AI 中文总结
研究两个局部零和问题，对于给定不被\(n\)整除的整数及与\(n\)互质的整数，分别探讨选取部分整数使其和满足特定整除条件的最小个数。证明了在\(\mathrm{rad}(n)\mid\mathrm{rad}(k)\)等条件下相关最小个数的值，并确定了逆问题。
AI 中文摘要
在本文中，我们研究两个局部零和问题。设\(n,k\geq2\)。我们用\(\mathsf{D}^*(n,nk)\)（分别地，\(\eta^{*}(n,nk)\)）表示最小正整数\(\ell\)（若存在），使得从任意给定的\(\ell\)个不被\(n\)整除的整数中，能选出一些（分别地，至多\(n\)个）它们的和能被\(n\)整除但不被\(nk\)整除。我们证明若\(\mathrm{rad}(n)\mid\mathrm{rad}(k)\)，则\(\mathsf{D}^*(n,nk)\)和\(\eta^{*}(n,nk)\)都等于\(2n - 1\)，否则为无穷。还确定了相应的逆问题。我们用\(\mathsf{D}_n^{\times}\)（分别地，\(\eta_n^{\times}\)）表示最小正整数\(\ell\)，使得从任意给定的\(\ell\)个与\(n\)互质的整数中，能选出一些（分别地，至多\(n\)个）它们的和\(\sigma\)满足\(\gcd(\sigma,n^2)=n\)。我们证明若\(n\)是素数幂，则\(\mathsf{D}_n^{\times}=\eta_n^{\times}=2n - 1\)，并确定其逆问题。
英文摘要
In the present paper, we investigate two local zero-sum problems. Let $n,k\ge 2$. We denote by $\mathsf{D}^*(n,nk)$ (resp. $η^{*}(n,nk)$) 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 $η^{*}(n,nk)$ 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. $η_n^{\times}$) 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 $σ$ satisfies $\gcd(σ, n^2)=n$. We prove that $\mathsf{D}_n^{\times}=η_n^{\times}=2n-1$ if $n$ is a prime power, and determine its inverse problem.
