import random

_mrpt_num_trials = 5 # number of bases to test

def is_probable_prime(n):
    """
    Miller-Rabin primality test.

    A return value of False means n is certainly not prime. A return value of
    True means n is very likely a prime.

    >>> is_probable_prime(1)
    Traceback (most recent call last):
        ...
    AssertionError
    >>> is_probable_prime(2)
    True
    >>> is_probable_prime(3)
    True
    >>> is_probable_prime(4)
    False
    >>> is_probable_prime(5)
    True
    >>> is_probable_prime(123456789)
    False

    >>> primes_under_1000 = [i for i in range(2, 1000) if is_probable_prime(i)]
    >>> len(primes_under_1000)
    168
    >>> primes_under_1000[-10:]
    [937, 941, 947, 953, 967, 971, 977, 983, 991, 997]

    >>> is_probable_prime(6438080068035544392301298549614926991513861075340134\
3291807343952413826484237063006136971539473913409092293733259038472039\
7133335969549256322620979036686633213903952966175107096769180017646161\
851573147596390153)
    True

    >>> is_probable_prime(7438080068035544392301298549614926991513861075340134\
3291807343952413826484237063006136971539473913409092293733259038472039\
7133335969549256322620979036686633213903952966175107096769180017646161\
851573147596390153)
    False
    """
    assert n >= 2
    # special case 2
    if n == 2:
        return True
    # ensure n is odd
    if n % 2 == 0:
        return False
    # write n-1 as 2**s * d
    # repeatedly try to divide n-1 by 2
    s = 0
    d = n-1
    while True:
        quotient, remainder = divmod(d, 2)
        if remainder == 1:
            break
        s += 1
        d = quotient
    assert(2**s * d == n-1)

    # test the base a to see whether it is a witness for the compositeness of n
    def try_composite(a):
        if pow(a, d, n) == 1:
            return False
        for i in range(s):
            if pow(a, 2**i * d, n) == n-1:
                return False
        return True # n is definitely composite

    for i in range(_mrpt_num_trials):
        a = random.randrange(2, n)
        if try_composite(a):
            return False

    return True # no base tested showed n as composite