




from collections.abc import Callable

import numpy as np







def      _a				(						SCREAMING_SNAKE_CASE_							:    Callable				,				SCREAMING_SNAKE_CASE_							:    float				,				SCREAMING_SNAKE_CASE_							:    float				,				SCREAMING_SNAKE_CASE_							:    float				,				SCREAMING_SNAKE_CASE_							:    float					):
					__lowerCAmelCase           =	int(np.ceil((x_end - xa) / step_size					)					)
					__lowerCAmelCase           =	np.zeros((n + 1,)					)
					__lowerCAmelCase           =	ya
					__lowerCAmelCase           =	xa

					for k in range(SCREAMING_SNAKE_CASE_					):
										__lowerCAmelCase           =	y[k] + step_size * ode_func(SCREAMING_SNAKE_CASE_				,				y[k]					)
										__lowerCAmelCase           =	y[k] + (
										    (step_size / 2) * (ode_func(SCREAMING_SNAKE_CASE_				,				y[k]					) + ode_func(x + step_size				,				SCREAMING_SNAKE_CASE_					))
										)
										x += step_size

					return y


if __name__ == "__main__":
					import doctest

					doctest.testmod()










import math







def      _a				(						SCREAMING_SNAKE_CASE_							:    int					):

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







def      _a				(						SCREAMING_SNAKE_CASE_							:    float = 0.1					):
					__lowerCAmelCase           =	3
					__lowerCAmelCase           =	3

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


if __name__ == "__main__":
					import doctest

					doctest.testmod()




