



'''simple docstring'''

import math
from collections.abc import Generator



def   __lowerCamelCase  (   lowerCAmelCase_						)    ->   Generator[int, None, None]:
		_a  :   Generator				 =							(i for i in range(1		,    (max_n + 1)						))
		for i in (n for n in numbers if n > 1):
				for j in range(2		,    lowerCAmelCase_						):
						if (i % j) == 0:
								break
				else:
						yield i



def   __lowerCamelCase  (   lowerCAmelCase_						)    ->   Generator[int, None, None]:
		_a  :   Generator				 =							(i for i in range(1		,    (max_n + 1)						))
		for i in (n for n in numbers if n > 1):
				# only need to check for factors up to sqrt(i)
				_a  :   List[str]				 =							int(math.sqrt(lowerCAmelCase_						)						) + 1
				for j in range(2		,    lowerCAmelCase_						):
						if (i % j) == 0:
								break
				else:
						yield i



def   __lowerCamelCase  (   lowerCAmelCase_						)    ->   Generator[int, None, None]:
		_a  :   Generator				 =							(i for i in range(1		,    (max_n + 1)		,    2						))
		# It's useless to test even numbers as they will not be prime
		if max_n > 2:
				yield 2  # Because 2 will not be tested, it's necessary to yield it now
		for i in (n for n in numbers if n > 1):
				_a  :   List[Any]				 =							int(math.sqrt(lowerCAmelCase_						)						) + 1
				for j in range(3		,    lowerCAmelCase_		,    2						):
						# As we removed the even numbers, we don't need them now
						if (i % j) == 0:
								break
				else:
						yield i



def   __lowerCamelCase  (   )    ->   List[Any]:
		from timeit import timeit

		_a  :   Optional[int]				 =							'from __main__ import slow_primes, primes, fast_primes'
		print(timeit('slow_primes(1_000_000_000_000)'		,    setup=lowerCAmelCase_		,    number=1000000						)						)
		print(timeit('primes(1_000_000_000_000)'		,    setup=lowerCAmelCase_		,    number=1000000						)						)
		print(timeit('fast_primes(1_000_000_000_000)'		,    setup=lowerCAmelCase_		,    number=1000000						)						)


if __name__ == "__main__":
							__lowerCAmelCase		    =     int(input('''Calculate primes up to:\n>> ''').strip())
							for ret in primes(number):
														print(ret)
							benchmark()









'''simple docstring'''

def   __lowerCamelCase  (   lowerCAmelCase_						)    ->   tuple[int, int]:
		# base case
		if len(lowerCAmelCase_						) == 1:
				return a[0], a[0]
		else:
				# split A into half.
				_a  :   Union[str, Any]				 =							a[: len(lowerCAmelCase_						) // 2]
				_a  :   List[Any]				 =							a[len(lowerCAmelCase_						) // 2 :]

				# 2 sub problems, 1/2 of original size.
				_a ,      _a  :   Any				 =							max_difference(lowerCAmelCase_						)
				_a ,      _a  :   Any				 =							max_difference(lowerCAmelCase_						)

				# get min of first and max of second
				# linear time
				_a  :   Union[str, Any]				 =							min(lowerCAmelCase_						)
				_a  :   Optional[int]				 =							max(lowerCAmelCase_						)

				# 3 cases, either (small1, big1),
				# (min_first, max_second), (small2, big2)
				# constant comparisons
				if biga - smalla > max_second - min_first and biga - smalla > biga - smalla:
						return smalla, biga
				elif biga - smalla > max_second - min_first:
						return smalla, biga
				else:
						return min_first, max_second


if __name__ == "__main__":
							import doctest

							doctest.testmod()




