


'''simple docstring'''


from __future__ import annotations

from math import gcd






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

		for _ in range(lowercase__      ):
				# These track the position within the cycle detection logic.
				_UpperCAmelCase						:				Optional[int]	    =					seed
				_UpperCAmelCase						:				Optional[Any]	    =					seed

				while True:
						# At each iteration, the tortoise moves one step and the hare moves two.
						_UpperCAmelCase						:				Any	    =					rand_fn(lowercase__					,       lowercase__					,       lowercase__      )
						_UpperCAmelCase						:				Optional[int]	    =					rand_fn(lowercase__					,       lowercase__					,       lowercase__      )
						_UpperCAmelCase						:				Union[str, Any]	    =					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``.
						_UpperCAmelCase						:				List[Any]	    =					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.
				_UpperCAmelCase						:				Optional[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

						lowerCamelCase__					=   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',
						)
						lowerCamelCase__					=   parser.parse_args()

						lowerCamelCase__					=   pollard_rho(args.num, attempts=args.attempts)
						if divisor is None:
												print(F'''{args.num} is probably prime''')
						else:
												lowerCamelCase__					=   args.num // divisor
												print(F'''{args.num} = {divisor} * {quotient}''')






import unittest

from .lib import (
    Matrix,
    Vector,
    axpy,
    square_zero_matrix,
    unit_basis_vector,
    zero_vector,
)







class   __lowercase      (unittest.TestCase ):
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						List[str]					=					Vector([1, 2, 3]					)
       self.assertEqual(x.component(0					)      ,							1					)
       self.assertEqual(x.component(2					)      ,							3					)
       __lowerCAmelCase					:						Dict					=					Vector()
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Any					=					Vector([0, 0, 0, 0, 0, 1]					)
       self.assertEqual(str(A_					)      ,							'''(0,0,0,0,0,1)'''					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Union[str, Any]					=					Vector([1, 2, 3, 4]					)
       self.assertEqual(len(A_					)      ,							4					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Dict					=					Vector([1, 2]					)
       __lowerCAmelCase					:						Optional[int]					=					Vector([1, 2, 3, 4, 5]					)
       __lowerCAmelCase					:						Optional[Any]					=					Vector([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]					)
       __lowerCAmelCase					:						str					=					Vector([1, -1, 1, -1, 2, -3, 4, -5]					)
       self.assertAlmostEqual(x.euclidean_length()      ,							2.236      ,							3					)
       self.assertAlmostEqual(y.euclidean_length()      ,							7.416      ,							3					)
       self.assertEqual(z.euclidean_length()      ,							0					)
       self.assertAlmostEqual(w.euclidean_length()      ,							7.616      ,							3					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Tuple					=					Vector([1, 2, 3]					)
       __lowerCAmelCase					:						List[str]					=					Vector([1, 1, 1]					)
       self.assertEqual((x + y).component(0					)      ,							2					)
       self.assertEqual((x + y).component(1					)      ,							3					)
       self.assertEqual((x + y).component(2					)      ,							4					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Any					=					Vector([1, 2, 3]					)
       __lowerCAmelCase					:						List[str]					=					Vector([1, 1, 1]					)
       self.assertEqual((x - y).component(0					)      ,							0					)
       self.assertEqual((x - y).component(1					)      ,							1					)
       self.assertEqual((x - y).component(2					)      ,							2					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						str					=					Vector([1, 2, 3]					)
       __lowerCAmelCase					:						List[Any]					=					Vector([2, -1, 4]					)  # for test of dot product
       __lowerCAmelCase					:						Optional[int]					=					Vector([1, -2, -1]					)
       self.assertEqual(str(x * 3.0					)      ,							'''(3.0,6.0,9.0)'''					)
       self.assertEqual((a * b)      ,							0					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       self.assertEqual(str(zero_vector(10					)					).count('''0'''					)      ,							10					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       self.assertEqual(str(unit_basis_vector(3      ,							1					)					)      ,							'''(0,1,0)'''					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						str					=					Vector([1, 2, 3]					)
       __lowerCAmelCase					:						Any					=					Vector([1, 0, 1]					)
       self.assertEqual(str(axpy(2      ,							A_      ,							A_					)					)      ,							'''(3,4,7)'''					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Tuple					=					Vector([1, 0, 0, 0, 0, 0]					)
       __lowerCAmelCase					:						Optional[Any]					=					x.copy()
       self.assertEqual(str(A_					)      ,							str(A_					)					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						List[str]					=					Vector([1, 0, 0]					)
       x.change_component(0      ,							0					)
       x.change_component(1      ,							1					)
       self.assertEqual(str(A_					)      ,							'''(0,1,0)'''					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Union[str, Any]					=					Matrix([[1, 2, 3], [2, 4, 5], [6, 7, 8]]      ,							3      ,							3					)
       self.assertEqual('''|1,2,3|\n|2,4,5|\n|6,7,8|\n'''      ,							str(A_					)					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						List[Any]					=					Matrix([[1, 2, 3], [2, 4, 5], [6, 7, 8]]      ,							3      ,							3					)
       __lowerCAmelCase					:						str					=					[[-3, -14, -10], [-5, -10, -5], [-2, -1, 0]]
       for x in range(a.height()					):
             for y in range(a.width()					):
                   self.assertEqual(minors[x][y]      ,							a.minor(A_      ,							A_					)					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Optional[int]					=					Matrix([[1, 2, 3], [2, 4, 5], [6, 7, 8]]      ,							3      ,							3					)
       __lowerCAmelCase					:						Tuple					=					[[-3, 14, -10], [5, -10, 5], [-2, 1, 0]]
       for x in range(a.height()					):
             for y in range(a.width()					):
                   self.assertEqual(cofactors[x][y]      ,							a.cofactor(A_      ,							A_					)					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Dict					=					Matrix([[1, 2, 3], [2, 4, 5], [6, 7, 8]]      ,							3      ,							3					)
       self.assertEqual(-5      ,							a.determinant()					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Tuple					=					Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]      ,							3      ,							3					)
       __lowerCAmelCase					:						Union[str, Any]					=					Vector([1, 2, 3]					)
       self.assertEqual('''(14,32,50)'''      ,							str(a * x					)					)
       self.assertEqual('''|2,4,6|\n|8,10,12|\n|14,16,18|\n'''      ,							str(a * 2					)					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Any					=					Matrix([[1, 2, 3], [2, 4, 5], [6, 7, 8]]      ,							3      ,							3					)
       a.change_component(0      ,							2      ,							5					)
       self.assertEqual('''|1,2,5|\n|2,4,5|\n|6,7,8|\n'''      ,							str(A_					)					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Tuple					=					Matrix([[1, 2, 3], [2, 4, 5], [6, 7, 8]]      ,							3      ,							3					)
       self.assertEqual(7      ,							a.component(2      ,							1					)      ,							0.01					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Any					=					Matrix([[1, 2, 3], [2, 4, 5], [6, 7, 8]]      ,							3      ,							3					)
       __lowerCAmelCase					:						Dict					=					Matrix([[1, 2, 7], [2, 4, 5], [6, 7, 10]]      ,							3      ,							3					)
       self.assertEqual('''|2,4,10|\n|4,8,10|\n|12,14,18|\n'''      ,							str(a + b					)					)
 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       __lowerCAmelCase					:						Dict					=					Matrix([[1, 2, 3], [2, 4, 5], [6, 7, 8]]      ,							3      ,							3					)
       __lowerCAmelCase					:						str					=					Matrix([[1, 2, 7], [2, 4, 5], [6, 7, 10]]      ,							3      ,							3					)
       self.assertEqual('''|0,0,-4|\n|0,0,0|\n|0,0,-2|\n'''      ,							str(a - b					)					)





 def 	UpperCamelCase__  (							self					)					->None:



       '''simple docstring'''





       self.assertEqual(
           '''|0,0,0,0,0|\n|0,0,0,0,0|\n|0,0,0,0,0|\n|0,0,0,0,0|\n|0,0,0,0,0|\n'''      ,							str(square_zero_matrix(5					)					)      ,							)


if __name__ == "__main__":
 unittest.main()


