Sums and differences of power-free numbers

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