
'''simple docstring'''





from math import sqrt







def snake_case_     (					lowercase__		):
  assert isinstance(lowercase__   ,       lowercase__		) and (
      number >= 0
  ), "'number' must been an int and positive"

  UpperCAmelCase__      :					Optional[Any]				       =							True

  # 0 and 1 are none primes.
  if number <= 1:
    UpperCAmelCase__      :					Union[str, Any]				       =							False

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

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

  return status







def snake_case_     (					lowercase__		):
  assert isinstance(lowercase__   ,       lowercase__		) and (n > 2), "'N' must been an int and > 2"

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

  UpperCAmelCase__      :					Dict				       =							[]  # this list will be returns.

  # actual sieve of erathostenes
  for i in range(len(lowercase__		)		):
    for j in range(i + 1   ,       len(lowercase__		)		):
      if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
        UpperCAmelCase__      :					Dict				       =							0

    # filters actual prime numbers.
  UpperCAmelCase__      :					Optional[int]				       =							[x for x in begin_list if x != 0]

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

  return ans







def snake_case_     (					lowercase__		):
  assert isinstance(lowercase__   ,       lowercase__		) and (n > 2), "'N' must been an int and > 2"

  UpperCAmelCase__      :					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(lowercase__		):
      ans.append(lowercase__		)

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

  return ans







def snake_case_     (					lowercase__		):
  assert isinstance(lowercase__   ,       lowercase__		) and number >= 0, "'number' must been an int and >= 0"

  UpperCAmelCase__      :					int				       =							[]  # this list will be returns of the function.

  # potential prime number factors.

  UpperCAmelCase__      :					Union[str, Any]				       =							2

  UpperCAmelCase__      :					int				       =							number

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

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

  else:
    ans.append(lowercase__		)

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

  return ans







def snake_case_     (					lowercase__		):
  assert isinstance(lowercase__   ,       lowercase__		) and (
      number >= 0
  ), "'number' bust been an int and >= 0"

  UpperCAmelCase__      :					int				       =							0

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

  UpperCAmelCase__      :					Optional[int]				       =							max(lowercase__		)

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

  return ans







def snake_case_     (					lowercase__		):
  assert isinstance(lowercase__   ,       lowercase__		) and (
      number >= 0
  ), "'number' bust been an int and >= 0"

  UpperCAmelCase__      :					str				       =							0

  # prime factorization of 'number'
  UpperCAmelCase__      :					Dict				       =							prime_factorization(lowercase__		)

  UpperCAmelCase__      :					Union[str, Any]				       =							min(lowercase__		)

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

  return ans







def snake_case_     (					lowercase__		):
  assert isinstance(lowercase__   ,       lowercase__		), "'number' must been an int"
  assert isinstance(number % 2 == 0   ,       lowercase__		), "compare bust been from type bool"

  return number % 2 == 0







def snake_case_     (					lowercase__		):
  assert isinstance(lowercase__   ,       lowercase__		), "'number' must been an int"
  assert isinstance(number % 2 != 0   ,       lowercase__		), "compare bust been from type bool"

  return number % 2 != 0







def snake_case_     (					lowercase__		):
  assert (
      isinstance(lowercase__   ,       lowercase__		) and (number > 2) and is_even(lowercase__		)
  ), "'number' must been an int, even and > 2"

  UpperCAmelCase__      :					Union[str, Any]				       =							[]  # this list will returned

  # creates a list of prime numbers between 2 up to 'number'
  UpperCAmelCase__      :					str				       =							get_prime_numbers(lowercase__		)
  UpperCAmelCase__      :					List[Any]				       =							len(lowercase__		)

  # run variable for while-loops.
  UpperCAmelCase__      :					Dict				       =							0
  UpperCAmelCase__      :					List[Any]				       =							None

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

  while i < len_pn and loop:
    UpperCAmelCase__      :					Dict				       =							i + 1

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

      j += 1

    i += 1

  # precondition
  assert (
      isinstance(lowercase__   ,       lowercase__		)
      and (len(lowercase__		) == 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 snake_case_     (					lowercase__   ,       lowercase__		):
  assert (
      isinstance(lowercase__   ,       lowercase__		)
      and isinstance(lowercase__   ,       lowercase__		)
      and (numbera >= 0)
      and (numbera >= 0)
  ), "'number1' and 'number2' must been positive integer."

  UpperCAmelCase__      :					List[str]				       =							0

  while numbera != 0:
    UpperCAmelCase__      :					Optional[Any]				       =							numbera % numbera
    UpperCAmelCase__      :					Dict				       =							numbera
    UpperCAmelCase__      :					Optional[int]				       =							rest

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

  return numbera







def snake_case_     (					lowercase__   ,       lowercase__		):
  assert (
      isinstance(lowercase__   ,       lowercase__		)
      and isinstance(lowercase__   ,       lowercase__		)
      and (numbera >= 1)
      and (numbera >= 1)
  ), "'number1' and 'number2' must been positive integer."

  UpperCAmelCase__      :					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'
    UpperCAmelCase__      :					List[Any]				       =							prime_factorization(lowercase__		)
    UpperCAmelCase__      :					Optional[Any]				       =							prime_factorization(lowercase__		)

  elif numbera == 1 or numbera == 1:
    UpperCAmelCase__      :					Union[str, Any]				       =							[]
    UpperCAmelCase__      :					int				       =							[]
    UpperCAmelCase__      :					Dict				       =							max(lowercase__   ,       lowercase__		)

  UpperCAmelCase__      :					Union[str, Any]				       =							0
  UpperCAmelCase__      :					Dict				       =							0

  UpperCAmelCase__      :					Any				       =							[]  # 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:
        UpperCAmelCase__      :					Dict				       =							prime_fac_a.count(lowercase__		)
        UpperCAmelCase__      :					int				       =							prime_fac_a.count(lowercase__		)

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

      else:
        UpperCAmelCase__      :					Dict				       =							prime_fac_a.count(lowercase__		)

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

      done.append(lowercase__		)

    # iterates through primeFac2
  for n in prime_fac_a:
    if n not in done:
      UpperCAmelCase__      :					Union[str, Any]				       =							prime_fac_a.count(lowercase__		)

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

      done.append(lowercase__		)

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

  return ans







def snake_case_     (					lowercase__		):
  assert isinstance(lowercase__   ,       lowercase__		) and (n >= 0), "'number' must been a positive int"

  UpperCAmelCase__      :					Any				       =							0
  UpperCAmelCase__      :					Optional[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(lowercase__		):
      ans += 1

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

  return ans







def snake_case_     (					lowercase__   ,       lowercase__		):
  assert (
      is_prime(lowercase__		) and is_prime(lowercase__		) and (p_number_a < p_number_a)
  ), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

  UpperCAmelCase__      :					List[str]				       =							p_number_a + 1  # jump to the next number

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

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

  while number < p_number_a:
    ans.append(lowercase__		)

    number += 1

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

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

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







def snake_case_     (					lowercase__		):
  assert isinstance(lowercase__   ,       lowercase__		) and (n >= 1), "'n' must been int and >= 1"

  UpperCAmelCase__      :					str				       =							[]  # will be returned.

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

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

  return ans







def snake_case_     (					lowercase__		):
  assert isinstance(lowercase__   ,       lowercase__		) and (
      number > 1
  ), "'number' must been an int and >= 1"

  UpperCAmelCase__      :					Optional[int]				       =							get_divisors(lowercase__		)

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

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







def snake_case_     (					lowercase__   ,       lowercase__		):
  assert (
      isinstance(lowercase__   ,       lowercase__		)
      and isinstance(lowercase__   ,       lowercase__		)
      and (denominator != 0)
  ), "The arguments must been from type int and 'denominator' != 0"

  # build the greatest common divisor of numerator and denominator.
  UpperCAmelCase__      :					Optional[Any]				       =							gcd(abs(lowercase__		)   ,       abs(lowercase__		)		)

  # precondition
  assert (
      isinstance(lowercase__   ,       lowercase__		)
      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 snake_case_     (					lowercase__		):
  assert isinstance(lowercase__   ,       lowercase__		) and (n >= 0), "'n' must been a int and >= 0"

  UpperCAmelCase__      :					Any				       =							1  # this will be return.

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

  return ans







def snake_case_     (					lowercase__		):
  assert isinstance(lowercase__   ,       lowercase__		) and (n >= 0), "'n' must been an int and >= 0"

  UpperCAmelCase__      :					Union[str, Any]				       =							0
  UpperCAmelCase__      :					Optional[Any]				       =							1
  UpperCAmelCase__      :					Dict				       =							1  # this will be return

  for _ in range(n - 1		):
    UpperCAmelCase__      :					Optional[Any]				       =							ans
    ans += fiba
    UpperCAmelCase__      :					Optional[Any]				       =							tmp

  return ans





'''simple docstring'''





import math







def snake_case_     (					lowercase__		):

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







def snake_case_     (					lowercase__ = 0.1		):
  UpperCAmelCase__      :					Dict				       =							3
  UpperCAmelCase__      :					Tuple				       =							3

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


if __name__ == "__main__":
      import doctest

      doctest.testmod()



