On Thu, Jul 12, 2018 at 2:22 PM Chris Angelico <rosuav@gmail.com> wrote:
On Fri, Jul 13, 2018 at 4:11 AM, Steven D'Aprano <steve@pearwood.info> wrote:
> There is no reason why primality testing can't be deterministic up to
> 2**64, and probabilistic with a ludicrously small chance of false
> positives beyond that. The implementation I use can be expected to fail
> on average once every 18 thousand years if you did nothing but test
> primes every millisecond of the day, 24 hours a day. That's good enough
> for most purposes :-)

What about false negatives? Guaranteed none? The failure mode of the
function should, IMO, be a defined and documented aspect of it.

Miller-Rabin or other pseudo-primality tests do not produce false negatives IIUC.

I'm more concerned in the space/time tradeoff in the primes() generator. I like this implementation for teaching purposes, but I'm well aware that it grows in memory usage rather quickly (the one Tim Peters posted is probably a better balance; but it depends on how many of these you create and how long you run them).

from math import sqrt, ceil

def up_to(seq, lim):
    for n in seq:
        if n < lim:
            yield n

def sieve_generator():
    "Pretty good Sieve; skip the even numbers, stop at sqrt(candidate)"
    yield 2
    candidate = 3
    found = []
    while True:
        lim = int(ceil(sqrt(candidate)))
        if all(candidate % prime != 0 for prime in up_to(found, lim)):
            yield candidate
        candidate += 2

So then how do you implement isprime().  One naive way is to compare it against elements of sieve_generator() until we are equal or larger than the test element.  But that's not super fast.   A pseudo-primality test is much faster (except for in the first few hundred thousand primes, maybe).

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