
'''simple docstring'''
from __future__ import annotations

from math import gcd







def        A_     (						__SCREAMING_SNAKE_CASE						:  int			,     __SCREAMING_SNAKE_CASE						:  int = 2			,     __SCREAMING_SNAKE_CASE						:  int = 1			,     __SCREAMING_SNAKE_CASE						:  int = 3			,     )				->       int | None:





  """simple docstring"""

  if num < 2:
    raise ValueError("""The input value cannot be less than 2""" )

  # Because of the relationship between ``f(f(x))`` and ``f(x)``, this
  # algorithm struggles to find factors that are divisible by two.
  # As a workaround, we specifically check for two and even inputs.
  #   See: https://math.stackexchange.com/a/2856214/165820
  if num > 2 and num % 2 == 0:
    return 2

  # Pollard's Rho algorithm requires a function that returns pseudorandom
  # values between 0 <= X < ``num``.  It doesn't need to be random in the
  # sense that the output value is cryptographically secure or difficult
  # to calculate, it only needs to be random in the sense that all output
  # values should be equally likely to appear.
  # For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
  # However, the success of Pollard's algorithm isn't guaranteed and is
  # determined in part by the initial seed and the chosen random function.
  # To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
  # where ``C`` is a value that we can modify between each attempt.
  def rand_fn(__SCREAMING_SNAKE_CASE						:  int			,     __SCREAMING_SNAKE_CASE						:  int			,     __SCREAMING_SNAKE_CASE						:  int ) -> int:
    return (pow(__SCREAMING_SNAKE_CASE			,     2 ) + step) % modulus

  for _ in range(__SCREAMING_SNAKE_CASE ):
    # These track the position within the cycle detection logic.
    __A		:     int												=		seed
    __A		:     Tuple												=		seed

    while True:
      # At each iteration, the tortoise moves one step and the hare moves two.
      __A		:     Optional[Any]												=		rand_fn(__SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE )
      __A		:     List[Any]												=		rand_fn(__SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE )
      __A		:     List[Any]												=		rand_fn(__SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE )

      # At some point both the tortoise and the hare will enter a cycle whose
      # length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
      # the tortoise and hare will end up on the same value modulo ``p``.
      # We can detect when this happens because the position difference between
      # the tortoise and the hare will share a common divisor with ``num``.
      __A		:     Dict												=		gcd(hare - tortoise			,     __SCREAMING_SNAKE_CASE )

      if divisor == 1:
        # No common divisor yet, just keep searching.
        continue
      else:
        # We found a common divisor!
        if divisor == num:
          # Unfortunately, the divisor is ``num`` itself and is useless.
          break
        else:
          # The divisor is a nontrivial factor of ``num``!
          return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
    __A		:     List[Any]												=		hare

    # The new step value for the random function can just be incremented.
    # At first the results will be similar to what the old function would
    # have produced, but the value will quickly diverge after a bit.
    step += 1

  # We haven't found a divisor within the requested number of attempts.
  # We were unlucky or ``num`` itself is actually prime.
  return None


if __name__ == "__main__":
     import argparse

     A__   :		str 		=argparse.ArgumentParser()
     parser.add_argument(
         'num',
         type=int,
         help='The value to find a divisor of',
     )
     parser.add_argument(
         '--attempts',
         type=int,
         default=3,
         help='The number of attempts before giving up',
     )
     A__   :		Optional[Any] 		=parser.parse_args()

     A__   :		Optional[Any] 		=pollard_rho(args.num, attempts=args.attempts)
     if divisor is None:
          print(F"{args.num} is probably prime")
     else:
          A__   :		Union[str, Any] 		=args.num // divisor
          print(F"{args.num} = {divisor} * {quotient}")





'''simple docstring'''
def        A_     (						__SCREAMING_SNAKE_CASE						:  int )				->       None:





  """simple docstring"""

  __A		:     Tuple												=		generate_pascal_triangle(__SCREAMING_SNAKE_CASE )
  for row_idx in range(__SCREAMING_SNAKE_CASE ):
    # Print left spaces
    for _ in range(num_rows - row_idx - 1 ):
      print(end=""" """ )
    # Print row values
    for col_idx in range(row_idx + 1 ):
      if col_idx != row_idx:
        print(triangle[row_idx][col_idx]			,     end=""" """ )
      else:
        print(triangle[row_idx][col_idx]			,     end="""""" )
    print()







def        A_     (						__SCREAMING_SNAKE_CASE						:  int )				->       list[list[int]]:





  """simple docstring"""

  if not isinstance(__SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE ):
    raise TypeError("""The input value of 'num_rows' should be 'int'""" )

  if num_rows == 0:
    return []
  elif num_rows < 0:
    raise ValueError(
        """The input value of 'num_rows' should be greater than or equal to 0""" )

  __A		:     list[list[int]]												=		[]
  for current_row_idx in range(__SCREAMING_SNAKE_CASE ):
    __A		:     Tuple												=		populate_current_row(__SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE )
    triangle.append(__SCREAMING_SNAKE_CASE )
  return triangle







def        A_     (						__SCREAMING_SNAKE_CASE						:  list[list[int]]			,     __SCREAMING_SNAKE_CASE						:  int )				->       list[int]:





  """simple docstring"""

  __A		:     Union[str, Any]												=		[-1] * (current_row_idx + 1)
  # first and last elements of current row are equal to 1
  __A							,					__A		:     Tuple												=		1, 1
  for current_col_idx in range(1			,     __SCREAMING_SNAKE_CASE ):
    calculate_current_element(
        __SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE )
  return current_row







def        A_     (						__SCREAMING_SNAKE_CASE						:  list[list[int]]			,     __SCREAMING_SNAKE_CASE						:  list[int]			,     __SCREAMING_SNAKE_CASE						:  int			,     __SCREAMING_SNAKE_CASE						:  int			,     )				->       None:





  """simple docstring"""

  __A		:     str												=		triangle[current_row_idx - 1][current_col_idx - 1]
  __A		:     Union[str, Any]												=		triangle[current_row_idx - 1][current_col_idx]
  __A		:     Any												=		above_to_left_elt + above_to_right_elt







def        A_     (						__SCREAMING_SNAKE_CASE						:  int )				->       list[list[int]]:





  """simple docstring"""

  if not isinstance(__SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE ):
    raise TypeError("""The input value of 'num_rows' should be 'int'""" )

  if num_rows == 0:
    return []
  elif num_rows < 0:
    raise ValueError(
        """The input value of 'num_rows' should be greater than or equal to 0""" )

  __A		:     list[list[int]]												=		[[1]]

  for row_index in range(1			,     __SCREAMING_SNAKE_CASE ):
    __A		:     Optional[Any]												=		[0] + result[-1] + [0]
    __A		:     Optional[Any]												=		row_index + 1
    # Calculate the number of distinct elements in a row
    __A		:     int												=		sum(divmod(__SCREAMING_SNAKE_CASE			,     2 ) )
    __A		:     str												=		[
        temp_row[i - 1] + temp_row[i] for i in range(1			,     distinct_elements + 1 )
    ]
    __A		:     Tuple												=		row_first_half[: (row_index + 1) // 2]
    row_second_half.reverse()
    __A		:     Any												=		row_first_half + row_second_half
    result.append(__SCREAMING_SNAKE_CASE )

  return result







def        A_     (						)				->       None:





  """simple docstring"""

  from collections.abc import Callable
  from timeit import timeit

  def benchmark_a_function(__SCREAMING_SNAKE_CASE						:  Callable			,     __SCREAMING_SNAKE_CASE						:  int ) -> None:
    __A		:     List[Any]												=		F"{func.__name__}({value})"
    __A		:     Dict												=		timeit(F"__main__.{call}"			,     setup="""import __main__""" )
    # print(f"{call:38} = {func(value)} -- {timing:.4f} seconds")
    print(F"{call:38} -- {timing:.4f} seconds" )

  for value in range(15 ):  # (1, 7, 14):
    for func in (generate_pascal_triangle, generate_pascal_triangle_optimized):
      benchmark_a_function(__SCREAMING_SNAKE_CASE			,     __SCREAMING_SNAKE_CASE )
    print()


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     benchmark()



