An extension of the Frobenius coin-exchange problem

TL;DR

This paper extends the classical Frobenius coin-exchange problem by exploring the smallest integer beyond which every number is representable more than k times, focusing on the case of two integers.

Contribution

It introduces a new generalization of the Frobenius problem, analyzing the minimal integer where multiple representations occur more than k times for two numbers.

Findings

Derived properties of g_k(a,b) similar to classical Frobenius results

Established bounds and formulas for the generalized Frobenius number

Focused on the case of two integers, providing new insights

Abstract

Given positive integers $a_1,...,a_n$ with $\gcd(a_1,...,a_n) = 1$, 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).