1. Introduction

The thesis investigates various aspects of an extensive number-theoreti-cal problem first raised by SYLVESTER [30] in 1884: Given a set $A=\{a_1 < a_2 < \ldots < a_n\}$ of relatively prime positive integers, which positive integers $K$ can be represented as $K=\sum _{i=1}^n x_ia_i$, where $x_i$ are non-negative integers. In Chapter 2 we introduce the notions and show the possibility of applications at school through a sequence of exercices. In Chapter 3, after presenting results on the greatest non-representable number and on the corresponding extremal version, our main result is a generalization of a theorem by ERDŐS and GRAHAM from 1972. In Chapter 4 we summarize the results on the number of the non-representable integers and we give a complete solution of the corresponding extremal problem. The last chapter deals with the sum of powers of the non-representable integers.