

from __future__ import annotations



def    lowerCAmelCase__							(		SCREAMING_SNAKE_CASE_:							int		,    SCREAMING_SNAKE_CASE_:							int		,    SCREAMING_SNAKE_CASE_:							int )     -> tuple[float, float]:




				'''simple docstring'''





				assert (
				    c % greatest_common_divisor(SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_ ) == 0
				)  # greatest_common_divisor(a,b) function implemented below
				((A__)      ,			(A__)      ,			(A__))            =     extended_gcd(SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_ )  # extended_gcd(a,b) function implemented below
				A__            =     c / d
				return (r * x, r * y)



def    lowerCAmelCase__							(		SCREAMING_SNAKE_CASE_:							int		,    SCREAMING_SNAKE_CASE_:							int		,    SCREAMING_SNAKE_CASE_:							int		,    SCREAMING_SNAKE_CASE_:							int = 2 )     -> None:




				'''simple docstring'''





				((A__)      ,			(A__))            =     diophantine(SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_ )  # Initial value
				A__            =     greatest_common_divisor(SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_ )
				A__            =     a // d
				A__            =     b // d

				for i in range(SCREAMING_SNAKE_CASE_ ):
								A__            =     xa + i * q
								A__            =     ya - i * p
								print(SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_ )



def    lowerCAmelCase__							(		SCREAMING_SNAKE_CASE_:							int		,    SCREAMING_SNAKE_CASE_:							int )     -> int:




				'''simple docstring'''





				if a < b:
								A__      ,			A__            =     b, a

				while a % b != 0:
								A__      ,			A__            =     b, a % b

				return b



def    lowerCAmelCase__							(		SCREAMING_SNAKE_CASE_:							int		,    SCREAMING_SNAKE_CASE_:							int )     -> tuple[int, int, int]:




				'''simple docstring'''





				assert a >= 0 and b >= 0

				if b == 0:
								A__      ,			A__      ,			A__            =     a, 1, 0
				else:
								((A__)      ,			(A__)      ,			(A__))            =     extended_gcd(SCREAMING_SNAKE_CASE_		,    a % b )
								A__            =     q
								A__            =     p - q * (a // b)

				assert a % d == 0 and b % d == 0
				assert d == a * x + b * y

				return (d, x, y)


if __name__ == "__main__":
	from doctest import testmod

	testmod(name="""diophantine""", verbose=True)
	testmod(name="""diophantine_all_soln""", verbose=True)
	testmod(name="""extended_gcd""", verbose=True)
	testmod(name="""greatest_common_divisor""", verbose=True)




import math



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



def    lowerCAmelCase__							(		SCREAMING_SNAKE_CASE_:							int = 2_0_0_0_0_0_0 )     -> int:




				'''simple docstring'''





				return sum(num for num in range(3		,    SCREAMING_SNAKE_CASE_		,    2 ) if is_prime(SCREAMING_SNAKE_CASE_ ) ) + 2 if n > 2 else 0


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

