Improved Bounds for the Index Conjecture in Zero-Sum Theory

2025/10/13 by Pendleton, Andrew
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.11976

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>10²⁰. In this paper, we prove that said upper bound can be reduced to 4.6⋅10¹³, and lower under certain coprimality conditions. Further, we verify the conjecture for n<1.8⋅10⁶ through the application of High Performance Computing (HPC).

Citations

• Solution to the index conjecture in zero-sum theory
• Note on the index conjecture in zero-sum theory and its connection to a Dedekind-type sum
• On the index of length four minimal zero-sum sequences
• On the index-conjecture on the length four minimal zero-sum sequences
• Heegaard Floer correction terms and Dedekind-Rademacher sums
• Minimal zero-sum sequences of length five over finite cyclic groups
• Approximate formulas for some functions of prime numbers

Related

• Solution to the index conjecture in zero-sum theory
• Note on the index conjecture in zero-sum theory and its connection to a Dedekind-type sum
• On the index-conjecture on the length four minimal zero-sum sequences
• On the index of length four minimal zero-sum sequences
• Further improvement on index bounds