Improved Bounds for the Index Conjecture in Zero-Sum Theory

Abstract

The Index Conjecture in zero-sum theory states that when $n$ is coprime to $6$ and $k$ equals $4$, every minimal zero-sum sequence of length $k$ modulo $n$ has index $1$. While other values of $(k,n)$ have been studied thoroughly in the last 30 years, it is only recently that the conjecture has been proven for $n>10^{20}$. In this paper, we prove that said upper bound can be reduced to $4.6\cdot 10^{13}$, and lower under certain coprimality conditions. Further, we verify the conjecture for $n<1.8\cdot 10^6$ through the application of High Performance Computing (HPC).