


'''simple docstring'''

def __snake_case					(      UpperCAmelCase_						:						int		):

 if isinstance(UpperCAmelCase_					,				UpperCAmelCase_		):
  raise TypeError("'float' object cannot be interpreted as an integer"		)
 if isinstance(UpperCAmelCase_					,				UpperCAmelCase_		):
  raise TypeError("'str' object cannot be interpreted as an integer"		)

 if num == 0:
  return "0b0"

 lowerCamelCase_      =  False

 if num < 0:
  lowerCamelCase_      =  True
  lowerCamelCase_      =  -num

 lowerCamelCase_      =  []
 while num > 0:
  binary.insert(0					,				num % 2		)
  num >>= 1

 if negative:
  return "-0b" + "".join(str(UpperCAmelCase_		) for e in binary		)

 return "0b" + "".join(str(UpperCAmelCase_		) for e in binary		)


if __name__ == "__main__":
     import doctest

     doctest.testmod()







'''simple docstring'''

import math





def __snake_case					(      UpperCAmelCase_						:						int		):
 lowerCamelCase_      =  math.loga(math.sqrt(4 * positive_integer + 1		) / 2 + 1 / 2		)

 return exponent == int(UpperCAmelCase_		)





def __snake_case					(      UpperCAmelCase_						:						float = 1 / 12345		):
 lowerCamelCase_      =  0
 lowerCamelCase_      =  0

 lowerCamelCase_      =  3
 while True:
  lowerCamelCase_      =  (integer**2 - 1) / 4
  # if candidate is an integer, then there is a partition for k
  if partition_candidate == int(UpperCAmelCase_		):
   lowerCamelCase_      =  int(UpperCAmelCase_		)
   total_partitions += 1
   if check_partition_perfect(UpperCAmelCase_		):
    perfect_partitions += 1
  if perfect_partitions > 0:
   if perfect_partitions / total_partitions < max_proportion:
    return int(UpperCAmelCase_		)
  integer += 1


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



