On Bialostocki's conjecture for zero-sum sequences

TL;DR

This paper confirms Bialostocki's conjecture for zero-sum sequences when the weights form an arithmetic progression with an even common difference, advancing understanding of permutation-based zero-sum properties.

Contribution

The paper proves Bialostocki's conjecture for a specific class of weights, namely arithmetic progressions with even common difference, which was previously unresolved.

Findings

Conjecture holds for weights forming an arithmetic progression with even common difference.

Provides new proof techniques for zero-sum sequence problems.

Enhances understanding of permutation sums in modular arithmetic.

Abstract

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