ON A PROBLEM OF CHEN AND LEV. Issue 1 (28th November 2018)

• Record Type: Journal Article
• Authors: CHEN, SHI-QIANG; TANG, MIN; YANG, QUAN-HUI
• Is Part Of: Bulletin of the Australian Mathematical Society. Volume 99:Issue 1(2019)
• Page Start: 15
• Page End: 22
• Publication Date: 2018-11-28
• Subjects: primary 11B34, -- secondary 11B13; partition, -- representation function, -- characteristic function; Mathematics -- Societies, etc; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0004972718001107
• Languages: English
• ISSNs: 0004-9727

Abstract

Abstract : For a given set $S\subset \mathbb{N}$, $R_{S}(n)$ is the number of solutions of the equation $n=s+s^{\prime }, s<s^{\prime }, s, s^{\prime }\in S$ . Suppose that $m$ and $r$ are integers with $m>r\geq 0$ and that $A$ and $B$ are sets with $A\cup B=\mathbb{N}$ and $A\cap B=\{r+mk:k\in \mathbb{N}\}$ . We prove that if $R_{A}(n)=R_{B}(n)$ for all positive integers $n$, then there exists an integer $l\geq 1$ such that $r=2^{2l}-1$ and $m=2^{2l+1}-1$ . This solves a problem of Chen and Lev ['Integer sets with identical representation functions', Integers 16 (2016), A36] under the condition $m>r$ .