Sums and differences of power-free numbers

TL;DR

This paper uses advanced sieve techniques to estimate solutions to equations involving sums and differences of numbers with different power-free properties, a novel approach for the first time applied to distinct powers.

Contribution

It introduces a generalized sieve method to analyze solutions involving sums and differences of numbers with different power-free exponents, extending prior work.

Findings

Established asymptotic estimates for solutions to a+b=n and a-b=n with k-free and l-free numbers

First study of this problem with distinct powers k and l

Demonstrated the effectiveness of generalized sieve techniques in this context

Abstract

We employ a generalised version of Heath-Brown's square sieve in order to establish an asymptotic estimate of the number of solutions $a, b \in \mathbb N$ to the equations $a+b=n$ and $a-b=n$, where $a$ is $k$-free and $b$ is $l$-free. This is the first time that this problem has been studied with distinct powers $k$ and $l$.