OP-12065OpenNumber theory

Davenport constant for $C_n^3$ ( $C_{53}$ )

Davenport constant for $C_n^3$ : determine $C_{53}$ or improve its bounds; the repository records 3 (lower) and 4 (upper).

MSC 11P70 · Inverse problems of additive number theory, including sumsets

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓

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Problem statement

In zero-sum theory, the Davenport constant $D(G)$ of a finite abelian group $G$ is defined as the smallest integer $l\in\mathbb{N}$ such that every sequence $S$ over $G$ of length $\lvert S\rvert\ge l$ has a non-empty zero-sum subsequence. [GG2006-def-D]

For $n\ge 2$ , let $C_n$ denote the cyclic group of order $n$ , and write

C_n^3\ :=\ C_n\oplus C_n\oplus C_n.

[GG2006-def-Cn]

We define

C_{53}\ :=\ \sup_{n\ge 2}\ \frac{D(C_n^3)-1}{n-1},

the maximal normalized Davenport constant among the rank- $3$ groups $C_n^3$ .

The following explicit uniform inequality is obtained from the inductive method and known multi-wise Davenport estimates over elementary prime groups:

3(n-1)+1\ \le\ D(C_n^3)\ \le\ 4n-P(n)-2 \qquad (n\ge 2),

where

P(n)=\max_{p^a\parallel n}p^a

is the largest primary component of $n$ . Since $P(n)\ge 2$ , this gives

D(C_n^3)\le 4n-4=4(n-1),

and hence

C_{53}\le 4.

The previously recorded published bounds included

3(n-1)+1\ \le\ D(C_n^3)\ \le\ \min\{20369,\ 3^{\omega(n)}\}(n-1)+1,

where $\omega(n)$ is the number of distinct prime factors of $n$ . [Zak2019-omega-def] [Zak2019-cor3.11] Here, the $3^{\omega(n)}$ term is inherited from earlier published work, while the uniform numeric constant $20369$ is the explicit contribution in [Zak2019]. [Zak2019-prev-3omega] [CMMPT2012]

In particular,

3\ \le\ C_{53}\ \le\ 4.

A long-standing conjecture (in a stronger, pointwise form) predicts that for all $n\ge 2$ one has

D(C_n^3)\ =\ 3(n-1)+1,

equivalently $C_{53}=3$ . [GG2006-conj3.5] [GG2006-D-equals-1-plus-d] [GG2006-def-dstar]

One unconditional family of exact evaluations is given by prime powers: if $n$ is a prime power, then $C_n^3$ is a $p$ -group, and Theorem 3.1 implies $d(C_n^3)=d^\*(C_n^3)=3(n-1)$ and hence $D(C_n^3)=3(n-1)+1$ . [GG2006-thm3.1] [GG2006-D-equals-1-plus-d] [GG2006-def-dstar]

The general determination of $D(G)$ for rank- $3$ groups (and in particular the pointwise determination of $D(C_n^3)$ for all $n$ ) remains open. [Zak2019-open-rank3]

Published work before the 2019 preprint includes Gao (2000) on rank- $3$ groups and Chintamani--Moriya--Gao--Paul--Thangadurai (2012), which gives the bound $D(C_n^3)\le 3^{\omega(n)}(n-1)+1$ used in Corollary 3.11. [Zak2019-ref-Gao2000] [Zak2019-prev-3omega] [CMMPT2012] [Gao2000]

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Background

Zero-sumfree reformulation. If $d(G)$ denotes the maximal length of a zero-sumfree sequence over $G$ , then $D(G)=1+d(G)$ (Definition 2.1 in [GG2006]). In these terms, the conjecture for $C_n^3$ is $d(C_n^3)=3(n-1)$ . [GG2006-D-equals-1-plus-d] [GG2006-conj3.5]
Trivial lower bound. For $G\cong C_{n_1}\oplus\cdots\oplus C_{n_r}$ with $1<n_1\mid\cdots\mid n_r$ , the quantity $d^\*(G)=\sum_{i=1}^r(n_i-1)$ is the standard lower bound for $d(G)$ ; for $C_n^3$ this gives $d^\*(C_n^3)=3(n-1)$ and $D^\*(C_n^3)=3(n-1)+1$ . [GG2006-def-dstar]
Wikipedia page on the Davenport constant

References (from the repository page)

[GG2006] Gao, Weidong; Geroldinger, Alfred. Zero-sum problems in finite abelian groups: A survey. Expositiones Mathematicae 24 (2006), 337–369. DOI: https://doi.org/10.1016/j.exmath.2006.07.002. Publisher entry (DOI). Mirror PDF. Google Scholar
[GG2006-def-Cn] loc: Expositiones Mathematicae PDF p.2, Section 2 “Preliminaries” quote: “For $n\in\mathbb{N}$ , let $C_n$ denote a cyclic group with $n$ elements.”
[GG2006-def-dstar] loc: Expositiones Mathematicae PDF p.2, Section 2 “Preliminaries” quote: “ $d^\*(G)=\sum_{i=1}^r (n_i-1)$ .”
[GG2006-def-D] loc: Expositiones Mathematicae PDF p.4, Definition 2.1 quote: “the smallest integer $l\in\mathbb{N}$ such that every sequence $S$ over $G$ of length $|S|\ge l$ has a non-empty zero-sum subsequence.”
[GG2006-D-equals-1-plus-d] loc: Expositiones Mathematicae PDF p.4, Definition 2.1 quote: “ $1+d(G)=D(G)$ .”
[GG2006-thm3.1] loc: Expositiones Mathematicae PDF p.5, Theorem 3.1 quote: “If $G$ is a $p$ -group or $r(G)\le 2$ , then $d(G)=d^\*(G)$ .”
[GG2006-d-ge-dstar] loc: Expositiones Mathematicae PDF p.5, Section 3, just before Theorem 3.1 quote: “the crucial inequality $d(G)\ge d^\*(G)$ .”
[GG2006-conj3.5] loc: Expositiones Mathematicae PDF p.5, Section 3, Conjecture 3.5 quote: “If $G=C_n^r$ , where $n,r\in\mathbb{N}_{\ge 3}$ , or $r(G)=3$ , then $d(G)=d^\*(G)$ .”
[Zak2019] Zakarczemny, Maciej. Note on the Davenport’s constant for finite abelian groups with rank three. (2019). PDF: https://arxiv.org/pdf/1910.10984. DOI: https://doi.org/10.48550/arXiv.1910.10984. Google Scholar
[Zak2019-open-rank3] loc: arXiv v1 PDF p.1, Introduction quote: “The exact value of the Davenport constant for groups of rank three is still unknown and this is an open and well-studied problem.”
[Zak2019-omega-def] loc: arXiv v1 PDF p.5, Corollary 3.11 quote: “let $\omega(n)$ denote the number of distinct prime factors of $n$ .”
[Zak2019-cor3.11] loc: arXiv v1 PDF p.5, Corollary 3.11 (Eq. (17)) quote: “ $3(n-1)+1\le D(C_n^3)\le \min\{20369,3^{\omega(n)}\}(n-1)+1$ .”
[Zak2019-prev-3omega] loc: arXiv v1 PDF p.5, proof of Corollary 3.11 quote: “By [3, Theorem 1.2], we get $D(C_n^3)\le 3^{\omega(n)}(n-1)+1$ .”
[Zak2019-ref-Gao2000] loc: arXiv v1 PDF p.7, References [9] quote: “[9] W. D. Gao, On Davenport's constant of finite abelian groups with rank three, Discrete Mathematics 222 (2000), pages 111-124.”
[CMMPT2012] Chintamani, M. N.; Moriya, B. K.; Gao, W. D.; Paul, P.; Thangadurai, R. New upper bounds for the Davenport and for the Erdős--Ginzburg--Ziv constants. Archiv der Mathematik 98 (2012), no. 2, 133–142. DOI: https://doi.org/10.1007/s00013-011-0345-z. Google Scholar
[Gao2000] Gao, W. D. On Davenport's constant of finite abelian groups with rank three. Discrete Mathematics 222 (2000), no. 1--3, 111–124. DOI: https://doi.org/10.1016/S0012-365X(00)00010-8. Google Scholar
[Grinsztajn2026] Grinsztajn, Max. An upper bound for the Davenport constant of $C_n^3$ . Proof PDF: https://github.com/maaxgrin/davenport-cn3-bound/blob/main/davenport_cn3_bound.pdf

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 3; best known upper bound 4. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound	Reference	Comments
$20369$	[Zak2019]	From Corollary 3.11: $D(C_n^3)\le 20369(n-1)+1$ for all $n\ge 2$ , hence $C_{53}\le 20369$ . [Zak2019-cor3.11]
$4$	[Grinsztajn2026]	From the pointwise estimate $D(C_n^3)\le 4n-P(n)-2$ , where $P(n)=\max_{p^a\parallel n}p^a$ . Since $P(n)\ge2$ , this gives $C_{53}\le4$ .

Known lower bounds

Bound	Reference	Comments
$3$	[GG2006]	Using $d(C_n^3)\ge d^\*(C_n^3)=3(n-1)$ and $D(G)=1+d(G)$ gives $D(C_n^3)\ge 3(n-1)+1$ , hence $C_{53}\ge 3$ . [GG2006-d-ge-dstar] [GG2006-D-equals-1-plus-d] [GG2006-def-dstar]

References

[1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Davenport constant for $C_n^3$ ($C_{53}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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Source & evidence

https://teorth.github.io/optimizationproblems/constants/53a.html

Constant page from github.com/teorth/optimizationproblems at commit 2c1968c, Apache-2.0. The maintainers review submissions for plausibility and replicability but do not certify correctness; asterisked bounds have minimal verification.

Source date: Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differ
Imported: 2026-10-10

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Difficulty: HardResisted sustained effort by specialists.
Posed by: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers) and contributors
Posed: 2026
Source: Curated problem listOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOriginal statement ↗
Citation: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Davenport constant for $C_n^3$ ($C_{53}$)
Added: 5 Nov 2026
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Fields

Number theory