An Extreme Family of Generalized Frobenius Numbers

Matthias Beck; Curtis Kifer

paper

An Extreme Family of Generalized Frobenius Numbers

arXiv:1005.2692

Abstract

We study a generalization of the \emph{Frobenius problem}: given $k$ positive relatively prime integers, what is the largest integer $g_{0}$ that cannot be represented as a nonnegative integral linear combination of these parameters? More generally, what is the largest integer $g_{s}$ that has exactly $s$ such representations? We illustrate a family of parameters, based on a recent paper by Tripathi, whose generalized Frobenius numbers $g_{0}, g_{1}, g_{2}, ...$ exhibit unnatural jumps; namely, $g_{0}, g_{1}, g_{k}, g_{(k - 1 k + 1)}, g_{(k - 1 k + 2)}, ...$ form an arithmetic progression, and any integer larger than $g_{(k - 1 k + j)}$ has at least $(k - k + j + 1)$ representations. Along the way, we introduce a variation of a generalized Frobenius number and prove some basic results about it.

arXiv comment5 pages, to appear in Integers: the Electronic Journal of Combinatorial Number Theory