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

TL;DR

This paper improves the upper bound on the number of square-full solutions to the equation x + y = z up to B, using new bounds for related cubic equations, advancing understanding in additive number theory.

Contribution

It provides the first improvement over the trivial exponent 3/5 for counting square-full solutions to x + y = z, introducing a new uniform bound for cubic equations.

Findings

Established an upper bound of O(B^{3/5 - 3/1555 + ε}) for solutions

Introduced a strong uniform bound for cubic equation counting functions

First improvement over previous exponent for this problem

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 $aX^3+bY^3=cZ^3$.