Browning–Verzobio: Sums of three powerful numbers

Full paper in Markdown.

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Full paper in Markdown.

Tim Browning, Matteo Verzobio, "Sums of three powerful numbers," arXiv:2608.24512 (2026).

Overview

Question and setup. Browning and Verzobio study primitive positive solutions of $a+b=c$ in which $a,b,c$ are respectively $p$-full, $q$-full, and $r$-full. Via $(a,b,c)\mapsto(a:c)$, these are Campana points on $(\mathbb P^1,\Delta_{p,q,r})$. Their counting function is

$$N(B)=\#\{(a,b,c)\in(\mathcal S_p\times\mathcal S_q\times\mathcal S_r)\cap[1,B]^3:\gcd(a,b,c)=1,\ a+b=c\},$$

as defined in (1.1). For $p\ge q\ge r$, the elementary estimate is $N(B)\ll B^{1/p+1/q}$, equation (1.3). The paper concentrates on obtaining a power saving, especially when the three exponents are comparable. The log-general-type condition is $1/p+1/q+1/r<1$, equation (1.2). Finiteness in that range is presented only as a consequence of cited Campana conjectures or of the $abc$ conjecture, not as a theorem of the paper.

Principal results. Theorem 1.1 proves that, for fixed integers $u\ge v\ge0$, with $p=r+u$ and $q=r+v$, all sufficiently large $r$ admit an explicit $\eta_{u,v}(r)>0$ such that

$$N(B)\ll_{\varepsilon,u,v,r}B^{1/p+1/q-\eta_{u,v}(r)+\varepsilon},\qquad \eta_{u,v}(r)=r^{-2}+O_{u,v}(r^{-5/2}).$$

This includes the diagonal case $p=q=r$ for sufficiently large $r$. Remark 4.5 separately shows that the method is nontrivial for every $r\ge2$ when $(p,q,r)=(r+2,r+1,r)$, and gives exact small-$r$ values of the relevant savings.

The main analytic theorem concerns the generalized Fermat surface

$$f(x,y,z)=a_1x^p+a_2y^q+a_3z^r=0$$

from (1.4), with primitive integral points in the rectangular box defined in (1.5). With $W=\exp(\sqrt{\log X\log Y/r})$, as in (1.6), Theorem 1.2 gives the coefficient-uniform estimate

$$\#S(X,Y,Z,f)\ll_{\varepsilon,p,q,r}(XYZ)^\varepsilon\left(W^2+W\max\{X,Y\}^{2/\sqrt{\max\{p,q,36\}}}+W\max\{X,Y\}^{1/r}\right).$$

The distinguished pair $(z,r)$ may be replaced by either of the other variable-exponent pairs. In particular, for $B=\max\{X,Y,Z\}$, the introduction records the coarser symmetric consequence

$$\#S(X,Y,Z,f)\ll B^{1/\sqrt p+1/\sqrt q+1/\sqrt r+\varepsilon}.$$

Remark 3.1 sharpens the middle exponent when $r\ge3$, replacing $\max\{p,q,36\}$ by $\max\{2(r-1)p/r,2(r-1)q/r,36\}$.

Slice estimate and proof mechanism. The key intermediate result is Theorem 2.1. For an irreducible plane curve $c(x,y)=0$ of degree $d$, it bounds the primitive points satisfying both $f=0$ and $c=0$. For $d=1,2$, equation (2.2) gives $\ll B^\varepsilon\max\{X,Y\}^{1/r}$; for $d\ge3$, equations (2.3)–(2.4) give

$$\ll \max\{X,Y\}^{2/\sqrt{\max\{p,q\}}+\varepsilon}+B^\varepsilon\max\{X,Y\}^{1/(dr)}.$$

The proof uses Capelli's criterion (Lemma 2.5) and the Brownawell–Masser function-field $abc$ theorem (Lemma 2.6) to show in Lemma 2.11 that reducibility of the induced polynomial $F(T)$ from (2.5) forces $\max\{p,q\}\le4d^2$. Lines and conics require separate arguments in Lemmas 2.12–2.16. In the irreducible case, Lemma 2.17 projects to an absolutely irreducible plane curve and applies a lopsided integral-point estimate; the reducible case is handled by Bombieri–Pila through Lemma 2.18.

Section 3 combines Theorem 2.1 with the Salberger determinant method. The surface points outside an exceptional set of size $\ll B^\varepsilon W^2$ lie on $\ll B^\varepsilon W$ auxiliary sections. Resultants reduce these sections to plane curves: Theorem 2.1 bounds the factors of degree below $D=\max\{p,q,r\}$, and a Binyamini–Cluckers–Novikov bound ([6, Theorem 2]) bounds those of degree at least $D$, yielding Theorem 1.2.

Transference to full numbers. Theorem 4.1 isolates the combinatorial input. It assumes a coefficient-uniform bound with exponent $\kappa$ for primitive solutions of $\lambda x^p+\mu y^q=\nu z^{r+k}$, uniformly over $0\le k\le r-1$, and converts it into a saving over (1.3), with every

$$\delta<\frac{1/p+1/q-(1/r-1/(qr)+\kappa/q)}{p+q+1+1/r}.$$

Its proof uses the unique factorization of every $m$-full integer as $v_0^m\prod_{s=1}^{m-1}v_s^{m+s}$, with the latter factors squarefree and pairwise coprime, followed by dyadic decomposition; the controlling inequalities are (4.2)–(4.7). Lemma 4.3 supplies a refined weighted optimization. Corollary 4.4 combines two permutations of Theorem 1.2 and gives the explicit bound

$$N(B)\ll_{\varepsilon,p}B^{1/p+1/q-\eta(p,q,r)+\varepsilon},$$

where $\eta=\max\{0,\eta_0,\eta_1\}$ and $\eta_0,\eta_1$ are defined immediately before the corollary. The final proof evaluates $\eta_1(r+u,r+v,r)$ asymptotically to obtain Theorem 1.1.

The results are upper bounds for primitive ternary additive relations and are uniform in the nonzero coefficients of the generalized Fermat equation. They neither establish the conjectural boundedness of $N(B)$ in the log-general-type range nor give asymptotics. The discussion of Darmon–Granville, Beukers, and genus-one cases in Section 1 is cited background concerning fixed coefficients, not part of the paper's new results.

Relation to E940

This source bears on Problem 940.

Write $R\ge3$ for the exponent in E940 and let $\mathcal P_R$ be the set of positive $R$-powerful integers. E940 asks whether infinitely many integers are not sums of at most $R$ elements of $\mathcal P_R$, and whether

$$E_R(X)=\#\{n\le X:n=a_1+\cdots+a_t\text{ for some }1\le t\le R,\,a_i\in\mathcal P_R\}$$

satisfies $E_R(X)=o(X)$.

On the diagonal $p=q=r=R$, the paper instead studies

$$N_R(B)=\#\{(a,b,c)\in\mathcal P_R^3\cap[1,B]^3:\gcd(a,b,c)=1,\ a+b=c\}.$$

Thus its “three powerful numbers” are the two summands and a powerful value of their sum. E940 permits an arbitrary target $n$, allows as many as $R$ summands, and imposes no primitivity condition. Even the diagonal form of Theorem 1.1, namely

$$N_R(B)\ll B^{2/R-\eta_R+\varepsilon},\qquad \eta_R=R^{-2}+O(R^{-5/2})$$

for sufficiently large $R$, controls only primitive representations whose sum is itself $R$-powerful. It therefore gives no upper bound for $E_R(X)$: removing the condition $c\in\mathcal P_R$ enlarges the target set drastically, and an upper bound for representation triples does not by itself exclude many integers having one representation. In particular, the paper does not address the density of sums of three cubes that forms the stated obstruction at $R=3$.