




def      lowerCamelCase_						(			UpperCamelCase__							: int					,	UpperCamelCase__							: bool = 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 > 3_3170_4406_4679_8873_8596_1981 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									=       [
				    2047,
				    137_3653,
				    2532_6001,
				    32_1503_1751,
				    2_1523_0289_8747,
				    3_4747_4966_0383,
				    341_5500_7172_8321,
				    1,
				    382_5123_0565_4641_3051,
				    1,
				    1,
				    3186_6585_7834_0311_5116_7461,
				    3_3170_4406_4679_8873_8596_1981,
				]

				__lowerCamelCase									=       [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
												__lowerCamelCase									=       primes[:idx]
												break
				__lowerCamelCase			,		__lowerCamelCase									=       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									=       False
								for r in range(UpperCamelCase__ ):
												__lowerCamelCase									=       pow(UpperCamelCase__					,	d * 2**r					,	UpperCamelCase__ )
												# see article for analysis explanation for m
												if (r == 0 and m == 1) or ((m + 1) % n == 0):
																__lowerCamelCase									=       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_						(			)	->			None:





				"""simple docstring"""


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

				assert not miller_rabin(83_8201 )
				assert miller_rabin(83_8207 )
				# 1_373_653

				assert not miller_rabin(1731_6001 )
				assert miller_rabin(1731_6017 )
				# 25_326_001

				assert not miller_rabin(30_7838_6641 )
				assert miller_rabin(30_7838_6653 )
				# 3_215_031_751

				assert not miller_rabin(1_7130_4557_4801 )
				assert miller_rabin(1_7130_4557_4819 )
				# 2_152_302_898_747

				assert not miller_rabin(2_7797_9972_8307 )
				assert miller_rabin(2_7797_9972_8327 )
				# 3_474_749_660_383

				assert not miller_rabin(113_8500_2390_9441 )
				assert miller_rabin(113_8500_2390_9527 )
				# 341_550_071_728_321

				assert not miller_rabin(127_5041_0188_4880_4351 )
				assert miller_rabin(127_5041_0188_4880_4391 )
				# 3_825_123_056_546_413_051

				assert not miller_rabin(796_6646_4458_5077_8779_1867 )
				assert miller_rabin(796_6646_4458_5077_8779_1951 )
				# 318_665_857_834_031_151_167_461

				assert not miller_rabin(5528_4067_7446_6478_9766_0333 )
				assert miller_rabin(5528_4067_7446_6478_9766_0359 )
				# 3_317_044_064_679_887_385_961_981
				# upper limit for probabilistic test


if __name__ == "__main__":
					test_miller_rabin()








def      lowerCamelCase_						(			)	->			List[str]:





				"""simple docstring"""


				__lowerCamelCase									=       0
				for i in range(1					,	1001 ):
								total += i**i
				return str(UpperCamelCase__ )[-10:]


if __name__ == "__main__":
					print(solution())


