An extension of the Frobenius coin-exchange problem

Matthias Beck; Sinai Robins

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An extension of the Frobenius coin-exchange problem

Matthias Beck·Sinai Robins

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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).

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Matthias Beck, Sinai Robins. 2002-04-02. An extension of the Frobenius coin-exchange problem. https://arxiv.org/abs/math/0204037

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