An extension of the Frobenius coin-exchange problem

Matthias Beck; Sinai Robins

paper

An extension of the Frobenius coin-exchange problem

arXiv:math/0204037

Abstract

Given positive integers $a_{1}, ..., a_{n}$ with $g cd (a_{1}, ..., a_{n}) = 1$ , we call an integer t representable if there exist nonnegative integers $m_{1}, ..., m_{n}$ such that $t = m_{1} a_{1} + ... + m_{n} a_{n}$ . In this paper, we discuss the linear diophantine problem of Frobenius: namely, find the largest integer which is not representable. We call this largest integer the Frobenius number $g (a_{1}, ..., a_{n})$ . We extend this problem to asking for the smallest integer $g_{k} (a_{1}, ..., a_{d})$ beyond which every integer is represented more than k times. We concentrate on the case d=2 and prove statements about $g_{k} (a, b)$ similar in spirit to classical results known about g(a,b).

8 pages