On the range of a covering function

Abstract

Let {a_s(mod n_s)}_{s=1}^k (k>1) be a finite system of residue classes with the moduli n_1,...,n_k distinct. By means of algebraic integers we show that the range of the covering function w(x)=|{1\le s\le k: x=a_s (mod n_s)}| is not contained in any residue class with modulus greater one. In particular, the values of w(x) cannot have the same parity.

7 pages; to appear in J. Number Theory