On the sum of two powered numbers

Abstract

Fix a positive real number $\theta$. The natural numbers $m$ with largest square-free divisor not exceeding $m^{\theta }$ form a set $\mathcal{A}$, say. It is shown that whenever $\theta >1/2$ then all large natural numbers $n$ are the sum of two elements of $\mathcal{A}$. This is nearly best possible.

3 pages. Submitted