This is to announce the release of my paper "Ultimate Prime Sieve --
Sieve of Zakiiya (SoZ)" in which I show and explain the development of
a class of Number Theory Sieves to generate prime numbers. I used
Ruby 1.9.0-1 as my development environment on a P4 2.8 Ghz laptop.
You can get the pdf of my paper and Ruby and Python source from here:
Below is a sample of one of the simple prime generators. I did a
Python version of this in my paper (see Python source too). The Ruby
version below is the minimum array size version, while the Python has
array of size N (I made no attempt to optimize its implementation,
it's to show the method). See my paper for what/why is going on here.
class Integer
def primesP3a
# all prime candidates 3 are of form 6*k+1 and 6*k+5
# initialize sieve array with only these candidate values
# where sieve contains the odd integers representatives
# convert integers to array indices/vals by i = (n-3)>>1
=(n>>1)-1
n1, n2 = -1, 1; lndx= (self-1) >>1; sieve = []
while n2 < lndx
n1 +=3; n2 += 3; sieve[n1] = n1; sieve[n2] = n2
end
#now initialize sieve array with (odd) primes < 6, resize array
sieve[0] =0; sieve[1]=1; sieve=sieve[0..lndx-1]
5.step(Math.sqr t(self).to_i, 2) do |i|
next unless sieve[(i>>1) - 1]
# p5= 5*i, k = 6*i, p7 = 7*i
# p1 = (5*i-3)>>1; p2 = (7*i-3)>>1; k = (6*i)>>1
i6 = 6*i; p1 = (i6-i-3)>>1; p2 = (i6+i-3)>>1; k = i6>>1
while p1 < lndx
sieve[p1] = nil; sieve[p2] = nil; p1 += k; p2 += k
end
end
return [2] if self < 3
[2]+([nil]+sieve).compact !.map {|i| (i<<1) +3 }
end
end
def primesP3(val):
# all prime candidates 3 are of form 6*k+(1,5)
# initialize sieve array with only these candidate values
n1, n2 = 1, 5
sieve = [False]*(val+6)
while n2 < val:
n1 += 6; n2 += 6; sieve[n1] = n1; sieve[n2] = n2
# now load sieve with seed primes 3 < pi < 6, in this case just 5
sieve[5] = 5
for i in range( 5, int(ceil(sqrt(v al))), 2) :
if not sieve[i]: continue
# p1= 5*i, k = 6*i, p2 = 7*i,
p1 = 5*i; k = p1+i; p2 = k+i
while p2 <= val:
sieve[p1] = False; sieve[p2] = False; p1 += k; p2 += k
if p1 <= val: sieve[p1] = False
primes = [2,3]
if val < 3 : return [2]
primes.extend( i for i in range(5, val+(val&1), 2) if sieve[i] )
return primes
Now to generate an array of the primes up to some N just do:
Ruby: 10000001.primes P3a
Python: primesP3a(10000 001)
The paper presents benchmarks with Ruby 1.9.0-1 (YARV). I would love
to see my various prime generators benchmarked with optimized
implementations in other languages. I'm hoping C/C++ gurus will do
good implementations . The methodology is very simple, since all I do
is additions, multiplications , and array reads/writes.
I would also like to the C implementations benchmarked against the
versions create by Daniel J Bernstein of the Sieve of Atkin (SoA). The
C code is here:
Have fun with the code. ;-)
Jabari Zakiya
Sieve of Zakiiya (SoZ)" in which I show and explain the development of
a class of Number Theory Sieves to generate prime numbers. I used
Ruby 1.9.0-1 as my development environment on a P4 2.8 Ghz laptop.
You can get the pdf of my paper and Ruby and Python source from here:
Below is a sample of one of the simple prime generators. I did a
Python version of this in my paper (see Python source too). The Ruby
version below is the minimum array size version, while the Python has
array of size N (I made no attempt to optimize its implementation,
it's to show the method). See my paper for what/why is going on here.
class Integer
def primesP3a
# all prime candidates 3 are of form 6*k+1 and 6*k+5
# initialize sieve array with only these candidate values
# where sieve contains the odd integers representatives
# convert integers to array indices/vals by i = (n-3)>>1
=(n>>1)-1
n1, n2 = -1, 1; lndx= (self-1) >>1; sieve = []
while n2 < lndx
n1 +=3; n2 += 3; sieve[n1] = n1; sieve[n2] = n2
end
#now initialize sieve array with (odd) primes < 6, resize array
sieve[0] =0; sieve[1]=1; sieve=sieve[0..lndx-1]
5.step(Math.sqr t(self).to_i, 2) do |i|
next unless sieve[(i>>1) - 1]
# p5= 5*i, k = 6*i, p7 = 7*i
# p1 = (5*i-3)>>1; p2 = (7*i-3)>>1; k = (6*i)>>1
i6 = 6*i; p1 = (i6-i-3)>>1; p2 = (i6+i-3)>>1; k = i6>>1
while p1 < lndx
sieve[p1] = nil; sieve[p2] = nil; p1 += k; p2 += k
end
end
return [2] if self < 3
[2]+([nil]+sieve).compact !.map {|i| (i<<1) +3 }
end
end
def primesP3(val):
# all prime candidates 3 are of form 6*k+(1,5)
# initialize sieve array with only these candidate values
n1, n2 = 1, 5
sieve = [False]*(val+6)
while n2 < val:
n1 += 6; n2 += 6; sieve[n1] = n1; sieve[n2] = n2
# now load sieve with seed primes 3 < pi < 6, in this case just 5
sieve[5] = 5
for i in range( 5, int(ceil(sqrt(v al))), 2) :
if not sieve[i]: continue
# p1= 5*i, k = 6*i, p2 = 7*i,
p1 = 5*i; k = p1+i; p2 = k+i
while p2 <= val:
sieve[p1] = False; sieve[p2] = False; p1 += k; p2 += k
if p1 <= val: sieve[p1] = False
primes = [2,3]
if val < 3 : return [2]
primes.extend( i for i in range(5, val+(val&1), 2) if sieve[i] )
return primes
Now to generate an array of the primes up to some N just do:
Ruby: 10000001.primes P3a
Python: primesP3a(10000 001)
The paper presents benchmarks with Ruby 1.9.0-1 (YARV). I would love
to see my various prime generators benchmarked with optimized
implementations in other languages. I'm hoping C/C++ gurus will do
good implementations . The methodology is very simple, since all I do
is additions, multiplications , and array reads/writes.
I would also like to the C implementations benchmarked against the
versions create by Daniel J Bernstein of the Sieve of Atkin (SoA). The
C code is here:
Have fun with the code. ;-)
Jabari Zakiya
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