bpo-40028: Add is_prime, factorise, previous_prime and next_prime

[remilapeyre]

https://bugs.python.org/issue40028

[sweeneyde]

Something like this would help to not exceed maximum recursion depth:

def _factorise(n):
    stack = [n]
    while stack:
        n = stack.pop()
        if is_prime(n):
            yield n
        else:
            d = _rho_pollard(n)
            stack += [n//d, d]

_factorise could perhaps then be in-lined, since it seems like factorise doesn't do too much on its own.
This then passes the test set(factorise(2**10000)) == {2}.

[remilapeyre]

Thanks!

[sweeneyde]

Is there a reason to recompute u here when its value should be the same as q?

[sweeneyde]

The following should be exactly the same algorithm, just refactored a little bit, and it runs a little faster for a few tests by my measurements.

def _rho_pollard(n):
    # From https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm
    # and https://en.wikipedia.org/wiki/Cycle_detection#Brent.27s_algorithm
    tortoise = hare = 2
    power = 1
    _gcd = gcd
    _repeat = itertools.repeat
    while True:
        # Double the length of the run each time
        for _ in _repeat(None, power):
            hare = (hare**2 + 1) % n
            d = _gcd(hare-tortoise, n)
            if d == 1:
                continue
            elif d == n:
                # failure; start another
                tortoise = hare = randrange(n)
                power = 1
                break
            else:
                return d
        else: # no break
            tortoise = hare
            power *= 2

Changes (feel free to pick and choose as you desire):

• starting at 2 rather than random seems to do good things
• Limit the Python int comparisons to one per inner loop
  • Factor out the inner loop as a repeat (a little faster than range()), making c code do the comparisons
• making _gcd and _repeat local variables helps a little.

[tiran]

The BPO was closed by @mdickinson

I suggest that your PR into a PyPI package and get feedback from the community.