






"""simple docstring"""





def      lowerCamelCase__       (							_lowerCamelCase       :		Any )      ->				int:
  lowerCamelCase_         =						len(__lowerCAmelCase )
  lowerCamelCase_         =						len(matrix[0] )
  lowerCamelCase_         =						min(__lowerCAmelCase		,				__lowerCAmelCase )

  for row in range(__lowerCAmelCase ):
    # Check if diagonal element is not zero
    if matrix[row][row] != 0:
      # Eliminate all the elements below the diagonal
      for col in range(row + 1		,				__lowerCAmelCase ):
        lowerCamelCase_         =						matrix[col][row] / matrix[row][row]
        for i in range(__lowerCAmelCase		,				__lowerCAmelCase ):
          matrix[col][i] -= multiplier * matrix[row][i]
    else:
      # Find a non-zero diagonal element to swap rows
      lowerCamelCase_         =						True
      for i in range(row + 1		,				__lowerCAmelCase ):
        if matrix[i][row] != 0:
          lowerCamelCase_         =						matrix[i], matrix[row]
          lowerCamelCase_         =						False
          break
      if reduce:
        rank -= 1
        for i in range(__lowerCAmelCase ):
          lowerCamelCase_         =						matrix[i][rank]

            # Reduce the row pointer by one to stay on the same row
      row -= 1

  return rank


if __name__ == "__main__":
  import doctest

  doctest.testmod()









import random

from .binary_exp_mod import bin_exp_mod



def 			_lowerCAmelCase       (					__lowerCAmelCase  ,  __lowerCAmelCase=1000      )   ->					str:


				"""simple docstring"""

				if n < 2:
								return False

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

				# this means n is odd
				snake_case__	:      Tuple     		=	n - 1
				snake_case__	:      Tuple     		=	0
				while d % 2 == 0:
								d /= 2
								exp += 1

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


if __name__ == "__main__":
				A__								=						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)))

