Square-Weighted Zero-Sum Constants

Let $A\subseteq \mathbb{Z}_n$ be a subset. A sequence $S=(x_1,\ldots,x_k)$ in $\mathbb{Z}_n$ is said to be an $A$

-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ such that $a_1x_1+\cdots+a_kx_k=0$ . By a square, we mean a non-zero square in $\mathbb{Z}_n$ . We determine the smallest natural number $k$

, such that every sequence in $\mathbb{Z}_n$ whose length is $k$

has a square-weighted zero-sum subsequence. We also determine the smallest natural number $k$

, such that every sequence in $\mathbb{Z}_n$ whose length is $k$

has a square-weighted zero-sum subsequence whose terms are consecutive terms of the given sequence.

Square-Weighted Zero-Sum Constants

Krishnendu Paul and Shameek Paul School of Mathematical Sciences Ramakrishna Mission Vivekananda Educational and Research Institute P.O. Belur Math, Dist. Howrah, West Bengal 711202 India

Krishnendu Paul and Shameek Paul
School of Mathematical Sciences
Ramakrishna Mission Vivekananda Educational and Research Institute
P.O. Belur Math, Dist. Howrah, West Bengal 711202
India