def        __UpperCAmelCase    (       UpperCAmelCase,		UpperCAmelCase = False       )->       bool:



			"""simple docstring"""







			if n == 2:
						return True
			if not n % 2 or n < 2:
						return False
			if n > 5 and n % 10 not in (1, 3, 7, 9):  # can quickly check last digit
						return False
			if n > 3317044064679887385961981 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
			lowercase							=		[
			    2047,
			    1373653,
			    25326001,
			    3215031751,
			    2152302898747,
			    3474749660383,
			    341550071728321,
			    1,
			    3825123056546413051,
			    1,
			    1,
			    318665857834031151167461,
			    3317044064679887385961981,
			]

			lowercase							=		[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
			for idx, _p in enumerate(UpperCAmelCase,		1       ):
						if n < _p:
									# then we have our last prime to check
									lowercase							=		primes[:idx]
									break
			lowercase		,lowercase							=		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:
						lowercase							=		False
						for r in range(UpperCAmelCase       ):
									lowercase							=		pow(UpperCAmelCase,		d * 2**r,		UpperCAmelCase       )
									# see article for analysis explanation for m
									if (r == 0 and m == 1) or ((m + 1) % n == 0):
												lowercase							=		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        __UpperCAmelCase    (       )->       None:



			"""simple docstring"""







			assert not miller_rabin(561       )
			assert miller_rabin(563       )
			# 2047

			assert not miller_rabin(838201       )
			assert miller_rabin(838207       )
			# 1_373_653

			assert not miller_rabin(17316001       )
			assert miller_rabin(17316017       )
			# 25_326_001

			assert not miller_rabin(3078386641       )
			assert miller_rabin(3078386653       )
			# 3_215_031_751

			assert not miller_rabin(1713045574801       )
			assert miller_rabin(1713045574819       )
			# 2_152_302_898_747

			assert not miller_rabin(2779799728307       )
			assert miller_rabin(2779799728327       )
			# 3_474_749_660_383

			assert not miller_rabin(113850023909441       )
			assert miller_rabin(113850023909527       )
			# 341_550_071_728_321

			assert not miller_rabin(1275041018848804351       )
			assert miller_rabin(1275041018848804391       )
			# 3_825_123_056_546_413_051

			assert not miller_rabin(79666464458507787791867       )
			assert miller_rabin(79666464458507787791951       )
			# 318_665_857_834_031_151_167_461

			assert not miller_rabin(552840677446647897660333       )
			assert miller_rabin(552840677446647897660359       )
			# 3_317_044_064_679_887_385_961_981
			# upper limit for probabilistic test


if __name__ == "__main__":
						test_miller_rabin()






from __future__ import annotations

from math import pi






def        __UpperCAmelCase    (       UpperCAmelCase,		UpperCAmelCase,		UpperCAmelCase       )->       dict[str, float]:



			"""simple docstring"""







			if (inductance, frequency, reactance).count(0       ) != 1:
						raise ValueError('''One and only one argument must be 0'''       )
			if inductance < 0:
						raise ValueError('''Inductance cannot be negative'''       )
			if frequency < 0:
						raise ValueError('''Frequency cannot be negative'''       )
			if reactance < 0:
						raise ValueError('''Inductive reactance cannot be negative'''       )
			if inductance == 0:
						return {"inductance": reactance / (2 * pi * frequency)}
			elif frequency == 0:
						return {"frequency": reactance / (2 * pi * inductance)}
			elif reactance == 0:
						return {"reactance": 2 * pi * frequency * inductance}
			else:
						raise ValueError('''Exactly one argument must be 0'''       )


if __name__ == "__main__":
						import doctest

						doctest.testmod()





