

"""simple docstring"""




import numpy as np
from numpy import floataa
from numpy.typing import NDArray



def 						UpperCAmelCase__ (					A__						,     A__ )   ->   NDArray[floataa]:




   """simple docstring"""
   lowerCamelCase__       ,							lowerCamelCase__						       =      np.shape(A__ )

   lowerCamelCase__						       =      np.zeros((rows, 1)						,     dtype=A__ )
   for row in reversed(range(A__ ) ):
      lowerCamelCase__						       =      0
      for col in range(row + 1						,     A__ ):
         total += coefficients[row, col] * x[col]

      lowerCamelCase__						       =      (vector[row] - total) / coefficients[row, row]

   return x



def 						UpperCAmelCase__ (					A__						,     A__ )   ->   NDArray[floataa]:




   """simple docstring"""
   lowerCamelCase__       ,							lowerCamelCase__						       =      np.shape(A__ )
   if rows != columns:
      return np.array(()						,     dtype=A__ )

   # augmented matrix
   lowerCamelCase__						       =      np.concatenate((coefficients, vector)						,     axis=1 )
   lowerCamelCase__						       =      augmented_mat.astype("float64" )

   # scale the matrix leaving it triangular
   for row in range(rows - 1 ):
      lowerCamelCase__						       =      augmented_mat[row, row]
      for col in range(row + 1						,     A__ ):
         lowerCamelCase__						       =      augmented_mat[col, row] / pivot
         augmented_mat[col, :] -= factor * augmented_mat[row, :]

   lowerCamelCase__						       =      retroactive_resolution(
       augmented_mat[:, 0:columns]						,     augmented_mat[:, columns : columns + 1] )

   return x


if __name__ == "__main__":
     import doctest

     doctest.testmod()







"""simple docstring"""




from __future__ import annotations

import math
from itertools import permutations



def 						UpperCAmelCase__ (					A__ )   ->   bool:




   """simple docstring"""

   if 1 < number < 4:
      # 2 and 3 are primes
      return True
   elif number < 2 or number % 2 == 0 or number % 3 == 0:
      # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
      return False

   # All primes number are in format of 6k +/- 1
   for i in range(5						,     int(math.sqrt(A__ ) + 1 )						,     6 ):
      if number % i == 0 or number % (i + 2) == 0:
         return False
   return True



def 						UpperCAmelCase__ (					A__ = 7 )   ->   int:




   """simple docstring"""
   lowerCamelCase__						       =      "".join(str(A__ ) for i in range(1						,     n + 1 ) )
   lowerCamelCase__						       =      [int("".join(A__ ) ) for i in permutations(A__						,     A__ )]
   lowerCamelCase__						       =      [num for num in perm_list if is_prime(A__ )]
   return max(A__ ) if pandigitals else 0


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




