


"""simple docstring"""
import numpy as np






def 		lowercase							(     _snake_case						:      int     ,   _snake_case						:      Optional[Any]     ,   _snake_case						:      Optional[int]     ,   _snake_case						:      int     ,   _snake_case						:      Union[str, Any]     )     ->Dict:
	"""simple docstring"""





	__snake_case	:						Union[str, Any]						=				int(np.ceil((x_end - xa) / h     )     )
	__snake_case	:						Dict						=				np.zeros((n + 1,)     )
	__snake_case	:						List[Any]						=				ya
	__snake_case	:						int						=				xa

	for k in range(_snake_case     ):
		__snake_case	:						Any						=				f(_snake_case     ,   y[k]     )
		__snake_case	:						List[Any]						=				f(x + 0.5 * h     ,   y[k] + 0.5 * h * ka     )
		__snake_case	:						int						=				f(x + 0.5 * h     ,   y[k] + 0.5 * h * ka     )
		__snake_case	:						Optional[int]						=				f(x + h     ,   y[k] + h * ka     )
		__snake_case	:						Optional[int]						=				y[k] + (1 / 6) * h * (ka + 2 * ka + 2 * ka + ka)
		x += h

	return y


if __name__ == "__main__":
			import doctest

			doctest.testmod()









"""simple docstring"""
import math






def 		lowercase							(     _snake_case						:      int     )     ->bool:
	"""simple docstring"""





	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(_snake_case     ) + 1     )     ,   6     ):
		if number % i == 0 or number % (i + 2) == 0:
			return False
	return True






def 		lowercase							(     _snake_case						:      float = 0.1     )     ->int:
	"""simple docstring"""





	__snake_case	:						Tuple						=				3
	__snake_case	:						Any						=				3

	while primes / (2 * j - 1) >= ratio:
		for i in range(j * j + j + 1     ,   (j + 2) * (j + 2)     ,   j + 1     ):
			primes += is_prime(_snake_case     )
		j += 2
	return j


if __name__ == "__main__":
			import doctest

			doctest.testmod()





