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

Abstract

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