The thesis investigates various aspects of the following problem: Given a set of relatively prime positive integers, which positive integers can be represented as , where are non-negative integers.
The short introduction is followed by Chapter 2, which plays a double role. On the one hand, we present here all notions necessary for the understanding of the later parts: the greatest non-representable number , the number of non-representable numbers , and the extremal versions of these: max and max where On the other hand, our presentation, including 15 characteristic problems, is planned to serve also educational purposes. It can be used for teaching talented students; besides the solutions of the above-mentioned problems we include historical remarks, comment on new results and mention also open qestions.
Chapter 3 is devoted to be investigation of and . Our main result is finding the exact value of for two residue classes mod . This is a generalization of a theorem by ERDŐS and GRAHAM from 1972. Before the proof we present in detail some known upper and lower bounds for these numbers, including a theorem by DIXMIER from 1990, on which the proof of our result is based.
In Chapter 4 we prove that the extremal number is obtained when we choose the largest integers not exceeding . This was a conjecture by ERDŐS and GRAHAM from 1980. We also prove that for infinitely many values of and , the extremal value is achieved also for another set differing from the set of the greatest numbers up to .
In the last chapter we deal with the sum of powers of the non-representable numbers. First we illustrate, how analysis can be used for determining when . Then we discuss some special cases of a general result by RÖDSETH for higher powers. Finally we show a completely elementary method applicable not only to the previous cases, but also for some problems with 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.
The thesis is based on the following publications of the author:  (Chapter 2),  (Sections 3.3 - 3.4) and  (Section 4.2). (The results of Sections 5.4 - 5.5 are unpublished.)