'''simple docstring'''





import numpy as np





def 	lowerCAmelCase_				(					snake_case_       : int					,    snake_case_       : Optional[Any]					,    snake_case_       : Optional[int]					,    snake_case_       : int					,    snake_case_       : Union[str, Any]			)    ->      Dict:


     '''simple docstring'''




     UpperCAmelCase_           =   int(np.ceil((x_end - xa) / h			)			)
     UpperCAmelCase_           =   np.zeros((n + 1,)			)
     UpperCAmelCase_           =   ya
     UpperCAmelCase_           =   xa

     for k in range(_snake_case			):
          UpperCAmelCase_           =   f(_snake_case					,    y[k]			)
          UpperCAmelCase_           =   f(x + 0.5 * h					,    y[k] + 0.5 * h * ka			)
          UpperCAmelCase_           =   f(x + 0.5 * h					,    y[k] + 0.5 * h * ka			)
          UpperCAmelCase_           =   f(x + h					,    y[k] + h * ka			)
          UpperCAmelCase_           =   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()





