论文标题
具有相同表示功能的非负整数的分区
Partitions of nonnegative integers with identical representation functions
论文作者
论文摘要
令$ \ mathbb {n} $为所有非负整数的集合。对于任何整数$ r $和$ m $，让$ r+m \ mathbb {n} = \ {r+mk：k \ in \ mathbb {n} \} $。对于$ s \ subseteq \ mathbb {n} $和$ n \ in \ mathbb {n} $，让$ r_ {s}（n）$表示等式$ n = s+s'$的解决方案的$ n = s'$ with $ s，s in s $ in s $和$ s <s'$。令$ r_ {1}，r_ {2}，m $为$ 0 <r_ {1} <r_ {2} <m $和$ 2 \ mid r_ {1} $的整数。在本文中，我们证明存在两组$ c $和$ d $，带有$ c \ cup d = \ mathbb {n} $和$ c \ cap d =（r_ {1}+m \ mathbb {n}）对于所有$ n \ in \ mathbb {n} $，并且仅当存在一个正整数$ l $时，以便$ r_ {1} = 2^{2L+1} -2，r_ {2} = 2} = 2^{2l+1} -1} -1，m = 2^{2l+2} {2l+2} -2 $。
Let $\mathbb{N}$ be the set of all nonnegative integers. For any integer $r$ and $m$, let $r+m\mathbb{N}=\{r+mk: k\in\mathbb{N}\}$. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s<s'$. Let $r_{1}, r_{2}, m$ be integers with $0<r_{1}<r_{2}<m$ and $2\mid r_{1}$. In this paper, we prove that there exist two sets $C$ and $D$ with $C\cup D=\mathbb{N}$ and $C\cap D=(r_{1}+m\mathbb{N})\cup (r_{2}+m\mathbb{N})$ such that $R_{C}(n)=R_{D}(n)$ for all $n\in\mathbb{N}$ if and only if there exists a positive integer $l$ such that $r_{1}=2^{2l+1}-2, r_{2}=2^{2l+1}-1, m=2^{2l+2}-2$.
