The abc Conjecture is True Almost Always

Let rad(n) denote the product of distinct prime factors of an integer n≥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 rad(abc)>Kε c1−ε, for some constant Kε>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 N2/3 many triples of coprime integers in a cube (a,b,c)∈{1,…,N}3 satisfying a+b=c and rad(abc)<c1−ε. 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 Bernert, Browning, Lichtman, and Teräväinen, who prove a refined estimate N3/5, giving the first power savings since 1962.