Two local zero-sum problems

Abstract

In the present paper, we investigate two local zero-sum problems. Let $n,k\ge 2$. We denote by $\mathsf{D}^*(n,nk)$ (resp. $\eta^{*}(n,nk)$) the smallest positive integer $\ell$ (if exists) such that, from any given $\ell$ 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 $\mathsf{D}^*(n,nk)$ and $\eta^{*}(n,nk)$ are equal to $2n-1$ if $\mathrm{rad}(n) \mid \mathrm{rad}(k)$ and infinite otherwise. The corresponding inverse problem is also determined. We denote by $\mathsf{D}_n^{\times}$ (resp. $\eta_n^{\times}$) the smallest positive integer $\ell$ such that, from any given $\ell$ integers coprime to $n$, one can select some (resp. at most $n$) of them whose sum $\sigma$ satisfies $\gcd(\sigma, n^2)=n$. We prove that $\mathsf{D}_n^{\times}=\eta_n^{\times}=2n-1$ if $n$ is a prime power, and determine its inverse problem.