Sums and differences of power-free numbers

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 $l$.