Solution to the index conjecture in zero-sum theory

TL;DR

This paper proves an important case of the index conjecture in zero-sum theory, showing that for sufficiently large n coprime to 6, all minimal zero-sum sequences of length 4 have index 1.

Contribution

The paper establishes the index conjecture for all sufficiently large n coprime to 6, resolving a longstanding open problem in zero-sum theory.

Findings

Confirmed the index conjecture for all n > N coprime to 6

Provided an explicit bound N beyond which the conjecture holds

Advanced understanding of minimal zero-sum sequences in modular arithmetic

Abstract

A problem in zero-sum theory is to determine all pairs $(k,n)$ for which every minimal zero-sum sequence of length $k$ modulo $n$ has index $1$. While all other cases have been solved more than a decade ago, the case when $k$ equals $4$ and $n$ is coprime to $6$ remains open. Precisely, The Index Conjecture in this subject states that if $n$ is coprime to $6$ then every minimal zero-sum sequence of length $4$ modulo $n$ has index $1$. In this paper, we prove an equivalent version of this conjecture for all $n>N$ for some absolute constant $N$.