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Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(x_1g)cdotldotscdot(x_lg)$ where $gin G$ and $x_1, ldots, x_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(x_1+cdots+x_l)/ord(g)$ over all possible $gin G$ such that $langle g rangle =G$. Recently the second and the third authors determined the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime order where $S=g^2(x_2g)(x_3g)(x_4g)$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime power order. It is shown that if $G=langle grangle$ is a cyclic group of prime power order $n=p^mu$ with $p geq 7$ and $mugeq 2$, and $S=(x_1g)(x_2g)(x_2g)(x_3g)(x_4g)$ with $x_1=x_2$ is a minimal zero-sum sequence with $gcd(n,x_1,x_2,x_3,x_4,x_5)=1$, then $ind(S)=2$ if and only if $S=(mg)(mg)(mfrac{n-1}{2}g)(mfrac{n+3}{2}g)(m(n-3)g)$ where $m$ is a positive integer such that $gcd(m,n)=1$.
Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(n_1g)cdotldotscdot(n_lg)$ where $gin G$ and $n_1, ldots, n_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1
Let $p > 155$ be a prime and let $G$ be a cyclic group of order $p$. Let $S$ be a minimal zero-sum sequence with elements over $G$, i.e., the sum of elements in $S$ is zero, but no proper nontrivial subsequence of $S$ has sum zero. We call $S$ is uns
Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)cdotldotscdot(n_lg)$ where $gin G$ and $n_1, ldots, n_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1+cdots+n_l)/or
We study the maximal cross number $mathsf{K}(G)$ of a minimal zero-sum sequence and the maximal cross number $mathsf{k}(G)$ of a zero-sum free sequence over a finite abelian group $G$, defined by Krause and Zahlten. In the first part of this paper, w
In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are our main ingredients.