How to implement an efficient infinite generator of prime numbers in Python?

Question

This is not a homework, I am just curious.

INFINITE is the key word here.

I wish to use it as for p in primes(). I believe that this is a built-

Answer 1

Here's a pretty fast infinite generator, written in Python2 but easily adjusted to Python3. To use it to add the primes up to 10**9, use the following:

from itertools import takewhile
from functools import partial
from operator import gt
print (sum(takewhile(partial(gt, 10**9), prime_gen_inf())))

It's a segmented sieve, faster but obviously less elegant than Will Ness's algorithm.

from operator import mul
from functools import reduce
def prod(x): return reduce(mul, x, 1)


def build_sieve(wheel):
    w = prod(wheel)
    w_phi = prod([p-1 for p in wheel])
    rems = [a for a in range(w) if all(a % p for p in wheel)]
    assert len(rems) == w_phi
    inv = {a:pow(a, w_phi - 1, w) for a in rems}
    try:
        known_p = wheel + rems[1 : rems.index(rems[1]*rems[1])]
    except ValueError:
        known_p = wheel + rems[1:]
    return wheel, w, w_phi, rems, inv, known_p

#Adjust the chunk variable based on your computer's architecture.
#
#Adjust the line with #! if you don't need "true" infinite.  If you don't need
#primes larger than 1<<32, use array('H', []), if 1<<64 use 'L', if 1<<128 (in
#Python3) use 'Q', otherwise use empty list [].
#To save memory, comment out the lines with #*, and uncomment the commented-out
#lines 
import itertools
from itertools import islice, count, compress, izip
chain_f = itertools.chain.from_iterable
from array import array
def prime_gen_inf(chunk=250000, sieve_info = build_sieve([2,3,5,7])):
    """    Indefinitely yields primes    """
    wheel, w, w_phi, rems, inv, known_p = sieve_info
    for p in known_p: yield p
    new_n = 0;
    while True:
        size = min(chunk, (p * p - new_n) / w)
        sieve = bytearray([1]) * size * w_phi
        n, new_n = new_n, new_n + size * w
        if not n:
            zero = bytearray([0])
            seen = len(known_p) - len(wheel) + 1
            sieve[:seen:1] = zero * seen
            p_gen = islice(prime_gen_inf(), len(wheel), None)
            new_p = next(p_gen)
            ps = []                                         #! array('H', [])
            p_invs = bytearray([])                                         #*
        while new_p * new_p < new_n:
            ps.append(new_p)
            p_invs.append(inv[new_p % w])                                  #*
            new_p = next(p_gen)
        for p, p_inv, modp in izip(ps, p_invs, [-n % p for p in ps]):      #*
            s = [(modp + p * (p_inv * (r - modp) % w)) / w for r in rems]  #*
        #for p in ps:
        #    s = [(-n%p + p * (inv[p%w] * (r - -n%p) % w)) / w for r in rems]
            for i, start in enumerate(s):
                slice_size = ((size - start - 1) / p + 1)
                sieve[i + start * w_phi :: p * w_phi] = zero * slice_size
        for p in compress(chain_f(izip(*[count(n+r, w) for r in rems])), sieve):
            yield p