






from __future__ import annotations

lowerCamelCase      :       str    =						[True] * 1000001
lowerCamelCase      :       int    =						2
while i * i <= 1000000:
 if seive[i]:
  for j in range(i * i, 1000001, i):
   lowerCamelCase      :       List[str]    =						False
 i += 1





def    __lowerCAmelCase   (    __snake_case							):
     return seive[n]





def    __lowerCAmelCase   (    __snake_case							):
     return any(digit in "02468" for digit in str(__snake_case							)							)





def    __lowerCAmelCase   (    __snake_case = 100_0000							):
     __lowerCAmelCase             =       [2]  # result already includes the number 2.
     for num in range(3       ,			limit + 1       ,			2							):
          if is_prime(__snake_case							) and not contains_an_even_digit(__snake_case							):
               __lowerCAmelCase             =       str(__snake_case							)
               __lowerCAmelCase             =       [int(str_num[j:] + str_num[:j]							) for j in range(len(__snake_case							)							)]
               if all(is_prime(__snake_case							) for i in list_nums							):
                    result.append(__snake_case							)
     return result





def    __lowerCAmelCase   (    ):
     return len(find_circular_primes()							)


if __name__ == "__main__":
 print(F'''{len(find_circular_primes()) = }''')














import numpy as np





def    __lowerCAmelCase   (    __snake_case       ,			__snake_case       ,			__snake_case = 1E-12       ,			__snake_case = 100       ,			):
     assert np.shape(__snake_case							)[0] == np.shape(__snake_case							)[1]
     # Ensure proper dimensionality.
     assert np.shape(__snake_case							)[0] == np.shape(__snake_case							)[0]
     # Ensure inputs are either both complex or both real
     assert np.iscomplexobj(__snake_case							) == np.iscomplexobj(__snake_case							)
     __lowerCAmelCase             =       np.iscomplexobj(__snake_case							)
     if is_complex:
          # Ensure complex input_matrix is Hermitian
          assert np.array_equal(__snake_case       ,			input_matrix.conj().T							)

     # Set convergence to False. Will define convergence when we exceed max_iterations
     # or when we have small changes from one iteration to next.

     __lowerCAmelCase             =       False
     __lowerCAmelCase             =       0
     __lowerCAmelCase             =       0
     __lowerCAmelCase             =       1E12

     while not convergence:
          # Multiple matrix by the vector.
          __lowerCAmelCase             =       np.dot(__snake_case       ,			__snake_case							)
          # Normalize the resulting output vector.
          __lowerCAmelCase             =       w / np.linalg.norm(__snake_case							)
          # Find rayleigh quotient
          # (faster than usual b/c we know vector is normalized already)
          __lowerCAmelCase             =       vector.conj().T if is_complex else vector.T
          __lowerCAmelCase             =       np.dot(__snake_case       ,			np.dot(__snake_case       ,			__snake_case							)							)

          # Check convergence.
          __lowerCAmelCase             =       np.abs(lambda_ - lambda_previous							) / lambda_
          iterations += 1

          if error <= error_tol or iterations >= max_iterations:
               __lowerCAmelCase             =       True

          __lowerCAmelCase             =       lambda_

     if is_complex:
          __lowerCAmelCase             =       np.real(lambda_							)

     return lambda_, vector





def    __lowerCAmelCase   (    ):
     __lowerCAmelCase             =       np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]							)
     __lowerCAmelCase             =       np.array([41, 4, 20]							)
     __lowerCAmelCase             =       real_input_matrix.astype(np.complexaaa							)
     __lowerCAmelCase             =       np.triu(1J * complex_input_matrix       ,			1							)
     complex_input_matrix += imag_matrix
     complex_input_matrix += -1 * imag_matrix.T
     __lowerCAmelCase             =       np.array([41, 4, 20]							).astype(np.complexaaa							)

     for problem_type in ["real", "complex"]:
          if problem_type == "real":
               __lowerCAmelCase             =       real_input_matrix
               __lowerCAmelCase             =       real_vector
          elif problem_type == "complex":
               __lowerCAmelCase             =       complex_input_matrix
               __lowerCAmelCase             =       complex_vector

          # Our implementation.
          __lowerCAmelCase      ,			__lowerCAmelCase             =       power_iteration(__snake_case       ,			__snake_case							)

          # Numpy implementation.

          # Get eigenvalues and eigenvectors using built-in numpy
          # eigh (eigh used for symmetric or hermetian matrices).
          __lowerCAmelCase      ,			__lowerCAmelCase             =       np.linalg.eigh(__snake_case							)
          # Last eigenvalue is the maximum one.
          __lowerCAmelCase             =       eigen_values[-1]
          # Last column in this matrix is eigenvector corresponding to largest eigenvalue.
          __lowerCAmelCase             =       eigen_vectors[:, -1]

          # Check our implementation and numpy gives close answers.
          assert np.abs(eigen_value - eigen_value_max							) <= 1E-6
          # Take absolute values element wise of each eigenvector.
          # as they are only unique to a minus sign.
          assert np.linalg.norm(np.abs(__snake_case							) - np.abs(__snake_case							)							) <= 1E-6


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 test_power_iteration()






