# Prime sieve optimization

The prime sieve of Atkin (http://en.wikipedia.org/wiki/Sieve_of_Atkin)
is, if done right, faster than the sieve of Eratosthenes. I’ve
included a reimplementation of the mathn Prime class using this sieve,
but it’s slower than the one in ruby 1.9.

If anyone wants to try their hand at improving this (or starting anew
if this is a bad start), feel free. Perhaps we can beat the one in
ruby 1.9 so we can replace it before 1.9 (1.10 ?) becomes the stable
version.

class Prime_Atkin
include Enumerable

# @@next_to_check is a multiple of 12.

@@next_to_check = 240

# @@primes should contain all primes in 1…@@next_to_check

@@primes = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,
73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,
163,167,173,179,181,191,193,197,199,211,223,227,229,233,239]

# @@primes_squared = @@primes.map { |prime| prime**2 }[2…-1]

@@primes_squared = [25,49,121,169,289,361,529,841,961,1369,1681,
1849,2209,2809,3481,3721,4489,5041,5329,6241,6889,7921,9409,
10201,10609,11449,11881,12769,16129,17161,18769,19321,22201,
22801,24649,26569,27889,29929,32041,32761,36481,37249,38809,
39601,44521,49729,51529,52441,54289,57121]

# Ensure that @@next_to_check + SieveWidth <= @@primes_squared.last

SieveWidth = 56880
@@new_primes = Array.new(SieveWidth, false)
def initialize
@index = -1
end

def succ
@index += 1
while @index >= @@primes.length
range_end = @@next_to_check + SieveWidth
high_x = Math.sqrt(range_end).floor

``````  1.upto(high_x) do |x|
x_sq = x**2
low_y = @@next_to_check - 4*x_sq
if low_y < 1
low_y = 1
else
low_y = Math.sqrt(low_y).ceil
end
high_y = [Math.sqrt(3*x_sq - @@next_to_check),
``````

Math.sqrt(range_end - 3*x_sq)].max.floor

``````    high_y.downto(low_y) do |y|
y_sq = y**2
n = 4*x_sq + y_sq - @@next_to_check
@@new_primes[n] = (not @@new_primes.at(n)) if (n >= 0 and n <
``````

SieveWidth and (n % 12 == 1 or n % 12 == 5))
n -= x_sq
@@new_primes[n] = (not @@new_primes.at(n)) if (n >= 0 and n <
SieveWidth and n % 12 == 7)
n -= 2*y_sq
@@new_primes[n] = (not @@new_primes.at(n)) if (n >= 0 and n <
SieveWidth and x > y and n % 12 == 11)
end
end

``````  @@primes_squared.each do |prime_squared|
multiple_index = prime_squared *
``````

(@@next_to_check/prime_squared).ceil - @@next_to_check
while multiple_index < SieveWidth
@@new_primes[multiple_index] = false
multiple_index += prime_squared
end
end

``````  @@new_primes.each_with_index do |prime_test, index|
if prime_test
prime = @@next_to_check + index
@@primes << prime
@@primes_squared << prime**2
end
end

@@next_to_check = range_end
@@new_primes.fill false
end
@@primes.at @index
``````

end
alias next succ

def each
loop do
yield succ
end
end
end

If I do all the sieving in one stretch, a naive implementation of the
Atkin sieve is actually faster than the Prime class in ruby 1.9. The
Prime class has to do it in blocks (it never knows quite how many
primes someone will ask for), and it seems that I’ve handled that

Here is a naive, all-in-one-stretch version (faster than Prime in 1.9)
:

limit = 500000
primes = Array.new(limit + 1) { false }

(1…limit ** 0.5).each do |x|
x_sq = x2
(1…limit ** 0.5).each do |y|
y_sq = y
2
n = 4x_sq + y_sq
primes[n] = (not primes[n]) if (n <= limit and (n % 12 == 1 or n %
12 == 5))
n = 3
x_sq + y_sq
primes[n] = (not primes[n]) if (n <= limit and n % 12 == 7)
n = 3*x_sq - y_sq
primes[n] = (not primes[n]) if (n <= limit and x > y and n % 12 ==
11)
end
end

primes.each_index do |i|
primes[i] =
if primes[i]
i_sq = i**2
i_sq.step(limit, i_sq) do |prime_square_mult|
primes[prime_square_mult] = false
end
i
else
nil
end
end

primes[2] = 2
primes[3] = 3
primes.compact!