Counting Square-full Solutions to $x+y=z$

We show that there are $O(B^{3\mathrm{/}5-3\mathrm{/}1555+\text{ep}})$ triples $(x,y,z)$ of square-full integesr up to $B$ satisfying the equation $x+y=z$ for any fixed $\text{ep}>0$. This is the first improvement over the `easy’ exponent $3\mathrm{/}5$, given by Browning and Van Valckenborgh. One new tool is a strong uniform bound for the counting function for equations $a{X}^3+b{Y}^3=c{Z}^3$.