primecount

primecount is a command-line program and C/C++ library that counts the number of primes ≤ x (maximum 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 install primecount
Arch Linux:	sudo pacman -S primecount
Debian/Ubuntu:	sudo apt install primecount
Fedora:	sudo dnf install primecount
FreeBSD:	pkg 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
sudo ldconfig

• Detailed build instructions

Usage examples

# Count the primes ≤ 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	0.01s	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	0.02s	0.01s	0.01s
1013	346,065,536,839	0.09s	0.06s	0.03s	0.02s	0.03s
1014	3,204,941,750,802	0.44s	0.20s	0.08s	0.08s	0.04s
1015	29,844,570,422,669	2.33s	0.89s	0.29s	0.16s	0.11s
1016	279,238,341,033,925	15.49s	5.10s	1.26s	0.58s	0.38s
1017	2,623,557,157,654,233	127.10s	39.39s	5.62s	2.26s	1.34s
1018	24,739,954,287,740,860	1,071.14s	366.93s	27.19s	9.96s	5.35s
1019	234,057,667,276,344,607	NaN	NaN	NaN	40.93s	20.16s
1020	2,220,819,602,560,918,840	NaN	NaN	NaN	167.64s	81.98s
1021	21,127,269,486,018,731,928	NaN	NaN	NaN	706.70s	353.01s
1022	201,467,286,689,315,906,290	NaN	NaN	NaN	3,012.10s	1,350.47s

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 $\pi(10^{25})$ 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 $\pi(10^{25})$ 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 about 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 -march=native option.

CXXFLAGS="-march=native" cmake .
make -j

By default primecount scales nicely up until 10²³ 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	$\pi(x)=\pi(\sqrt{x})+\phi(x,\pi(\sqrt{x}))-1$
Meissel's Formula	$\pi(x)=\pi(\sqrt[3]{x})+\phi(x,\pi(\sqrt[3]{x}))-\mathrm{P_2}(x,\pi(\sqrt[3]{x}))-1$
Lehmer's Formula	$\pi(x)=\pi(\sqrt[4]{x})+\phi(x,\pi(\sqrt[4]{x}))-\mathrm{P_2}(x,\pi(\sqrt[4]{x}))-\mathrm{P_3}(x,\pi(\sqrt[4]{x}))-1$
LMO Formula	$\pi(x)=\pi(\sqrt[3]{x})+\mathrm{S_1}(x,\pi(\sqrt[3]{x}))+\mathrm{S_2}(x,\pi(\sqrt[3]{x}))-\mathrm{P_2}(x,\pi(\sqrt[3]{x}))-1$

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 $O(x\log{\log{x}})$ 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 $O(x)$ operations and uses $O(\sqrt{x}/\log{x})$ space. In 1870 E. D. F. Meissel improved Legendre's formula by setting $a=\pi(\sqrt[3]{x})$ and by adding the correction term $\mathrm{P_2}(x,a)$, Meissel's formula has a running time of $O(x/\log^3{x})$ operations and uses $O(\sqrt[3]{x})$ space. In 1959 D. H. Lehmer extended Meissel's formula and slightly improved the running time to $O(x/\log^4{x})$ operations and $O(x^{\frac{3}{8}})$ 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 $O(x^{\frac{2}{3}}/\log{x})$ operations and which uses only $O(\sqrt[3]{x}\ \log^2{x})$ 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 so-called hard special leaves 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 counter array in combination with the POPCNT instruction which is more cache efficient and much faster. The Hard-Special-Leaves.md document contains more information. primecount's easy special leaf implementation and its partial sieve function implementation also contain significant improvements.

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 $\mathrm{Li}^{-1}(n)$ or the inverse Riemann R function $\mathrm{R}^{-1}(n)$ 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 $O(x^{\frac{2}{3}}/\log^2{x})$ operations using $O(\sqrt{x})$ 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 <= 1000: %ld\n", pix);

    return 0;
}

• C API documentation
• libprimecount build instructions

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 <= 1000: " << pix << std::endl;

    return 0;
}

• C++ API documentation
• libprimecount build instructions

Bindings for other languages

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

Common Lisp:	cl-primecount
Julia:	primecount_jll.jl
Haskell:	primecount-haskell
Python:	primecountpy
Python:	primecount-python
Rust:	primecount-rs

Many thanks to the developers of these bindings!