






from functools import lru_cache

@lru_cache
def     lowerCAmelCase__  (  UpperCamelCase__       ):



  '''simple docstring'''
  if not isinstance(UpperCamelCase__  ,     UpperCamelCase__       ):
    raise TypeError("""k must be an integer."""       )
  if k < 0:
    raise ValueError("""k integer must be greater or equal to zero."""       )
  if k == 0:
    return 0
  elif k == 1:
    return 1
  else:
    return fibonacci(k - 1       ) + fibonacci(k - 2       )

def     lowerCAmelCase__  (  UpperCamelCase__  ,     UpperCamelCase__       ):



  '''simple docstring'''
  _a						: int    =			len(UpperCamelCase__       )
  # Find m such that F_m >= n where F_i is the i_th fibonacci number.
  _a						: Union[str, Any]    =			0
  while True:
    if fibonacci(UpperCamelCase__       ) >= len_list:
      _a						: Union[str, Any]    =			i
      break
    i += 1
  _a						: List[Any]    =			0
  while fibb_k > 0:
    _a						: List[Any]    =			min(
        offset + fibonacci(fibb_k - 1       )  ,     len_list - 1       )  # Prevent out of range
    _a						: Tuple    =			arr[index_k]
    if item_k_1 == val:
      return index_k
    elif val < item_k_1:
      fibb_k -= 1
    elif val > item_k_1:
      offset += fibonacci(fibb_k - 1       )
      fibb_k -= 2
  else:
    return -1


if __name__ == "__main__":
 import doctest

 doctest.testmod()









"""simple docstring"""




from __future__ import annotations

import math
from itertools import permutations

def     lowerCAmelCase__  (  UpperCamelCase__       ):



  '''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(UpperCamelCase__       ) + 1       )  ,     6       ):
    if number % i == 0 or number % (i + 2) == 0:
      return False
  return True

def     lowerCAmelCase__  (  UpperCamelCase__ = 7       ):



  '''simple docstring'''
  _a						: Tuple    =			"""""".join(str(UpperCamelCase__       ) for i in range(1  ,     n + 1       )       )
  _a						: str    =			[int("""""".join(UpperCamelCase__       )       ) for i in permutations(UpperCamelCase__  ,     UpperCamelCase__       )]
  _a						: Union[str, Any]    =			[num for num in perm_list if is_prime(UpperCamelCase__       )]
  return max(UpperCamelCase__       ) if pandigitals else 0


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

