# Ultimate Prime Sieve -- Sieve Of Zakiya (SoZ)

J

#### jzakiya

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:

http://www.4shared.com/dir/7467736/97bd7b71/sharing.html

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; sieve=1; sieve=sieve[0..lndx-1]

5.step(Math.sqrt(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  if self < 3
+([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

for i in range( 5, int(ceil(sqrt(val))), 2) :
if not sieve: 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 
primes.extend( i for i in range(5, val+(val&1), 2) if sieve )

return primes

Now to generate an array of the primes up to some N just do:

Ruby: 10000001.primesP3a
Python: primesP3a(10000001)

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

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:

http://cr.yp.to/primesgen.html

Have fun with the code. ;-)

Jabari Zakiya

J

#### jzakiya

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:

http://www.4shared.com/dir/7467736/97bd7b71/sharing.html

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;  sieve=1;  sieve=sieve[0..lndx-1]

5.step(Math.sqrt(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  if self < 3
+([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

for i in range( 5, int(ceil(sqrt(val))), 2) :
if not sieve:  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 
primes.extend( i for i in range(5, val+(val&1), 2)  if sieve )

return primes

Now to generate an array of the primes up to some N just do:

Ruby:      10000001.primesP3a
Python:   primesP3a(10000001)

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

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:

http://cr.yp.to/primesgen.html

Have fun with the code.  ;-)

Jabari Zakiya

CORRECTION:

http://cr.yp.to/primegen.html NOT "primesgen"

V

#### vippstar

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.
<snip>
Not related to ISO C. Try another newsgroup.

U

#### user923005

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:

http://www.4shared.com/dir/7467736/97bd7b71/sharing.html

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;  sieve=1;  sieve=sieve[0..lndx-1]

5.step(Math.sqrt(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  if self < 3
+([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

for i in range( 5, int(ceil(sqrt(val))), 2) :
if not sieve:  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 
primes.extend( i for i in range(5, val+(val&1), 2)  if sieve )

return primes

Now to generate an array of the primes up to some N just do:

Ruby:      10000001.primesP3a
Python:   primesP3a(10000001)

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

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:

http://cr.yp.to/primesgen.html

I might give it a go. Bernstein's primegen is not your best
competition. This is:
http://www.primzahlen.de/files/referent/kw/sieb.htm

Here is output using a single 3 GHz CPU:

Microsoft Windows [Version 6.0.6001]

C:\math\sieve\ecprime\x64\Release 64>ecprime 100000000000

100%
primes: 4118054813
time: 60.153 sec
C:\math\sieve\ecprime\x64\Release 64>

So your mark is to beat sieving through 100 billion in a minute.

I might give your algorithm a crack (no promises though). I set
follow-ups to news:comp.programming

U

#### user923005

... snip ...

Ruby and Python are not C.  This is comp.lang.c, and other
languages are off-topic.

That's true, and his request (to build a version in C to see how it
compares to other sieves) is also off topic.
However, Ruby and Python were not the gist of his post {more like an
aside}, so he may deserve censure, but certainly not for that.

In my previous post I made a forward to which is
more appropriate. There is a contests newsgroup also, but it does not
seem very active and this is not really a contest either.