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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
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