ðŸš€ Fast prime counting function implementations

Project README

primecount is a command-line program and C/C++ library
that counts the number of primesÂ â‰¤Â x (maximum 10^{31}) 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.

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` |

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
```

```
# Count the primes â‰¤ 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
```

```
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 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
```

x |
Prime Count |
Legendre |
Meissel |
LagariasMiller Odlyzko |
DelegliseRivat |
Gourdon |

10^{10} |
455,052,511 | 0.01s | 0.01s | 0.01s | 0.01s | 0.00s |

10^{11} |
4,118,054,813 | 0.01s | 0.01s | 0.01s | 0.01s | 0.01s |

10^{12} |
37,607,912,018 | 0.03s | 0.02s | 0.02s | 0.01s | 0.01s |

10^{13} |
346,065,536,839 | 0.09s | 0.06s | 0.03s | 0.02s | 0.03s |

10^{14} |
3,204,941,750,802 | 0.44s | 0.20s | 0.08s | 0.08s | 0.04s |

10^{15} |
29,844,570,422,669 | 2.33s | 0.89s | 0.29s | 0.16s | 0.11s |

10^{16} |
279,238,341,033,925 | 15.49s | 5.10s | 1.26s | 0.58s | 0.38s |

10^{17} |
2,623,557,157,654,233 | 127.10s | 39.39s | 5.62s | 2.26s | 1.34s |

10^{18} |
24,739,954,287,740,860 | 1,071.14s | 366.93s | 27.19s | 9.96s | 5.35s |

10^{19} |
234,057,667,276,344,607 | NaN | NaN | NaN | 40.93s | 20.16s |

10^{20} |
2,220,819,602,560,918,840 | NaN | NaN | NaN | 167.64s | 81.98s |

10^{21} |
21,127,269,486,018,731,928 | NaN | NaN | NaN | 706.70s | 353.01s |

10^{22} |
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 BÃ¼the mentions in [11] that he computed $\pi(10^{25})$ in 40,000 CPU core hours using the analytic prime counting function algorithm. BÃ¼the 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!

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 10^{23} 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'
```

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 TomÃ¡s 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.

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.

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;
}
```

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;
}
```

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!

Open Source Agenda is not affiliated with "Primecount" Project. README Source: kimwalisch/primecount

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