
from __future__ import annotations

lowercase__       =[True] * 1000001
lowercase__       =2
while i * i <= 1000000:
      if seive[i]:
            for j in range(i * i, 1000001, i):
                  lowercase__       =False
      i += 1




def    __UpperCamelCase     (				lowerCAmelCase__							:				int  ):
 return seive[n]




def    __UpperCamelCase     (				lowerCAmelCase__							:				int  ):
 return any(digit in '''02468''' for digit in str(lowerCAmelCase__  )  )




def    __UpperCamelCase     (				lowerCAmelCase__							:				int = 1_0_0_0_0_0_0  ):
 __a :  Tuple						=  [2]  # result already includes the number 2.
 for num in range(3  ,   limit + 1  ,   2  ):
  if is_prime(lowerCAmelCase__  ) and not contains_an_even_digit(lowerCAmelCase__  ):
   __a :  str						=  str(lowerCAmelCase__  )
   __a :  Optional[int]						=  [int(str_num[j:] + str_num[:j]  ) for j in range(len(lowerCAmelCase__  )  )]
   if all(is_prime(lowerCAmelCase__  ) for i in list_nums  ):
    result.append(lowerCAmelCase__  )
 return result




def    __UpperCamelCase     (				):
 return len(find_circular_primes()  )


if __name__ == "__main__":
      print(F"""{len(find_circular_primes()) = }""")


from __future__ import annotations

from fractions import Fraction
from math import gcd, sqrt




def    __UpperCamelCase     (				lowerCAmelCase__							:				int  ):
 __a :  int						=  int(number**0.5  )
 return number == sq * sq




def    __UpperCamelCase     (				lowerCAmelCase__							:				int  ,   lowerCAmelCase__							:				int  ,   lowerCAmelCase__							:				int  ,   lowerCAmelCase__							:				int  ,   lowerCAmelCase__							:				int  ,   lowerCAmelCase__							:				int  ):
 __a :  int						=  x_num * y_den * z_den + y_num * x_den * z_den + z_num * x_den * y_den
 __a :  int						=  x_den * y_den * z_den
 __a :  int						=  gcd(lowerCAmelCase__  ,   lowerCAmelCase__  )
 top //= hcf
 bottom //= hcf
 return top, bottom




def    __UpperCamelCase     (				lowerCAmelCase__							:				int = 3_5  ):
 __a :  set						=  set()
 __a :  int
 __a :  Fraction						=  Fraction(0  )
 __a :  tuple[int, int]

 for x_num in range(1  ,   order + 1  ):
  for x_den in range(x_num + 1  ,   order + 1  ):
   for y_num in range(1  ,   order + 1  ):
    for y_den in range(y_num + 1  ,   order + 1  ):
     # n=1
     __a :  Any						=  x_num * y_den + x_den * y_num
     __a :  Dict						=  x_den * y_den
     __a :  int						=  gcd(lowerCAmelCase__  ,   lowerCAmelCase__  )
     z_num //= hcf
     z_den //= hcf
     if 0 < z_num < z_den <= order:
      __a :  str						=  add_three(
          lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  )
      unique_s.add(lowerCAmelCase__  )

     # n=2
     __a :  Union[str, Any]						=  (
         x_num * x_num * y_den * y_den + x_den * x_den * y_num * y_num
     )
     __a :  Optional[Any]						=  x_den * x_den * y_den * y_den
     if is_sq(lowerCAmelCase__  ) and is_sq(lowerCAmelCase__  ):
      __a :  Any						=  int(sqrt(lowerCAmelCase__  )  )
      __a :  Optional[Any]						=  int(sqrt(lowerCAmelCase__  )  )
      __a :  int						=  gcd(lowerCAmelCase__  ,   lowerCAmelCase__  )
      z_num //= hcf
      z_den //= hcf
      if 0 < z_num < z_den <= order:
       __a :  str						=  add_three(
           lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  )
       unique_s.add(lowerCAmelCase__  )

                    # n=-1
     __a :  List[str]						=  x_num * y_num
     __a :  List[str]						=  x_den * y_num + x_num * y_den
     __a :  str						=  gcd(lowerCAmelCase__  ,   lowerCAmelCase__  )
     z_num //= hcf
     z_den //= hcf
     if 0 < z_num < z_den <= order:
      __a :  Optional[int]						=  add_three(
          lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  )
      unique_s.add(lowerCAmelCase__  )

     # n=2
     __a :  List[str]						=  x_num * x_num * y_num * y_num
     __a :  Optional[int]						=  (
         x_den * x_den * y_num * y_num + x_num * x_num * y_den * y_den
     )
     if is_sq(lowerCAmelCase__  ) and is_sq(lowerCAmelCase__  ):
      __a :  Union[str, Any]						=  int(sqrt(lowerCAmelCase__  )  )
      __a :  List[str]						=  int(sqrt(lowerCAmelCase__  )  )
      __a :  Optional[int]						=  gcd(lowerCAmelCase__  ,   lowerCAmelCase__  )
      z_num //= hcf
      z_den //= hcf
      if 0 < z_num < z_den <= order:
       __a :  int						=  add_three(
           lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  )
       unique_s.add(lowerCAmelCase__  )

 for num, den in unique_s:
  total += Fraction(lowerCAmelCase__  ,   lowerCAmelCase__  )

 return total.denominator + total.numerator


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