





'''simple docstring'''




from math import sqrt






def        _lowerCAmelCase			(  __snake_case  : int			)     ->     bool:
     assert isinstance(__snake_case ,							__snake_case			) and (
         number >= 0
     ), "'number' must been an int and positive"

     __A						: Union[str, Any]      =       True

     # 0 and 1 are none primes.
     if number <= 1:
          __A						: Optional[int]      =       False

     for divisor in range(2 ,							int(round(sqrt(__snake_case			)			)			) + 1			):
          # if 'number' divisible by 'divisor' then sets 'status'
          # of false and break up the loop.
          if number % divisor == 0:
               __A						: Optional[Any]      =       False
               break

    # precondition
     assert isinstance(__snake_case ,							__snake_case			), "'status' must been from type bool"

     return status






def        _lowerCAmelCase			(  __snake_case  : Tuple			)     ->     Optional[int]:
     assert isinstance(__snake_case ,							__snake_case			) and (n > 2), "'N' must been an int and > 2"

     # beginList: contains all natural numbers from 2 up to N
     __A						: Union[str, Any]      =       list(range(2 ,							n + 1			)			)

     __A						: Union[str, Any]      =       []  # this list will be returns.

     # actual sieve of erathostenes
     for i in range(len(__snake_case			)			):
          for j in range(i + 1 ,							len(__snake_case			)			):
               if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
                    __A						: List[str]      =       0

    # filters actual prime numbers.
     __A						: Tuple      =       [x for x in begin_list if x != 0]

     # precondition
     assert isinstance(__snake_case ,							__snake_case			), "'ans' must been from type list"

     return ans






def        _lowerCAmelCase			(  __snake_case  : List[Any]			)     ->     Union[str, Any]:
     assert isinstance(__snake_case ,							__snake_case			) and (n > 2), "'N' must been an int and > 2"

     __A						: List[Any]      =       []

     # iterates over all numbers between 2 up to N+1
     # if a number is prime then appends to list 'ans'
     for number in range(2 ,							n + 1			):
          if is_prime(__snake_case			):
               ans.append(__snake_case			)

    # precondition
     assert isinstance(__snake_case ,							__snake_case			), "'ans' must been from type list"

     return ans






def        _lowerCAmelCase			(  __snake_case  : List[Any]			)     ->     Optional[int]:
     assert isinstance(__snake_case ,							__snake_case			) and number >= 0, "'number' must been an int and >= 0"

     __A						: Dict      =       []  # this list will be returns of the function.

     # potential prime number factors.

     __A						: Optional[Any]      =       2

     __A						: Optional[int]      =       number

     if number == 0 or number == 1:
          ans.append(__snake_case			)

     # if 'number' not prime then builds the prime factorization of 'number'
     elif not is_prime(__snake_case			):
          while quotient != 1:
               if is_prime(__snake_case			) and (quotient % factor == 0):
                    ans.append(__snake_case			)
                    quotient /= factor
               else:
                    factor += 1

     else:
          ans.append(__snake_case			)

     # precondition
     assert isinstance(__snake_case ,							__snake_case			), "'ans' must been from type list"

     return ans






def        _lowerCAmelCase			(  __snake_case  : Optional[Any]			)     ->     Optional[int]:
     assert isinstance(__snake_case ,							__snake_case			) and (
         number >= 0
     ), "'number' bust been an int and >= 0"

     __A						: Union[str, Any]      =       0

     # prime factorization of 'number'
     __A						: Any      =       prime_factorization(__snake_case			)

     __A						: Tuple      =       max(__snake_case			)

     # precondition
     assert isinstance(__snake_case ,							__snake_case			), "'ans' must been from type int"

     return ans






def        _lowerCAmelCase			(  __snake_case  : str			)     ->     List[str]:
     assert isinstance(__snake_case ,							__snake_case			) and (
         number >= 0
     ), "'number' bust been an int and >= 0"

     __A						: Optional[int]      =       0

     # prime factorization of 'number'
     __A						: Union[str, Any]      =       prime_factorization(__snake_case			)

     __A						: List[str]      =       min(__snake_case			)

     # precondition
     assert isinstance(__snake_case ,							__snake_case			), "'ans' must been from type int"

     return ans






def        _lowerCAmelCase			(  __snake_case  : int			)     ->     Any:
     assert isinstance(__snake_case ,							__snake_case			), "'number' must been an int"
     assert isinstance(number % 2 == 0 ,							__snake_case			), "compare bust been from type bool"

     return number % 2 == 0






def        _lowerCAmelCase			(  __snake_case  : int			)     ->     Dict:
     assert isinstance(__snake_case ,							__snake_case			), "'number' must been an int"
     assert isinstance(number % 2 != 0 ,							__snake_case			), "compare bust been from type bool"

     return number % 2 != 0






def        _lowerCAmelCase			(  __snake_case  : int			)     ->     Tuple:
     assert (
         isinstance(__snake_case ,							__snake_case			) and (number > 2) and is_even(__snake_case			)
     ), "'number' must been an int, even and > 2"

     __A						: Any      =       []  # this list will returned

     # creates a list of prime numbers between 2 up to 'number'
     __A						: Optional[int]      =       get_prime_numbers(__snake_case			)
     __A						: List[str]      =       len(__snake_case			)

     # run variable for while-loops.
     __A						: Tuple      =       0
     __A						: str      =       None

     # exit variable. for break up the loops
     __A						: List[str]      =       True

     while i < len_pn and loop:
          __A						: List[Any]      =       i + 1

          while j < len_pn and loop:
               if prime_numbers[i] + prime_numbers[j] == number:
                    __A						: Dict      =       False
                    ans.append(prime_numbers[i]			)
                    ans.append(prime_numbers[j]			)

               j += 1

          i += 1

     # precondition
     assert (
         isinstance(__snake_case ,							__snake_case			)
         and (len(__snake_case			) == 2)
         and (ans[0] + ans[1] == number)
         and is_prime(ans[0]			)
         and is_prime(ans[1]			)
     ), "'ans' must contains two primes. And sum of elements must been eq 'number'"

     return ans






def        _lowerCAmelCase			(  __snake_case  : Union[str, Any] ,							__snake_case  : Dict			)     ->     Tuple:
     assert (
         isinstance(__snake_case ,							__snake_case			)
         and isinstance(__snake_case ,							__snake_case			)
         and (numbera >= 0)
         and (numbera >= 0)
     ), "'number1' and 'number2' must been positive integer."

     __A						: str      =       0

     while numbera != 0:
          __A						: Dict      =       numbera % numbera
          __A						: str      =       numbera
          __A						: Union[str, Any]      =       rest

     # precondition
     assert isinstance(__snake_case ,							__snake_case			) and (
         numbera >= 0
     ), "'number' must been from type int and positive"

     return numbera






def        _lowerCAmelCase			(  __snake_case  : Optional[int] ,							__snake_case  : Any			)     ->     List[str]:
     assert (
         isinstance(__snake_case ,							__snake_case			)
         and isinstance(__snake_case ,							__snake_case			)
         and (numbera >= 1)
         and (numbera >= 1)
     ), "'number1' and 'number2' must been positive integer."

     __A						: Optional[Any]      =       1  # actual answer that will be return.

     # for kgV (x,1)
     if numbera > 1 and numbera > 1:
          # builds the prime factorization of 'number1' and 'number2'
          __A						: int      =       prime_factorization(__snake_case			)
          __A						: Tuple      =       prime_factorization(__snake_case			)

     elif numbera == 1 or numbera == 1:
          __A						: Tuple      =       []
          __A						: Dict      =       []
          __A						: Any      =       max(__snake_case ,							__snake_case			)

     __A						: str      =       0
     __A						: int      =       0

     __A						: List[str]      =       []  # captured numbers int both 'primeFac1' and 'primeFac2'

     # iterates through primeFac1
     for n in prime_fac_a:
          if n not in done:
               if n in prime_fac_a:
                    __A						: List[Any]      =       prime_fac_a.count(__snake_case			)
                    __A						: Tuple      =       prime_fac_a.count(__snake_case			)

                    for _ in range(max(__snake_case ,							__snake_case			)			):
                         ans *= n

               else:
                    __A						: Optional[Any]      =       prime_fac_a.count(__snake_case			)

                    for _ in range(__snake_case			):
                         ans *= n

               done.append(__snake_case			)

    # iterates through primeFac2
     for n in prime_fac_a:
          if n not in done:
               __A						: Tuple      =       prime_fac_a.count(__snake_case			)

               for _ in range(__snake_case			):
                    ans *= n

               done.append(__snake_case			)

    # precondition
     assert isinstance(__snake_case ,							__snake_case			) and (
         ans >= 0
     ), "'ans' must been from type int and positive"

     return ans






def        _lowerCAmelCase			(  __snake_case  : Tuple			)     ->     List[Any]:
     assert isinstance(__snake_case ,							__snake_case			) and (n >= 0), "'number' must been a positive int"

     __A						: str      =       0
     __A						: int      =       2  # this variable holds the answer

     while index < n:
          index += 1

          ans += 1  # counts to the next number

          # if ans not prime then
          # runs to the next prime number.
          while not is_prime(__snake_case			):
               ans += 1

    # precondition
     assert isinstance(__snake_case ,							__snake_case			) and is_prime(
         __snake_case			), "'ans' must been a prime number and from type int"

     return ans






def        _lowerCAmelCase			(  __snake_case  : Tuple ,							__snake_case  : str			)     ->     List[Any]:
     assert (
         is_prime(__snake_case			) and is_prime(__snake_case			) and (p_number_a < p_number_a)
     ), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

     __A						: Any      =       p_number_a + 1  # jump to the next number

     __A						: Optional[Any]      =       []  # this list will be returns.

     # if number is not prime then
     # fetch the next prime number.
     while not is_prime(__snake_case			):
          number += 1

     while number < p_number_a:
          ans.append(__snake_case			)

          number += 1

          # fetch the next prime number.
          while not is_prime(__snake_case			):
               number += 1

    # precondition
     assert (
         isinstance(__snake_case ,							__snake_case			)
         and ans[0] != p_number_a
         and ans[len(__snake_case			) - 1] != p_number_a
     ), "'ans' must been a list without the arguments"

     # 'ans' contains not 'pNumber1' and 'pNumber2' !
     return ans






def        _lowerCAmelCase			(  __snake_case  : Dict			)     ->     Optional[int]:
     assert isinstance(__snake_case ,							__snake_case			) and (n >= 1), "'n' must been int and >= 1"

     __A						: Dict      =       []  # will be returned.

     for divisor in range(1 ,							n + 1			):
          if n % divisor == 0:
               ans.append(__snake_case			)

    # precondition
     assert ans[0] == 1 and ans[len(__snake_case			) - 1] == n, "Error in function getDivisiors(...)"

     return ans






def        _lowerCAmelCase			(  __snake_case  : Union[str, Any]			)     ->     int:
     assert isinstance(__snake_case ,							__snake_case			) and (
         number > 1
     ), "'number' must been an int and >= 1"

     __A						: Optional[int]      =       get_divisors(__snake_case			)

     # precondition
     assert (
         isinstance(__snake_case ,							__snake_case			)
         and (divisors[0] == 1)
         and (divisors[len(__snake_case			) - 1] == number)
     ), "Error in help-function getDivisiors(...)"

     # summed all divisors up to 'number' (exclusive), hence [:-1]
     return sum(divisors[:-1]			) == number






def        _lowerCAmelCase			(  __snake_case  : Tuple ,							__snake_case  : List[Any]			)     ->     Any:
     assert (
         isinstance(__snake_case ,							__snake_case			)
         and isinstance(__snake_case ,							__snake_case			)
         and (denominator != 0)
     ), "The arguments must been from type int and 'denominator' != 0"

     # build the greatest common divisor of numerator and denominator.
     __A						: List[Any]      =       gcd(abs(__snake_case			) ,							abs(__snake_case			)			)

     # precondition
     assert (
         isinstance(__snake_case ,							__snake_case			)
         and (numerator % gcd_of_fraction == 0)
         and (denominator % gcd_of_fraction == 0)
     ), "Error in function gcd(...,...)"

     return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)






def        _lowerCAmelCase			(  __snake_case  : Dict			)     ->     Optional[int]:
     assert isinstance(__snake_case ,							__snake_case			) and (n >= 0), "'n' must been a int and >= 0"

     __A						: List[str]      =       1  # this will be return.

     for factor in range(1 ,							n + 1			):
          ans *= factor

     return ans







def        _lowerCAmelCase			(  __snake_case  : Tuple			)     ->     Optional[Any]:
     assert isinstance(__snake_case ,							__snake_case			) and (n >= 0), "'n' must been an int and >= 0"

     __A						: Any      =       0
     __A						: Optional[int]      =       1
     __A						: Tuple      =       1  # this will be return

     for _ in range(n - 1			):
          __A						: Tuple      =       ans
          ans += fiba
          __A						: List[str]      =       tmp

     return ans






'''simple docstring'''




import math






def        _lowerCAmelCase			(  __snake_case  : int			)     ->     bool:

     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        _lowerCAmelCase			(  __snake_case  : float = 0.1			)     ->     int:
     __A						: Tuple      =       3
     __A						: Optional[int]      =       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()