Partitions of natural numbers with the same representation functions
type: Item
instance of: scholarly article
title: Partitions of natural numbers with the same representation functions
zbMATH Open document ID: 1239.11016
author: Min Tang
author: Yonggao Chen
published in: Journal of Number Theory
publication date: 2009-09-15T00:00:00Z
review text: Let \(\mathbb{N}\) be the set of nonnegative integers. For a set \(A \subseteq \mathbb{N}\) let \(R_{2}(A, n)\) and \(R_{3}(A, n)\) be the number of representations of \(n\) as the sum of two terms \(a\), \(b\) from the set \(A\) with \(a < b\), \(a \leq b\) respectively. The authors prove the following theorems about the above functions. If \(R_{3}(A, n) = R_{3}(\mathbb{N} \setminus A, n)\) for all large enough \(n\), then \(R_{3}(A, n) = R_{3}(\mathbb{N} \setminus A, n) \geq 1\) for all large enough \(n\). They also proved that under the same conditions the same is true for the representation function \(R_{2}(A, n)\) except a few sets. The proofs are elementary and the paper contains some related open problems.
Mathematics Subject Classification ID: 11B34
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 5603994
zbMATH Keywords: partition
zbMATH Keywords: representation function
zbMATH Keywords: Erdős - Turán conjecture
zbMATH Keywords: binary representation
reviewed by: Sándor Kiss
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.04.020
OpenAlex ID: W1990813864
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DOI: 10.1016/J.JNT.2009.04.020
description: scientific article; zbMATH DE number 5603994