5. Summation of the non-representable numbers

In the last chapter we deal with the sum $S_k(A)$ of $k^{\rm th}$ powers of the non-representable munbers. First we illustrate, how analysis can be used for determining $S_1(A)=S(A)$ when $n=2$ [2]. The investigation is based on the generating function, which is the following in the case of our problem:

$$f(x)=\frac{1}{(1-x^{a_1})(1-x^{a_2})\cdots(1-x^{a_n})}.$$

$G(a_1, a_2, \ldots , a_n)$ is the greatest integer $k$, for which the coefficient of $x^k$ is zero in $f(x)$.

Then we discuss some special cases of a general result by RÖDSETH [27] for higher powers, namely we compute the exact value of $S_2(a,b)$ and $S_3(a,b)$ 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 $A$, if we know the smallest representable elements of the residue classes mod $a_1$.

THEOREM 5.4.1. Let $a_1 < a_2 < \ldots < a_n$ be relatively prime positive integers. Consider from each non-zero residue class mod $a_1$ the smallest integer representable by $a_2, a_3, \ldots ,a_n$, and denote these by $x_1, x_2, \ldots ,x_{a_1-1}$. Then

$$S(a_1, a_2, \ldots , a_n)= \sum\limits_{j=1}^{a_1-1}\left(\frac{x_j^2}{2a_1}-\frac{x_j}{2}\right) +\frac{a_1^2-1}{12}.$$

Thus we determined the sum of non-representable numbers by three different methods in the case $n=2$.

COROLLARY 5.4.2. Let $a$ and $b$ be relatively prime positive integers. Then

$$S(a, b)=\frac{1}{12}(a-1)(b-1)(2ab-a-b-1).$$

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 $n>2$, we compute the exact value of $S_1(ab, bc, ca)$ 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.