



import random

from .binary_exp_mod import bin_exp_mod



def       lowercase       (   SCREAMING_SNAKE_CASE__       :	str				,       SCREAMING_SNAKE_CASE__       :	Any=1_000				)  ->     Any:
 if n < 2:
  return False

 if n % 2 == 0:
  return n == 2

 # this means n is odd
 _snake_case    :  List[str]					=					n - 1
 _snake_case    :  Tuple					=					0
 while d % 2 == 0:
  d /= 2
  exp += 1

 # n - 1=d*(2**exp)
 _snake_case    :  Dict					=					0
 while count < prec:
  _snake_case    :  int					=					random.randint(2				,       n - 1				)
  _snake_case    :  str					=					bin_exp_mod(SCREAMING_SNAKE_CASE__				,       SCREAMING_SNAKE_CASE__				,       SCREAMING_SNAKE_CASE__				)
  if b != 1:
   _snake_case    :  Optional[int]					=					True
   for _ in range(SCREAMING_SNAKE_CASE__				):
    if b == n - 1:
     _snake_case    :  List[Any]					=					False
     break
    _snake_case    :  Union[str, Any]					=					b * b
    b %= n
   if flag:
    return False
   count += 1
 return True


if __name__ == "__main__":
  a__    =	abs(int(input("""Enter bound : """).strip()))
  print("""Here's the list of primes:""")
  print(""", """.join(str(i) for i in range(n + 1) if is_prime_big(i)))










from math import ceil, sqrt



def       lowercase       (   SCREAMING_SNAKE_CASE__       :	int = 1_000_000				)  ->     int:
 _snake_case    :  Tuple					=					0

 for outer_width in range(3				,       (limit // 4) + 2				):
  if outer_width**2 > limit:
   _snake_case    :  int					=					max(ceil(sqrt(outer_width**2 - limit				)				)				,       1				)
  else:
   _snake_case    :  Any					=					1
  if (outer_width - hole_width_lower_bound) % 2:
   hole_width_lower_bound += 1

  answer += (outer_width - hole_width_lower_bound - 2) // 2 + 1

 return answer


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