

import math







def 			A_      (    _UpperCAmelCase    ):

						if 1 < number < 4:
												# 2 and 3 are primes
												return True
						elif number < 2 or number % 2 == 0 or number % 3 == 0:
												# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
												return False

						# All primes number are in format of 6k +/- 1
						for i in range(5				,	int(math.sqrt(_UpperCAmelCase    ) + 1    )				,	6    ):
												if number % i == 0 or number % (i + 2) == 0:
																		return False
						return True







def 			A_      (    _UpperCAmelCase = 1_00_01    ):

						try:
												SCREAMING_SNAKE_CASE_: Any  		=       int(_UpperCAmelCase    )
						except (TypeError, ValueError):
												raise TypeError("Parameter nth must be int or castable to int."    ) from None
						if nth <= 0:
												raise ValueError("Parameter nth must be greater than or equal to one."    )
						SCREAMING_SNAKE_CASE_: list[int]  		=       []
						SCREAMING_SNAKE_CASE_: str  		=       2
						while len(_UpperCAmelCase    ) < nth:
												if is_prime(_UpperCAmelCase    ):
																		primes.append(_UpperCAmelCase    )
																		num += 1
												else:
																		num += 1
						return primes[len(_UpperCAmelCase    ) - 1]


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









import unittest

from transformers import is_torch_available
from transformers.testing_utils import require_torch


if is_torch_available():
						import torch

						from transformers.generation import DisjunctiveConstraint



@require_torch
class   __lowercase				( unittest.TestCase	):




			"""simple docstring"""







			def        _SCREAMING_SNAKE_CASE			( self  :     List[str]):
									# For consistency across different places the DisjunctiveConstraint is called,
									# dc.token_ids is a list of integers. It is also initialized only by integers.

									SCREAMING_SNAKE_CASE_: Optional[Any]  		=       [[1, 2, 4], [1, 2, 3, 4]]
									SCREAMING_SNAKE_CASE_: Any  		=       DisjunctiveConstraint(lowerCAmelCase__)
									self.assertTrue(isinstance(dc.token_ids   ,						lowerCAmelCase__))

									with self.assertRaises(lowerCAmelCase__):
															DisjunctiveConstraint(torch.LongTensor([[1, 2, 4], [1, 2, 3]]))

									with self.assertRaises(lowerCAmelCase__):
															DisjunctiveConstraint([torch.LongTensor([1, 2, 4]), torch.LongTensor([1, 2, 3, 4, 5])])



			def        _SCREAMING_SNAKE_CASE			( self  :     Optional[int]):
									# We can't have constraints that are complete subsets of another. This leads to a preverse
									# interpretation of "constraint fulfillment": does generating [1,2,3] fulfill the constraint?
									# It would mean that it generated [1,2] which fulfills it, but it's in the middle of potentially
									# fulfilling [1,2,3,4]. If we believe that [1,2,3] does fulfill the constraint, then the algorithm
									# will necessarily never reach [1,2,3,4], giving users a false sense of control (better to just not allow it).
									SCREAMING_SNAKE_CASE_: Union[str, Any]  		=       [[1, 2], [1, 2, 3, 4]]

									with self.assertRaises(lowerCAmelCase__):
															DisjunctiveConstraint(lowerCAmelCase__)  # fails here



			def        _SCREAMING_SNAKE_CASE			( self  :     Union[str, Any]):
									SCREAMING_SNAKE_CASE_: List[str]  		=       [[1, 2, 3], [1, 2, 4]]

									SCREAMING_SNAKE_CASE_: Tuple  		=       DisjunctiveConstraint(lowerCAmelCase__)

									SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_: int  		=       dc.update(1)
									SCREAMING_SNAKE_CASE_: Dict  		=       stepped is True and completed is False and reset is False
									self.assertTrue(lowerCAmelCase__)
									self.assertTrue(not dc.completed)
									self.assertTrue(dc.current_seq == [1])

									SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_: str  		=       dc.update(2)
									SCREAMING_SNAKE_CASE_: Optional[Any]  		=       stepped is True and completed is False and reset is False
									self.assertTrue(lowerCAmelCase__)
									self.assertTrue(not dc.completed)
									self.assertTrue(dc.current_seq == [1, 2])

									SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_: Tuple  		=       dc.update(3)
									SCREAMING_SNAKE_CASE_: Tuple  		=       stepped is True and completed is True and reset is False
									self.assertTrue(lowerCAmelCase__)
									self.assertTrue(dc.completed)  # Completed!
									self.assertTrue(dc.current_seq == [1, 2, 3])




			def        _SCREAMING_SNAKE_CASE			( self  :     Union[str, Any]):
									SCREAMING_SNAKE_CASE_: Union[str, Any]  		=       [[1, 2, 3], [1, 2, 4, 5], [1, 2, 5]]

									SCREAMING_SNAKE_CASE_: List[Any]  		=       DisjunctiveConstraint(lowerCAmelCase__)

									SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_: Optional[Any]  		=       dc.update(1)
									self.assertTrue(not dc.completed)
									self.assertTrue(dc.current_seq == [1])

									SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_: Dict  		=       dc.update(2)
									self.assertTrue(not dc.completed)
									self.assertTrue(dc.current_seq == [1, 2])

									SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_: Tuple  		=       dc.update(4)
									self.assertTrue(not dc.completed)
									self.assertTrue(dc.current_seq == [1, 2, 4])

									SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_: Optional[Any]  		=       dc.update(5)
									self.assertTrue(dc.completed)  # Completed!
									self.assertTrue(dc.current_seq == [1, 2, 4, 5])

									dc.reset()

									SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_: List[Any]  		=       dc.update(1)
									self.assertTrue(not dc.completed)
									self.assertTrue(dc.remaining() == 3)
									self.assertTrue(dc.current_seq == [1])

									SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_: Tuple  		=       dc.update(2)
									self.assertTrue(not dc.completed)
									self.assertTrue(dc.remaining() == 2)
									self.assertTrue(dc.current_seq == [1, 2])

									SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_: Union[str, Any]  		=       dc.update(5)
									self.assertTrue(dc.completed)  # Completed!
									self.assertTrue(dc.remaining() == 0)
									self.assertTrue(dc.current_seq == [1, 2, 5])






