






'''simple docstring'''




def       __UpperCAmelCase						(  lowerCamelCase_ = 600_851_475_143)							->     int:

		try:
				UpperCamelCase__				:					Dict					=					int(lowerCamelCase_)
		except (TypeError, ValueError):
				raise TypeError('Parameter n must be int or castable to int.')
		if n <= 0:
				raise ValueError('Parameter n must be greater than or equal to one.')
		UpperCamelCase__				:					Optional[int]					=					1
		UpperCamelCase__				:					Union[str, Any]					=					2
		while i * i <= n:
				while n % i == 0:
						UpperCamelCase__				:					List[str]					=					i
						n //= i
				i += 1
		if n > 1:
				UpperCamelCase__				:					str					=					n
		return int(lowerCamelCase_)


if __name__ == "__main__":
			print(f'''{solution() = }''')












'''simple docstring'''




import numpy as np
from numpy import ndarray
from scipy.optimize import Bounds, LinearConstraint, minimize







def       __UpperCAmelCase						(  lowerCamelCase_)							->     float:
		return np.dot(lowerCamelCase_     ,      lowerCamelCase_)







class       __lowercase					:

	def __init__(      self   :						Tuple  ,					*,
	    UpperCAmelCase_   :						float = np.inf  ,					UpperCAmelCase_   :						str = "linear"  ,					UpperCAmelCase_   :						float = 0.0  ,					):
			UpperCamelCase__				:					Union[str, Any]					=					regularization
			UpperCamelCase__				:					Optional[int]					=					gamma
			if kernel == "linear":
					UpperCamelCase__				:					List[str]					=					self.__linear
			elif kernel == "rbf":
					if self.gamma == 0:
							raise ValueError('rbf kernel requires gamma')
					if not isinstance(self.gamma  ,					(float, int)):
							raise ValueError('gamma must be float or int')
					if not self.gamma > 0:
							raise ValueError('gamma must be > 0')
					UpperCamelCase__				:					Union[str, Any]					=					self.__rbf
					# in the future, there could be a default value like in sklearn
					# sklear: def_gamma = 1/(n_features * X.var()) (wiki)
					# previously it was 1/(n_features)
			else:
					UpperCamelCase__				:					Optional[int]					=					F'Unknown kernel: {kernel}'
					raise ValueError(UpperCAmelCase_)

	def      __UpperCamelCase							(      self   :						Any  ,					UpperCAmelCase_   :						ndarray  ,					UpperCAmelCase_   :						ndarray):
			return np.dot(UpperCAmelCase_  ,					UpperCAmelCase_)

	def      __UpperCamelCase							(      self   :						Union[str, Any]  ,					UpperCAmelCase_   :						ndarray  ,					UpperCAmelCase_   :						ndarray):
			return np.exp(-(self.gamma * norm_squared(vectora - vectora)))

	def      __UpperCamelCase							(      self   :						Any  ,					UpperCAmelCase_   :						list[ndarray]  ,					UpperCAmelCase_   :						ndarray):
			UpperCamelCase__				:					Any					=					observations
			UpperCamelCase__				:					Tuple					=					classes

			# using Wolfe's Dual to calculate w.
			# Primal problem: minimize 1/2*norm_squared(w)
			#   constraint: yn(w . xn + b) >= 1
			#
			# With l a vector
			# Dual problem: maximize sum_n(ln) -
			#       1/2 * sum_n(sum_m(ln*lm*yn*ym*xn . xm))
			#   constraint: self.C >= ln >= 0
			#           and sum_n(ln*yn) = 0
			# Then we get w using w = sum_n(ln*yn*xn)
			# At the end we can get b ~= mean(yn - w . xn)
			#
			# Since we use kernels, we only need l_star to calculate b
			# and to classify observations

			((UpperCamelCase__),	)				:					Optional[Any]					=					np.shape(UpperCAmelCase_)

			def to_minimize(UpperCAmelCase_   :						ndarray) -> float:
					UpperCamelCase__				:					Union[str, Any]					=					0
					((UpperCamelCase__),	)				:					int					=					np.shape(UpperCAmelCase_)
					for i in range(UpperCAmelCase_):
							for j in range(UpperCAmelCase_):
									s += (
									    candidate[i]
									    * candidate[j]
									    * classes[i]
									    * classes[j]
									    * self.kernel(observations[i]  ,					observations[j])
									)
					return 1 / 2 * s - sum(UpperCAmelCase_)

			UpperCamelCase__				:					List[str]					=					LinearConstraint(UpperCAmelCase_  ,					0  ,					0)
			UpperCamelCase__				:					Dict					=					Bounds(0  ,					self.regularization)

			UpperCamelCase__				:					Any					=					minimize(
			    UpperCAmelCase_  ,					np.ones(UpperCAmelCase_)  ,					bounds=UpperCAmelCase_  ,					constraints=[ly_contraint]).x
			UpperCamelCase__				:					str					=					l_star

			# calculating mean offset of separation plane to points
			UpperCamelCase__				:					Any					=					0
			for i in range(UpperCAmelCase_):
					for j in range(UpperCAmelCase_):
							s += classes[i] - classes[i] * self.optimum[i] * self.kernel(
							    observations[i]  ,					observations[j])
			UpperCamelCase__				:					List[str]					=					s / n





	def      __UpperCamelCase							(      self   :						str  ,					UpperCAmelCase_   :						ndarray):
			UpperCamelCase__				:					Optional[int]					=					sum(
			    self.optimum[n]
			    * self.classes[n]
			    * self.kernel(self.observations[n]  ,					UpperCAmelCase_)
			    for n in range(len(self.classes)))
			return 1 if s + self.offset >= 0 else -1


if __name__ == "__main__":
		import doctest

		doctest.testmod()




