Two local zero-sum problems

Gao Weidong; Jiang Xiao; Mu Yucen

number theory

Two local zero-sum problems

arXiv:2607.11313

summary

The paper determines the smallest number of integers needed to guarantee a subset whose sum is divisible by n but not by nk (or whose sum has a prescribed gcd with n²), showing this number is 2n‑1 under certain rad‑divisibility or prime‑power conditions and infinite otherwise, and also solves the corresponding inverse problems.

Abstract

In the present paper, we investigate two local zero-sum problems. Let $n, k \geq 2$ . We denote by $D^{*} (n, nk)$ (resp. $I^\cdot^{*} (n, nk)$ ) the smallest positive integer $ℓ$ (if exists) such that, from any given $ℓ$ integers not divisible by $n$ , one can select some (resp. at most $n$ ) of them whose sum is divisible by $n$ but not by $nk$ . We prove that both $D^{*} (n, nk)$ and $I^\cdot^{*} (n, nk)$ are equal to $2 n - 1$ if $rad (n) ∣ rad (k)$ and infinite otherwise. The corresponding inverse problem is also determined. We denote by $D_{n}^{\times}$ (resp. $I^\cdot_{n}^{\times}$ ) the smallest positive integer $ℓ$ such that, from any given $ℓ$ integers coprime to $n$ , one can select some (resp. at most $n$ ) of them whose sum $I ¨ $ satisfies $g cd (I ¨ , n^{2}) = n$ . We prove that $D_{n}^{\times} = I^\cdot_{n}^{\times} = 2 n - 1$ if $n$ is a prime power, and determine its inverse problem.

Topics & keywords

#zero-sum problems #additive combinatorics #Davenport constant #inverse problems #prime power integers #rad functionD* (n,nk)η* (n,nk)D_n^×η_n^×rad(n)rad(k)gcd conditionsubset suminfinite threshold