Bounds on generalized Frobenius numbers

TL;DR

This paper establishes upper and lower bounds on the s-Frobenius number for any nonnegative integer s using Geometry of Numbers techniques, extending classical Frobenius number results.

Contribution

It introduces bounds on the s-Frobenius number for arbitrary s, generalizing the classical Frobenius problem with novel geometric methods.

Findings

Derived bounds for the s-Frobenius number.

Extended classical Frobenius results to the s-variant.

Applied Geometry of Numbers techniques to Frobenius problems.

Abstract

Let $N \geq 2$ and let $1 < a_1 < ... < a_N$ be relatively prime integers. The Frobenius number of this $N$-tuple is defined to be the largest positive integer that has no representation as $\sum_{i=1}^N a_i x_i$ where $x_1,...,x_N$ are non-negative integers. More generally, the $s$-Frobenius number is defined to be the largest positive integer that has precisely $s$ distinct representations like this. We use techniques from the Geometry of Numbers to give upper and lower bounds on the $s$-Frobenius number for any nonnegative integer $s$.