On Bialostocki's conjecture for zero-sum sequences

Abstract

Let $n$ be a positive even integer, and let $a_1,...,a_n$ and $w_1,...,w_n$ be integers satisfying ${\sum }_{k=1}^n{a}_k\equiv {\sum }_{k=1}^n{w}_k=0(modn)$. A conjecture of Bialostocki states that there is a permutation $\sigma$ on {1,...,n} such that ${\sum }_{k=1}^n{w}_k{a}_{\sigma (k)}=0(modn)$. In this paper we confirm the conjecture when $w_1,...,w_n$ form an arithmetic progression with even common difference.

6 pages