The $abc$ conjecture is true almost always

TL;DR

This paper demonstrates that almost all coprime triples satisfying $a+b=c$ in a large cube adhere to the $abc$ conjecture, with a classical estimate showing only a small subset violate it, and discusses recent improvements with power-savings.

Contribution

It provides an elementary, self-contained proof that almost all such triples satisfy the $abc$ conjecture, and contextualizes recent refined estimates with power-savings.

Findings

Most coprime triples satisfy the $abc$ conjecture in a quantitative sense.

Classical estimate bounds the number of triples violating the conjecture by $O(N^{2/3})$.

Recent work achieves a refined bound of $O(N^{33/50})$, the first power-savings since 1962.

Abstract

Let ${\rm rad}(n)$ denote the product of distinct prime factors of an integer $n\geq 1$. The celebrated $abc$ conjecture asks whether every solution to the equation $a+b=c$ in triples of coprime integers $(a,b,c)$ must satisfy ${\rm rad}(abc) > K_\varepsilon\, c^{1-\varepsilon}$, for some constant $K_\varepsilon>0$. In this expository note, we present a classical estimate of de Bruijn that implies almost all such triples satisfy the $abc$ conjecture, in a precise quantitative sense. Namely, there are at most $O(N^{2/3})$ many triples of coprime integers in a cube $(a,b,c)\in\{1,\ldots,N\}^3$ satisfying $a+b=c$ and ${\rm rad}(abc) < c^{1-\varepsilon}$. The proof is elementary and essentially self-contained. Beyond revisiting a classical argument for its own sake, this exposition is aimed to contextualize a new result of Browning, Lichtman, and Ter\"av\"ainen, who prove a refined estimate $O(N^{33/50})$, giving the first power-savings since 1962.