The Frobenius exchange problem on competitions and in classroom

Abstract

Let a_1, ..., a_n be relatively prime positive integers. The still unsolved Frobenius problem asks for the largest integer which cannot be represented as Σ x_i a_i with non-negative integers xi, and also for the number of non-representable positive integers. These and several related questions have been investigated by many prominent mathematicians, including Paul Erdős, and a wide range of partial results were obtained by various interesting methods differing both in character and difficulty. In this paper we give a self-contained introduction to this field through problems and comments suitable also for treatment in a class of talented students.