaN NaN 710.78s 355.13s
1022 201,467,286,689,315,906,290 NaN NaN NaN 3,114.37s 1,564.49s

The benchmarks above were run on an AMD 7R32 CPU (from 2020) with 16 cores/32 threads clocked at 3.30GHz. Note that Jan Büthe mentions in [11] that he computed pi(1025) in 40,000 CPU core hours using the analytic prime counting function algorithm. Büthe also mentions that by using additional zeros of the zeta function the runtime could have potentially been reduced to 4,000 CPU core hours. However using primecount and Xavier Gourdon's algorithm pi(1025) can be computed in only 460 CPU core hours on an AMD Ryzen 3950X CPU!

Performance tips

If you have an x64 CPU and you have installed primecount using the package manager of your Linux distribution, then it is possible that the POPCNT instruction has been disabled in order to ensure that primecount works on very old CPUs. Unfortunately this decreases performance by up to 30%. On the other hand, if you compile primecount from source the POPCNT instruction will be enabled by default. The fastest primecount binary can be built using the Clang compiler and the -march=native option.

CXX=clang++ CXXFLAGS="-march=native" cmake .
make -j

By default primecount scales nicely up until 1020 on current x64 CPUs. For larger values primecount's large memory usage causes many TLB (translation lookaside buffer) cache misses that significantly deteriorate primecount's performance. Fortunately the Linux kernel allows to enable transparent huge pages so that large memory allocations will automatically be done using huge pages instead of ordinary pages which dramatically reduces the number of TLB cache misses.

# Enable transparent huge pages until next reboot
sudo bash -c 'echo always > /sys/kernel/mm/transparent_hugepage/enabled'

Algorithms

Legendre's Formula
Meissel's Formula
Lehmer's Formula
LMO Formula

Up until the early 19th century the most efficient known method for counting primes was the sieve of Eratosthenes which has a running time of operations. The first improvement to this bound was Legendre's formula (1830) which uses the inclusion-exclusion principle to calculate the number of primes below x without enumerating the individual primes. Legendre's formula has a running time of operations and uses space. In 1870 E. D. F. Meissel improved Legendre's formula by setting and by adding the correction term . Meissel's formula has a running time of operations and uses space. In 1959 D. H. Lehmer extended Meissel's formula and slightly improved the running time to operations and space. In 1985 J. C. Lagarias, V. S. Miller and A. M. Odlyzko published a new algorithm based on Meissel's formula which has a lower runtime complexity of operations and which uses only space.

primecount's Legendre, Meissel and Lehmer implementations are based on Hans Riesel's book [5], its Lagarias-Miller-Odlyzko and Deleglise-Rivat implementations are based on Tomás Oliveira's paper [9] and the implementation of Xavier Gourdon's algorithm is based on Xavier Gourdon's paper [7]. primecount's implementation of the special leaves formula is different from the algorithms that have been described in any of the combinatorial prime counting papers so far. Instead of using a binary indexed tree for counting which is very cache inefficient primecount uses a linear counters array in combination with the POPCNT instruction which is more cache efficient and much faster. The Special-Leaves.md document contains more information. primecount's implementation of the easy special leaves is also noteworthy and described in Easy-Special-Leaves.md.

Fast nth prime calculation

The most efficient known method for calculating the nth prime is a combination of the prime counting function and a prime sieve. The idea is to closely approximate the nth prime (e.g. using the inverse logarithmic integral or the inverse Riemann R function ) and then count the primes up to this guess using the prime counting function. Once this is done one starts sieving (e.g. using the segmented sieve of Eratosthenes) from there on until one finds the actual nth prime. The author has implemented primecount::nth_prime(n) this way (option: --nth-prime), it finds the nth prime in operations using space.

C API

Include the <primecount.h> header to use primecount's C API. All functions that are part of primecount's C API return -1 in case an error occurs and print the corresponding error message to the standard error stream.

#include <primecount.h>
#include <stdio.h>

int main()
{
    int64_t pix = primecount_pi(1000);
    printf("primes below 1000 = %ld\n", pix);

    return 0;
}

C++ API

Include the <primecount.hpp> header to use primecount's C++ API. All functions that are part of primecount's C++ API throw a primecount_error exception (which is derived from std::exception) in case an error occurs.

#include <primecount.hpp>
#include <iostream>

int main()
{
    int64_t pix = primecount::pi(1000);
    std::cout << "primes below 1000 = " << pix << std::endl;

    return 0;
}

Bindings for other languages

primesieve natively supports C and C++ and has bindings available for:

Common Lisp: cl-primecount
Haskell: hprimecount
Python: primecount-python
Rust: primecount-rs

Many thanks to the developers of these bindings!