



"""simple docstring"""





def      a__			(   lowerCAmelCase__			,    lowerCAmelCase__			,    lowerCAmelCase__			,    lowerCAmelCase__   ):

						# 1. Validate that path exists between current and next vertices
						if graph[path[curr_ind - 1]][next_ver] == 0:
												return False

						# 2. Validate that next vertex is not already in path
						return not any(vertex == next_ver for vertex in path   )
def      a__			(   lowerCAmelCase__			,    lowerCAmelCase__			,    lowerCAmelCase__   ):

						# Base Case
						if curr_ind == len(lowerCAmelCase__   ):
												# return whether path exists between current and starting vertices
												return graph[path[curr_ind - 1]][path[0]] == 1

						# Recursive Step
						for next_ver in range(0			,    len(lowerCAmelCase__   )   ):
												if valid_connection(lowerCAmelCase__			,    lowerCAmelCase__			,    lowerCAmelCase__			,    lowerCAmelCase__   ):
																		# Insert current vertex  into path as next transition
																		UpperCAmelCase_    				=							next_ver
																		# Validate created path
																		if util_hamilton_cycle(lowerCAmelCase__			,    lowerCAmelCase__			,    curr_ind + 1   ):
																								return True
																		# Backtrack
																		UpperCAmelCase_    				=							-1
						return False


def      a__			(   lowerCAmelCase__			,    lowerCAmelCase__ = 0   ):
						UpperCAmelCase_    				=							[-1] * (len(lowerCAmelCase__   ) + 1)
						# initialize start and end of path with starting index
						UpperCAmelCase_    				=							UpperCAmelCase_    				=							start_index
						# evaluate and if we find answer return path either return empty array
						return path if util_hamilton_cycle(lowerCAmelCase__			,    lowerCAmelCase__			,    1   ) else []









"""simple docstring"""





from __future__ import annotations

import math
def      a__			(   lowerCAmelCase__   ):

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


lowerCamelCase      =		[num for num in range(3, 100_001, 2) if not is_prime(num)]
def      a__			(   lowerCAmelCase__   ):
      if not isinstance(lowerCAmelCase__			,    lowerCAmelCase__   ):
            raise ValueError("n must be an integer"   )
      if n <= 0:
            raise ValueError("n must be >= 0"   )

      UpperCAmelCase_    				=							[]
      for num in range(len(lowerCAmelCase__   )   ):
            UpperCAmelCase_    				=							0
            while 2 * i * i <= odd_composites[num]:
                  UpperCAmelCase_    				=							odd_composites[num] - 2 * i * i
                  if is_prime(lowerCAmelCase__   ):
                        break
                  i += 1
            else:
                  list_nums.append(odd_composites[num]   )
                  if len(lowerCAmelCase__   ) == n:
                        return list_nums

      return []
def      a__			(   ):
      return compute_nums(1   )[0]


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




