






from __future__ import annotations

from math import gcd






def 				lowerCamelCase__      (    __lowerCamelCase			:     int  ,		__lowerCamelCase			:     int = 2  ,		__lowerCamelCase			:     int = 1  ,		__lowerCamelCase			:     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(__lowerCamelCase			:     int  ,		__lowerCamelCase			:     int  ,		__lowerCamelCase			:     int		) -> int:
    return (pow(__lowerCamelCase  ,		2		) + step) % modulus

  for _ in range(__lowerCamelCase		):
    # These track the position within the cycle detection logic.
    __UpperCAmelCase							:	str           =  seed
    __UpperCAmelCase							:	List[Any]           =  seed

    while True:
      # At each iteration, the tortoise moves one step and the hare moves two.
      __UpperCAmelCase							:	str           =  rand_fn(__lowerCamelCase  ,		__lowerCamelCase  ,		__lowerCamelCase		)
      __UpperCAmelCase							:	Union[str, Any]           =  rand_fn(__lowerCamelCase  ,		__lowerCamelCase  ,		__lowerCamelCase		)
      __UpperCAmelCase							:	str           =  rand_fn(__lowerCamelCase  ,		__lowerCamelCase  ,		__lowerCamelCase		)

      # 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``.
      __UpperCAmelCase							:	Any           =  gcd(hare - tortoise  ,		__lowerCamelCase		)

      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.
    __UpperCAmelCase							:	List[str]           =  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		:       Dict			     = 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		:       Dict			     = parser.parse_args()

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










from __future__ import annotations

from math import pi






def 				lowerCamelCase__      (    __lowerCamelCase			:     float  ,		__lowerCamelCase			:     float  ,		__lowerCamelCase			:     float		):

  if (inductance, frequency, reactance).count(0		) != 1:
    raise ValueError("""One and only one argument must be 0"""		)
  if inductance < 0:
    raise ValueError("""Inductance cannot be negative"""		)
  if frequency < 0:
    raise ValueError("""Frequency cannot be negative"""		)
  if reactance < 0:
    raise ValueError("""Inductive reactance cannot be negative"""		)
  if inductance == 0:
    return {"inductance": reactance / (2 * pi * frequency)}
  elif frequency == 0:
    return {"frequency": reactance / (2 * pi * inductance)}
  elif reactance == 0:
    return {"reactance": 2 * pi * frequency * inductance}
  else:
    raise ValueError("""Exactly one argument must be 0"""		)


if __name__ == "__main__":
       import doctest

       doctest.testmod()


