

def        lowerCAmelCase__				(						SCREAMING_SNAKE_CASE_:				int						,							SCREAMING_SNAKE_CASE_:				Tuple = False   )      ->   str:




				'''simple docstring'''





				if n == 2:
								return True
				if not n % 2 or n < 2:
								return False
				if n > 5 and n % 1_0 not in (1, 3, 7, 9):  # can quickly check last digit
								return False
				if n > 3_3_1_7_0_4_4_0_6_4_6_7_9_8_8_7_3_8_5_9_6_1_9_8_1 and not allow_probable:
								raise ValueError(
								    "Warning: upper bound of deterministic test is exceeded. "
								    "Pass allow_probable=True to allow probabilistic test. "
								    "A return value of True indicates a probable prime."   )
				# array bounds provided by analysis
				A__          =      [
				    2_0_4_7,
				    1_3_7_3_6_5_3,
				    2_5_3_2_6_0_0_1,
				    3_2_1_5_0_3_1_7_5_1,
				    2_1_5_2_3_0_2_8_9_8_7_4_7,
				    3_4_7_4_7_4_9_6_6_0_3_8_3,
				    3_4_1_5_5_0_0_7_1_7_2_8_3_2_1,
				    1,
				    3_8_2_5_1_2_3_0_5_6_5_4_6_4_1_3_0_5_1,
				    1,
				    1,
				    3_1_8_6_6_5_8_5_7_8_3_4_0_3_1_1_5_1_1_6_7_4_6_1,
				    3_3_1_7_0_4_4_0_6_4_6_7_9_8_8_7_3_8_5_9_6_1_9_8_1,
				]

				A__          =      [2, 3, 5, 7, 1_1, 1_3, 1_7, 1_9, 2_3, 2_9, 3_1, 3_7, 4_1]
				for idx, _p in enumerate(__A						,							1   ):
								if n < _p:
												# then we have our last prime to check
												A__          =      primes[:idx]
												break
				A__     ,					A__          =      n - 1, 0
				# break up n -1 into a power of 2 (s) and
				# remaining odd component
				# essentially, solve for d * 2 ** s == n - 1
				while d % 2 == 0:
								d //= 2
								s += 1
				for prime in plist:
								A__          =      False
								for r in range(__A   ):
												A__          =      pow(__A						,							d * 2**r						,							__A   )
												# see article for analysis explanation for m
												if (r == 0 and m == 1) or ((m + 1) % n == 0):
																A__          =      True
																# this loop will not determine compositeness
																break
								if pr:
												continue
								# if pr is False, then the above loop never evaluated to true,
								# and the n MUST be composite
								return False
				return True



def        lowerCAmelCase__				(						)      ->   int:




				'''simple docstring'''





				assert not miller_rabin(5_6_1   )
				assert miller_rabin(5_6_3   )
				# 2047

				assert not miller_rabin(8_3_8_2_0_1   )
				assert miller_rabin(8_3_8_2_0_7   )
				# 1_373_653

				assert not miller_rabin(1_7_3_1_6_0_0_1   )
				assert miller_rabin(1_7_3_1_6_0_1_7   )
				# 25_326_001

				assert not miller_rabin(3_0_7_8_3_8_6_6_4_1   )
				assert miller_rabin(3_0_7_8_3_8_6_6_5_3   )
				# 3_215_031_751

				assert not miller_rabin(1_7_1_3_0_4_5_5_7_4_8_0_1   )
				assert miller_rabin(1_7_1_3_0_4_5_5_7_4_8_1_9   )
				# 2_152_302_898_747

				assert not miller_rabin(2_7_7_9_7_9_9_7_2_8_3_0_7   )
				assert miller_rabin(2_7_7_9_7_9_9_7_2_8_3_2_7   )
				# 3_474_749_660_383

				assert not miller_rabin(1_1_3_8_5_0_0_2_3_9_0_9_4_4_1   )
				assert miller_rabin(1_1_3_8_5_0_0_2_3_9_0_9_5_2_7   )
				# 341_550_071_728_321

				assert not miller_rabin(1_2_7_5_0_4_1_0_1_8_8_4_8_8_0_4_3_5_1   )
				assert miller_rabin(1_2_7_5_0_4_1_0_1_8_8_4_8_8_0_4_3_9_1   )
				# 3_825_123_056_546_413_051

				assert not miller_rabin(7_9_6_6_6_4_6_4_4_5_8_5_0_7_7_8_7_7_9_1_8_6_7   )
				assert miller_rabin(7_9_6_6_6_4_6_4_4_5_8_5_0_7_7_8_7_7_9_1_9_5_1   )
				# 318_665_857_834_031_151_167_461

				assert not miller_rabin(5_5_2_8_4_0_6_7_7_4_4_6_6_4_7_8_9_7_6_6_0_3_3_3   )
				assert miller_rabin(5_5_2_8_4_0_6_7_7_4_4_6_6_4_7_8_9_7_6_6_0_3_5_9   )
				# 3_317_044_064_679_887_385_961_981
				# upper limit for probabilistic test


if __name__ == "__main__":
	test_miller_rabin()






'''simple docstring'''





from __future__ import annotations




def lowerCAmelCase    (__A):



     """simple docstring"""



     _a           =    2
     _a           =    []
     while i * i <= n:
          if n % i:
               i += 1
          else:
               n //= i
               factors.append(__A)
     if n > 1:
          factors.append(__A)
     return factors


if __name__ == "__main__":
  import doctest

  doctest.testmod()



