
'''simple docstring'''

import random

from .binary_exp_mod import bin_exp_mod







def     snake_case__						(   UpperCamelCase      ,UpperCamelCase=10_00						)			->   Optional[int]:
		if n < 2:
				return False

		if n % 2 == 0:
				return n == 2

		# this means n is odd
		_UpperCamelCase							:			Optional[int]     =   n - 1
		_UpperCamelCase							:			Tuple     =   0
		while d % 2 == 0:
				d /= 2
				exp += 1

		# n - 1=d*(2**exp)
		_UpperCamelCase							:			int     =   0
		while count < prec:
				_UpperCamelCase							:			Tuple     =   random.randint(2      ,n - 1						)
				_UpperCamelCase							:			Tuple     =   bin_exp_mod(a_      ,a_      ,a_						)
				if b != 1:
						_UpperCamelCase							:			List[str]     =   True
						for _ in range(a_						):
								if b == n - 1:
										_UpperCamelCase							:			int     =   False
										break
								_UpperCamelCase							:			Optional[Any]     =   b * b
								b %= n
						if flag:
								return False
						count += 1
		return True


if __name__ == "__main__":
	_UpperCAmelCase :   Any			    =	abs(int(input("""Enter bound : """).strip()))
	print("""Here's the list of primes:""")
	print(""", """.join(str(i) for i in range(n + 1) if is_prime_big(i)))













import unittest

import numpy as np





def 		__UpperCAmelCase     (	a_				,	a_				,	a_				,	a_ = None				,	):
							snake_case_           =				np.shape(a_)
							snake_case_           =				np.shape(a_)
							snake_case_           =				np.shape(a_)

							if shape_a[0] != shape_b[0]:
														snake_case_           =				(
														    'Expected the same number of rows for A and B. '
														    f'''Instead found A of size {shape_a} and B of size {shape_b}'''
														)
														raise ValueError(a_)

							if shape_b[1] != shape_c[1]:
														snake_case_           =				(
														    'Expected the same number of columns for B and C. '
														    f'''Instead found B of size {shape_b} and C of size {shape_c}'''
														)
														raise ValueError(a_)

							snake_case_           =				pseudo_inv
							if a_inv is None:
														try:
																					snake_case_           =				np.linalg.inv(a_)
														except np.linalg.LinAlgError:
																					raise ValueError(
																					    'Input matrix A is not invertible. Cannot compute Schur complement.')

							return mat_c - mat_b.T @ a_inv @ mat_b




class 					UpperCamelCase_ (     unittest.TestCase							):



	'''simple docstring'''




	def 				_UpperCamelCase     (			self   )			->       None:
								snake_case_           =				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]   )
								snake_case_           =				np.array([[0, 3], [3, 0], [2, 3]]   )
								snake_case_           =				np.array([[2, 1], [6, 3]]   )

								snake_case_           =				schur_complement(a			,     a			,     a   )

								snake_case_           =				np.block([[a, b], [b.T, c]]   )

								snake_case_           =				np.linalg.det(a   )
								snake_case_           =				np.linalg.det(a   )
								snake_case_           =				np.linalg.det(a   )

								self.assertAlmostEqual(a			,     det_a * det_s   )




	def 				_UpperCamelCase     (			self   )			->       None:
								snake_case_           =				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]   )
								snake_case_           =				np.array([[0, 3], [3, 0], [2, 3]]   )
								snake_case_           =				np.array([[2, 1], [6, 3]]   )

								with self.assertRaises(a   ):
															schur_complement(a			,     a			,     a   )






	def 				_UpperCamelCase     (			self   )			->       None:
								snake_case_           =				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]   )
								snake_case_           =				np.array([[0, 3], [3, 0], [2, 3]]   )
								snake_case_           =				np.array([[2, 1, 3], [6, 3, 5]]   )

								with self.assertRaises(a   ):
															schur_complement(a			,     a			,     a   )


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	unittest.main()





