The index conjecture for length-four minimal zero-sum sequences

About 12 years old · traced to 2014

Let $G$ be a finite cyclic group of order $|G|$ , and let $S$ be a sequence over $G$ . Write

S=(n_1g)\cdot\ldots\cdot(n_kg),

where $g$ generates $G$ , and define

\operatorname{ind}(S)=\min\left\{\frac{n_1+\cdots+n_k}{\operatorname{ord}(g)}: g\in G,\ \langle g\rangle=G\right\}.

A sequence is minimal zero-sum if it has sum zero and no proper nontrivial subsequence has sum zero. The index conjecture. If $\gcd(|G|,6)=1$ , then every minimal zero-sum sequence $S$ over $G$ of length $|S|=4$ satisfies

\operatorname{ind}(S)=1.

The paper proves the conjecture under the additional assumption that $S$ is reduced and at least one coefficient in a representation of $S$ is coprime to $|G|$ ; the general length-four case is not resolved in the supplied text.

References

Primary source

Li-meng Xia, “On the index-conjecture on the length four minimal zero-sum sequences”, arXiv:1401.7979 (2014).

Additional references

A Multiplicative Fourier Proof of the Length-Four Index Conjecture — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An unrefereed 2026 preprint claims to settle the conjecture for every eligible cyclic group, but independent verification is absent.

The conjecture concerns length-four minimal zero-sum sequences over cyclic groups of order $n$ with $(n,6)=1$ , asserting that each has index $1$ . No proposer is identified in the supplied sources.

Known results

Ge (2021) proved the conjecture for all sufficiently large $n$ .
Pendleton reduced the explicit large- $n$ threshold to $4.6\times 10^{13}$ .
Earlier work proved reduced cases, cases satisfying condition (A1), and groups whose order is a product of at most three prime factors.
Computation verified the conjecture for $n\le 1000$ , later extended to $n<1.8\times 10^6$ under additional conditions.

August 2026 claimed proof

Hongjian Li, Pingzhi Yuan, Shijie Yuan, and Weilin Zhang claim an all- $n$ proof using multiplicative Fourier analysis on $(\mathbb{Z}/n\mathbb{Z})^\times$ , removing the large- $n$ restriction. The claim appears only as an unrefereed preprint and has no independent verification or reported gap in the supplied sources.

Current status (as of August 2026): The conjecture is claimed for all $n$ with $(n,6)=1$ , while the new proof remains unverified and the claim is not yet settled.

Sources

No solutions have been posted yet.