Unbounded discrepancy in Frobenius numbers

TL;DR

This paper investigates the properties of Frobenius numbers, demonstrating that the difference between certain largest integers with specific representation counts can be arbitrarily large, revealing unbounded discrepancies.

Contribution

It establishes the unbounded nature of differences in Frobenius numbers for fixed parameters and provides explicit examples for various dimensions.

Findings

g_0 - g_k is unbounded for n=5 and any k>0

Examples show g_0 > g_k for n≥6

g_0 > g_1 for n≥4

Abstract

Let g_j denote the largest integer that is represented exactly j times as a non-negative integer linear combination of { x_1, ... , x_n. We show that for any k > 0, and n = 5, the quantity g_0 - g_k is unbounded. Furthermore, we provide examples with g_0 > g_k for n >= 6 and g_0 > g_1 for n >= 4.