In the last chapter we deal with the sum of powers of the non-representable munbers. First we illustrate, how analysis can be used for determining when . The investigation is based on the generating function, which is the following in the case of our problem:
Then we discuss some special cases of a general result by RÖDSETH  for higher powers, namely we compute the exact value of and with the aid of this theorem.
Finally, we show, that these power suns can be computed also with a completely elementary method, moreover the sum can be calculated for any set , if we know the smallest representable elements of the residue classes mod .
THEOREM 5.4.1. Let be relatively prime positive integers. Consider from each non-zero residue class mod the smallest integer representable by , and denote these by . Then
Thus we determined the sum of non-representable numbers by three different methods in the case .
COROLLARY 5.4.2. Let and be relatively prime positive integers. Then
We obtain similar results also for the sum of second and third powers (Corollaries 5.5.3. and 5.5.6.). For illustrating that our elementary method is applicable also for , we compute the exact value of for the numbers in Problem 1 of Chapter 2 (Theorem 5.4.3.).
This final part can also be considered as a continuation of Chapter 2 in a certain sense, since it can be used as a material for continued training of teachers who know already the basic facts and methods of the topic.