无平方因子奇数DCS的一个必要条件

Necessary Conditions for Distinct Covering Systems with Square-free Moduli

设A=｛as(ms):s=1, ,k｝为无平方因子奇数不同模覆盖系,设模的最小公倍数N=［m1,m2, ,mk］=p1p2 pn,其中p1<p2< <pn均为奇素数.本文利用覆盖系问题中的CELL方法证明了不同模覆盖系的一个必要条件.

Let A=｛a_s(m_s):s=1,…,k｝ be a finite system of arithmetic sequences that cover the integers where m_1,…,m_k and a_1,…,a_k∈Z. N is the least common multiple of the moduli.One fascinating problem on distinct covering systems (DCS) is if a DCS exist with all modulus odd In this paper we shall prove a necessary condition for A to be DCS consisting of odd square-free moduli.