The $abc$ conjecture is true almost always

TL;DR

This note analyzes the frequency with which coprime triples $(a,b,c)$ in $\{1,\dots,N\}^3$ with $a+b=c$ violate the abc conjecture inequality $\mathrm{rad}(abc) > K_\varepsilon\, c^{1-\varepsilon}$. It uses a classical de Bruijn-type radical bound to show that only $O(N^{2/3})$ of the $O(N^2)$ such triples satisfy $\mathrm{rad}(abc) < c^{1-\varepsilon}$, i.e., the abc conjecture holds for almost all triples in this setting. The argument is elementary and self-contained, and it sets the stage for a sharper bound $O(N^{33/50})$ due to Browning, Lichtman, and Teravainen from 2024, representing the first power-saving since 1962. Overall, the work frames the abc conjecture as almost-true in a quantitative, finite-volume sense and motivates deeper techniques for stronger savings.

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 Teräväinen, who prove a refined estimate $O(N^{33/50})$, giving the first power-savings since 1962.

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2: br
• Lemma 2.1: Divisor bound
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Remark 2.4