What's the Fastest Algorithm to Generate All Prime Numbers Below a Given Integer N?

Fastest Way to List All Primes Below N: An Exploration

Problem:

Determine the fastest method to list all prime numbers less than a given integer N.

Question:

Can the given algorithm be optimized for faster execution?

Answer:

The provided algorithm can be significantly improved for speed. A comparison of various implementations reveals that rwh_primes1 with Psyco is the most efficient for generating primes less than 1,000,000.

Additional Findings:

• Without Psyco, rwh_primes2 emerges as the fastest method.
• Utilizing NumPy offers further performance enhancements, with primesfrom2to proving to be the fastest among all tested methods.

Implementation Details:

• ambi_sieve_plain: A straightforward sieve-based approach.
• rwh_primes, rwh_primes1, and rwh_primes2: Variations of Robert William Hanks' algorithms.
• sieve_wheel_30: A specialized algorithm optimized for 30-based calculations.
• sieveOfEratosthenes: The classic sieve method with bitset optimizations.
• sieveOfAtkin: A modern sieve utilizing modulo arithmetic.
• sundaram3: Sundaram's algorithm with optimizations for sets of smaller numbers.
• ambi_sieve: A sieve-based approach with NumPy optimizations.
• primesfrom3to and primesfrom2to: NumPy-based algorithms for efficiently generating primes.

Timings:

Method

Time (ms) with Psyco

Time (ms) without Psyco

rwh_primes1

43.0

93.7

sieveOfAtkin

46.4

314.0

rwh_primes

57.4

94.6

sieve_wheel_30

63.0

97.4

rwh_primes2

67.8

68.1

sieveOfEratosthenes

147.0

178.0

ambi_sieve_plain

152.0

286.0

sundaram3

194.0

416.0

primesfrom2to

Method	Time (ms) with Psyco	Time (ms) without Psyco
rwh_primes1	43.0	93.7
sieveOfAtkin	46.4	314.0
rwh_primes	57.4	94.6
sieve_wheel_30	63.0	97.4
rwh_primes2	67.8	68.1
sieveOfEratosthenes	147.0	178.0
ambi_sieve_plain	152.0	286.0
sundaram3	194.0	416.0
primesfrom2to	15.9	N/A
primesfrom3to	18.4	N/A
ambi_sieve	29.3	N/A

15.9

N/A

primesfrom3to

18.4

N/A

ambi_sieve

29.3

N/A

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