Table of Contents
ISRN Combinatorics
Volume 2013, Article ID 979487, 4 pages
http://dx.doi.org/10.1155/2013/979487
Research Article

Partitions of Natural Numbers with the Intersection Not Empty

School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, China

Received 14 May 2012; Accepted 19 July 2012

Academic Editors: A. Cossidente, M. Eliasi, L. Feng, and M. Gionfriddo

Copyright © 2013 Wen Yu and Min Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be the set of nonnegative integers. For a given set the representation functions , are defined as the number of solutions of the equation with condition , , respectively. In this paper, we prove that if and , then cannot hold for all sufficiently large integers where .

1. Introduction

Let be the set of nonnegative integers. For , let , , denote the number of solutions of respectively. Sárközy asked ever whether there exist two sets and of nonnegative integers with infinite symmetric difference, that is: for . As Dombi [1] has shown, the answer is negative for by the simple observation that is odd if and only if for some . For , Dombi presented a partition of the set of all positive integers into two disjoint subsets and such that for all sufficiently large . For , Chen and Wang [2] constructed sets such that for all integers , where is the set of all those positive integers such that the number of zeroes in the binary representation of is even, and is the set of all those positive integers such that the number of zeroes in the binary representation of is odd. For the other related results, the reader is referred to see [37].

It is natural to ask the following: for , are there subsets with for all large enough integers such that and ? Recently, the authors of this paper [8] obtained some results in this direction.

In this paper, we obtain the following results.

Theorem 1. If and , then cannot hold for all sufficiently large integers .

Theorem 2. If and , then cannot hold for all sufficiently large integers .

Remark 3. For , similarly, we can prove that if and , then cannot hold for all sufficiently large integers , where .

2. Proof of Theorem 1

Proof. Suppose that there exist integer and sets with and such that for all . Without loss of generality, we may assume that , is a positive integer; then there exists a polynomial of degree at most such that Let Then we have Moreover, Therefore,
Write By (3) and (7), we have Writing both sides of (9) in power series, then we have
Comparing the coefficients of on the both sides of (10), we have Then we have Since , we have . Then, by (12), has the following 30 cases: By (13), we have or Thus has the following 10 cases: By (14) and (25), we have which has the following 8 cases: By (15) and (26), we have which has the following 6 cases: By (16) and (27), we have which has the following 6 cases: By (17) and (28), we have which has the following 2 cases: By (18) and (29), we have which has the following 2 cases: By (19) and (30), we have which has the following 2 cases: By (20) and (31), we have which has the following 2 cases: By (21), we have or which contradicts (32).
This completes the proof of Theorem 1.

3. Proof of Theorem 2

Proof. Suppose that there exist integer and sets with and such that for all . Without loss of generality, we may assume that , is a positive integer then there exists a polynomial of degree at most such that
Define , and as in the proof of Theorem 1. Then we have Therefore,
Write By (35) and (37), we have We write both sides of (39) in power series; then we have
Comparing the coefficients of on the both sides of (40), we have
The remainder of the proof is the same as that of the proof of Theorem 1. We omit it here.

Acknowledgments

This work was supported by the National Natural Science Foundation of China, Grant no. 10901002 and Anhui Provincial Natural Science Foundation, Grant no. 1208085QA02.

References

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