The Davenport constant of an interval: a proof that $\mathsf{D}=\chi$

TL;DR

This paper proves a conjecture determining the Davenport constant of an integer interval, showing it equals a specific value based on the interval bounds and a gcd condition, advancing understanding of zero-sum sequences.

Contribution

It provides a proof that the Davenport constant for an interval equals m+M minus a specific integer r defined by a gcd condition, confirming a conjecture.

Findings

Davenport constant equals m+M-r for the interval [−m,M]

r is the minimal sum of two non-negative integers with gcd condition

The conjecture is formally proven

Abstract

For two positive integers $m$ and $M$, we study the Davenport constant of the interval of integers $[\![ -m,M ]\!]$, that is the maximal length of a minimal zero-sum sequence composed of elements from $[\![ -m,M ]\!]$. We prove the conjecture that it is equal to $m+M- r$ where $r$ is the smallest integer which can be decomposed as a sum of two non-negative integers $t_1$ and $t_2$ ($r=t_1+t_2$) having the property that $\gcd (M-t_1, m-t_2)=1$.