require 'prime'

def odd_composites
  enum = Enumerator.new do |y|
    a = 3
    loop do
      y << a unless Prime.prime?(a)
      a += 2
    end
  end
  enum
end

def squares
  enum = Enumerator.new do |y|
    n = 1
    loop do
      y << n ** 2
      n += 1
    end
  end
  enum
end


def find_sum(num)
  found = false
  primes = Prime::EratosthenesGenerator.new
  prime = 0
  squares_enum = squares
  while prime < num
    prime = primes.next
    squares_enum.rewind
    twice_square = squares_enum.next * 2
    while twice_square < num
      return [prime, twice_square / 2] if twice_square + prime == num
      twice_square = squares_enum.next * 2
    end
  end
  found
end

def property?(num)
  find_sum(num)
end

def find_incorrect_up_to(n)
  odd_composites.take_while { |x| x < n }.reject{ |x| property?(x) }
end