Partitions of the set of nonnegative integers with the same representation functions

TL;DR

This paper investigates how to partition natural numbers into two sets that produce identical representation functions, solving a recent open problem in the field.

Contribution

It provides a complete solution to the problem of partitioning natural numbers into two sets with matching representation functions, addressing a question posed by Lev and Chen.

Findings

Identified all partitions of natural numbers with identical representation functions.

Provided a constructive method for such partitions.

Resolved a previously open problem in additive number theory.

Abstract

For a set of nonnegative integers $S$ let $R_{S}(n)$ denote the number of unordered representations of the integer $n$ as the sum of two different terms from $S$. In this paper we focus on partitions of the natural numbers into two sets affording identical representation functions. We solve a recent problem of Lev and Chen.