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

TL;DR

The paper tackles counting square-full integers $u,v,w$ with $u+v=w$ and $w\le B$, improving the exponent in the upper bound to $n(B) \ll_\varepsilon B^{3/5-3/1555+\varepsilon}$, the first substantive improvement beyond the trivial $3/5$ barrier. It reduces the problem to bounding solutions of a ternary cubic form via a bi-homogeneous congruence modulo $y_3^3$, and it introduces a strong uniform bound for the cubic form counting function $\rho(B;a,b,c)$. The authors deploy a square-sieve, lattice coverings of the solution space, and sharp elliptic-curve rank bounds (via 3-descent) to control auxiliary counting problems, eventually combining all bounds to derive the stated exponent. The techniques illuminate the structure of square-full Diophantine solutions and may inspire further progress toward the predicted order of magnitude for primitive counts related to Campana orbifolds and rational points on related varieties.

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

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof : Proof of Theorem \ref{['cubic']}
• proof : Proof of Lemma \ref{['rank']}
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4