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Representation functions in the set of natural numbers
- Publication Year :
- 2023
-
Abstract
- Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s+s'$, $s, s'\in S$, $s<s'$. In this paper, we determine the structure of all sets $A$ and $B$ such that $A\cup B=\mathbb{N}\setminus\{r+mk:k\in\mathbb{N}\}$, $A\cap B=\emptyset$ and $R_{A}(n)=R_{B}(n)$ for every positive integer $n$, where $m$ and $r$ are two integers with $m\ge 2$ and $r\ge 0$.<br />Comment: 32 pages
- Subjects :
- Mathematics - Number Theory
11B34
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2312.17098
- Document Type :
- Working Paper