The $abc$ conjecture is true almost always

Abstract

Let $\mathrm{rad}(n)$ denote the product of distinct prime factors of an integer $n\ge 1$. The celebrated $abc$ conjecture asks whether every solution to the equation $a+b=c$ in triples of coprime integers $(a,b,c)$ must satisfy $\mathrm{rad}(abc)>K_{\epsilon }{c}^{1-\epsilon }$, for some constant $K_{\epsilon }>0$. 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 $(a,b,c)\in \{1,\dots ,N{\}}^3$ satisfying $a+b=c$ and $\mathrm{rad}(abc)<c^{1-\epsilon }$. 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.

4 pages. arXiv admin note: expository paper