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On the range of a covering function

Zhi-Wei Sun
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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.

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Zhi-Wei Sun. 2004-09-21. On the range of a covering function. https://arxiv.org/abs/math/0409279

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