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 ${\mathsf{D}}^{\ast }(n,nk)$ (resp. ${\eta }^{\ast }(n,nk)$) 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 ${\mathsf{D}}^{\ast }(n,nk)$ and ${\eta }^{\ast }(n,nk)$ 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.