A characterization of the Frobenius problem and its application to arithmetic progressions

Abstract

In the Frobenius problem we are given a set of coprime, positive integers $a_1,a_2,...,a_k$, and are interested in the set of positive numbers NR that have no representation by the linear form ${\sum }_i{a}_i{x}_i$ in nonnegative integers $x_1,x_2,...,x_k$. We give a functional relationship that completely characterizes the set NR, and apply it to the case when the numbers are in an arithmetic progression.

10 pages