techben / pyprimesieve

Many primes, very fast. Python sieve using primesieve.

pyprimesieve

Many primes, very fast. Uses primesieve.

primesieve, one of the fastest (if not the fastest) prime sieve implementaions available, is actively maintained by Kim Walisch.

It uses a segmented sieve of Eratosthenes with wheel factorization for a complexity of O(nloglogn) operations.

Performance

Regarding primesieve for C++:

primesieve generates the first 50,847,534 primes up to 10^9 in just 0.4 seconds on a single core of an Intel Core i7-920 2.66GHz, this is about 50 times faster than an ordinary C/C++ sieve of Eratosthenes implementation and about 10,000 times faster than trial-division. primesieve outperforms [Kim's] older ecprime (fastest from 2002 to 2010) by about 30 percent and also substantially outperforms primegen the fastest sieve of Atkin implementation on the web.

For comparison, on an Intel Core i7 2GHz, pyprimesieve populates an entire Python list of the first 50,847,534 primes in 1.40 seconds. It's expected that a Python implementation would be slower than C++ but, surprisingly, by only one second.

pyprimesieve outperforms all of the fastest prime sieving implementations for Python.

Time (ms) to generate the all primes below one million and iterate over them in Python:

algorithm	time
pyprimesieve	2.79903411865
primesfrom2to	13.1568908691
primesfrom3to	13.5800838470
ambi_sieve	16.1600112915
rwh_primes2	38.7749671936
rwh_primes1	48.5658645630
rwh_primes	52.0040988922
sieve_wheel_30	59.3869686127
sieveOfEratosthenes	59.4990253448
ambi_sieve_plain	161.740064621
sieveOfAtkin	232.724905014
sundaram3	251.194953918

It can be seen here that pyprimesieve is 4.7 times faster than the fastest Python alternative using Numpy and 13.85 times faster than the fastest pure Python sieve.

All benchmark scripts and algorithms are available for reproduction. Prime sieve algorithm implementations were taken from this discussion on SO.

Functions

primes(n): List of prime numbers up to n.

primes(start, n): List of prime numbers from start up to n.

primes_sum(n): The summation of prime numbers up to n. The optimal number of threads will be determined for the given number and system.

primes_sum(start, n): The summation of prime numbers from start up to n. The optimal number of threads will be determined for the given numbers and system.

primes_nth(n): The nth prime number.

factorize(n): List of tuples in the form of (prime, power) for the prime factorization of n.

Installation

pip install pyprimesieve

NOTE: To enable the parallelized version of prime summation, you must use a compiler that supports OpenMP. You may need to pass a valid compiler as an environment variable.

Testing

After installation, you can make sure everything is working by running the following inside the project root folder,

python tests

License

"Modified BSD License". See LICENSE for details. Copyright Jared Suttles, 2015.