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 [2]. 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 [27] 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.