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Cooking with Python, Part 2
Pages: 1, 2

Recipe 18.10: Computing Prime Numbers

Credit: David Eppstein, Tim Peters, Alex Martelli, Wim Stolker, Kazuo Moriwaka, Hallvard Furuseth, Pierre Denis, Tobias Klausmann, David Lees, Raymond Hettinger


You need to compute an unbounded sequence of all primes, or the list of all primes that are less than a certain threshold.


To compute an unbounded sequence, a generator is the natural Pythonic approach, and the Sieve of Eratosthenes, using a dictionary as the supporting data structure, is the natural algorithm to use:

import itertools
def eratosthenes( ):
    '''Yields the sequence of prime numbers via the Sieve of Eratosthenes.'''
    D = {  }  # map each composite integer to its first-found prime factor
    for q in itertools.count(2):     # q gets 2, 3, 4, 5, ... ad infinitum
        p = D.pop(q, None)
        if p is None:
            # q not a key in D, so q is prime, therefore, yield it
            yield q
            # mark q squared as not-prime (with q as first-found prime factor)
            D[q*q] = q
            # let x <- smallest (N*p)+q which wasn't yet known to be composite
            # we just learned x is composite, with p first-found prime factor,
            # since p is the first-found prime factor of q -- find and mark it
            x = p + q
            while x in D:
                x += p
            D[x] = p


To compute all primes up to a predefined threshold, rather than an unbounded sequence, it's reasonable to wonder if it's possible to use a faster way than good old Eratosthenes, even in the smart variant shown as the "Solution". Here is a function that uses a few speed-favoring tricks, such as a hard-coded tuple of the first few primes:

def primes_less_than(N):
    # make `primes' a list of known primes < N
    primes = [x for x in (2, 3, 5, 7, 11, 13) if x < N]
    if N <= 17: return primes
    # candidate primes are all odd numbers less than N and over 15,
    # not divisible by the first few known primes, in descending order
    candidates = [x for x in xrange((N-2)|1, 15, -2)
                  if x % 3 and x % 5 and x % 7 and x % 11 and x % 13]
    # make `top' the biggest number that we must check for compositeness
    top = int(N ** 0.5)
    while (top+1)*(top+1) <= N:
        top += 1
    # main loop, weeding out non-primes among the remaining candidates
    while True:
        # get the smallest candidate: it must be a prime
        p = candidates.pop( )
        if p > top:
        # remove all candidates which are divisible by the newfound prime
        candidates = filter(p._ _rmod_ _, candidates)
    # all remaining candidates are prime, add them (in ascending order)
    candidates.reverse( )
    return primes

On a typical small task such as looping over all primes up to 8,192, eratosthenes (on an oldish 1.2 GHz Athlon PC, with Python 2.4) takes 22 milliseconds, while primes_less_than takes 9.7; so, the slight trickery and limitations of primes_less_than can pay for themselves quite handsomely if generating such primes is a bottleneck in your program. Be aware, however, that eratosthenes scales better. If you need all primes up to 199,999, eratosthenes will deliver them in 0.88 seconds, while primes_less_than takes 0.65.

Since primes_less_than's little speed-up tricks can help, it's natural to wonder whether a perhaps simpler trick can be retrofitted into eratosthenes as well. For example, we might simply avoid wasting work on a lot of even numbers, concentrating on odd numbers only, beyond the initial 2. In other words:

def erat2( ):
    D = {  }
    yield 2
    for q in itertools.islice(itertools.count(3), 0, None, 2):
        p = D.pop(q, None)
        if p is None:
            D[q*q] = q
            yield q
            x = p + q
            while x in D or not (x&1):
                x += p
            D[x] = p

And indeed, erat2 takes 16 milliseconds, versus eratosthenes' 22, to get primes up to 8,192; 0.49 seconds, versus eratosthenes' 0.88, to get primes up to 199,999. In other words, erat2 scales just as well as eratosthenes and is always approximately 25% faster. Incidentally, if you're wondering whether it might be even faster to program at a slightly lower level, with q = 3 to start, a while True as the loop header, and a q += 2 at the end of the loop, don't worry—the slightly higher-level approach using itertools' count and islice functions is repeatedly approximately 4% faster. Other languages may impose a performance penalty for programming with higher abstraction, Python rewards you for doing that.

You might keep pushing the same idea yet further, avoiding multiples of 3 as well as even numbers, and so on. However, it would be an exercise in diminishing returns: greater and greater complication for smaller and smaller gain. It's better to quit while we're ahead!

If you're into one liners, you might want to study the following:

def primes_oneliner(N):
    aux = {  }
    return [aux.setdefault(p, p) for p in range(2, N)
            if 0 not in [p%d for d in aux if p>=d+d]]

Be aware that one liners, even clever ones, are generally anything but speed demons! primes_oneliner takes 2.9 seconds to complete the same small task (computing primes up to 8,192) which, eratosthenes does in 22 milliseconds, and primes_less_than in 9.7--so, you're slowing things down by 130 to 300 times just for the fun of using a clever, opaque one liner, which is clearly not a sensible tradeoff. Clever one liners can be instructive but should almost never be used in production code, not just because they're terse and make maintenance harder than straightforward coding (which is by far the main reason), but also because of the speed penalties they may entail.

While prime numbers, and number theory more generally, used to be considered purely theoretical problems, nowadays they have plenty of practical applications, starting with cryptography.

See Also

To explore both number theory and its applications, the best book is probably Kenneth Rosen, Elementary Number Theory and Its Applications (Addison-Wesley); for more information about prime numbers.

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