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 $aX^3+bY^3=cZ^3$.
Fields of Study & Subjects
Analytic Number Theory Research
Algebraic Geometry and Number Theory
Limits and Structures in Graph Theory
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 12 January 2026
- Document type
- Preprint