





from __future__ import annotations

from math import pi, sqrt


def 					A__	(       SCREAMING_SNAKE_CASE_							,       SCREAMING_SNAKE_CASE_		)   ->     tuple:

		if inductance <= 0:
				raise ValueError('''Inductance cannot be 0 or negative'''		)

		elif capacitance <= 0:
				raise ValueError('''Capacitance cannot be 0 or negative'''		)

		else:
				return (
				    "Resonant frequency",
				    float(1 / (2 * pi * (sqrt(inductance * capacitance		)))		),
				)


if __name__ == "__main__":
				import doctest

				doctest.testmod()








def 					A__	(       SCREAMING_SNAKE_CASE_							,       SCREAMING_SNAKE_CASE_ = False		)   ->     bool:
		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
		lowerCamelCase   : Optional[Any]								=[
		    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,
		]

		lowerCamelCase   : Dict								=[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(SCREAMING_SNAKE_CASE_							,       1		):
				if n < _p:
						# then we have our last prime to check
						lowerCamelCase   : Any								=primes[:idx]
						break
		lowerCamelCase		,   lowerCamelCase   : Any								=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:
				lowerCamelCase   : Union[str, Any]								=False
				for r in range(SCREAMING_SNAKE_CASE_		):
						lowerCamelCase   : List[str]								=pow(SCREAMING_SNAKE_CASE_							,       d * 2**r							,       SCREAMING_SNAKE_CASE_		)
						# see article for analysis explanation for m
						if (r == 0 and m == 1) or ((m + 1) % n == 0):
								lowerCamelCase   : List[str]								=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 					A__	(       )   ->     None:
		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()

