

"""simple docstring"""



from __future__ import annotations

from math import gcd





def  _snake_case     (    UpperCAmelCase_  :					Optional[Any]					,   UpperCAmelCase_  :					Dict = 2					,   UpperCAmelCase_  :					str = 1					,   UpperCAmelCase_  :					int = 3					,   ):
     # A value less than 2 can cause an infinite loop in the algorithm.
     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(UpperCAmelCase_  :					int					,   UpperCAmelCase_  :					List[Any]					,   UpperCAmelCase_  :					str			) -> int:
          return (pow(lowercase__					,   2			) + step) % modulus

     for _ in range(lowercase__			):
          # These track the position within the cycle detection logic.
          A__										=  seed
          A__										=  seed

          while True:
               # At each iteration, the tortoise moves one step and the hare moves two.
               A__										=  rand_fn(lowercase__					,   lowercase__					,   lowercase__			)
               A__										=  rand_fn(lowercase__					,   lowercase__					,   lowercase__			)
               A__										=  rand_fn(lowercase__					,   lowercase__					,   lowercase__			)

               # 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__										=  gcd(hare - tortoise					,   lowercase__			)

               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__										=  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

 SCREAMING_SNAKE_CASE_					:	int							  =						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',
 )
 SCREAMING_SNAKE_CASE_					:	str							  =						parser.parse_args()

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








"""simple docstring"""



import math
import sys





def  _snake_case     (    UpperCAmelCase_  :					int			):
     if number != int(UpperCAmelCase_			):
          raise ValueError("""the value of input must be a natural number"""			)
     if number < 0:
          raise ValueError("""the value of input must not be a negative number"""			)
     if number == 0:
          return 1
     A__										=  [-1] * (number + 1)
     A__										=  0
     for i in range(1					,   number + 1			):
          A__										=  sys.maxsize
          A__										=  int(math.sqrt(UpperCAmelCase_			)			)
          for j in range(1					,   root + 1			):
               A__										=  1 + answers[i - (j**2)]
               A__										=  min(UpperCAmelCase_					,   UpperCAmelCase_			)
          A__										=  answer
     return answers[number]


if __name__ == "__main__":
 import doctest

 doctest.testmod()






