





from __future__ import annotations

from math import gcd



def A_     (		A__		,				A__ = 2		,				A__ = 1		,				A__ = 3		,				)  ->		int | None:
       # 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(A__		,				A__		,				A__    ) -> int:
              return (pow(A__		,				2    ) + step) % modulus

       for _ in range(A__    ):
              # These track the position within the cycle detection logic.
              a__						:     Dict			=    seed
              a__						:     Optional[int]			=    seed

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

                     # 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__						:     str			=    gcd(hare - tortoise		,				A__    )

                     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__						:     Union[str, 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

    lowercase				:	Union[str, Any]      						=  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""",
    )
    lowercase				:	Optional[Any]      						=  parser.parse_args()

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









from __future__ import annotations

from collections.abc import Callable



def A_     (		A__		,				A__		,				A__		,				A__ = 100		,				)  ->		float:
       a__						:     Dict			=    x_start
       a__						:     Any			=    fnc(A__    )
       a__						:     Optional[int]			=    0.0
       for _ in range(A__    ):
              # Approximates small segments of curve as linear and solve
              # for trapezoidal area
              a__						:     Union[str, Any]			=    (x_end - x_start) / steps + xa
              a__						:     str			=    fnc(A__    )
              area += abs(fxa + fxa    ) * (xa - xa) / 2
              # Increment step
              a__						:     Optional[Any]			=    xa
              a__						:     Optional[int]			=    fxa
       return area


if __name__ == "__main__":



    def A_     (		A__    )  ->		List[str]:
           return x**3 + x**2

    print("""f(x) = x^3 + x^2""")
    print("""The area between the curve, x = -5, x = 5 and the x axis is:""")
    lowercase				:	Union[str, Any]      						=  1_0
    while i <= 1_0_0_0_0_0:
        print(F"""with {i} steps: {trapezoidal_area(f, -5, 5, i)}""")
        i *= 1_0


