Evaluating the Sharpness and Limitations of Bounds on the Frobenius Number

TL;DR

This paper analyzes various bounds on the Frobenius number, comparing their tightness, and proves that a significantly better general upper bound with sub-quadratic growth cannot exist, revealing fundamental limitations.

Contribution

It provides a comparative analysis of existing bounds on the Frobenius number and proves the impossibility of a universal sub-quadratic upper bound.

Findings

Several upper bounds are assessed for tightness.

A formal proof shows no sub-quadratic worst-case bound can exist.

Insights into structural properties of bounds are provided.

Abstract

In this paper we study the (classical) Frobenius problem, namely the problem of finding the largest integer that cannot be represented as a nonnegative integral combination of given relatively prime (strictly) positive integers (known as the Frobenius number). We firstly compare several upper bounds on the Frobenius number, assessing their relative tightness through both theoretical arguments and Monte Carlo simulations. We then explore whether a general upper bound with a worst-case exponent strictly less than quadratic can exist, and formally demonstrate that such an improvement is impossible. These findings offer new insights into the structural properties of established bounds and underscore inherent constraints for future refinement.