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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

### Problem

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

### Solution

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
else:
# 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``````

### Discussion

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( )
primes.append(p)
if p > top:
break
# 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( )
primes.extend(candidates)
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
else:
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.

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