Improved bounds for the index conjecture in zero-sum theory

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).