The abc conjecture is true almost always

2025/05/20 by Jared Duker Lichtman, Lichtman, Jared Duker · 1 citation
Computer Science · #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.2505.13991

Abstract

Let \rm 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 \rm rad(abc) > K_ε c^1-ε, for some constant K_ε>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)∈\1,…,N\³ satisfying a+b=c and \rm rad(abc) < c^1-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 Browning, Lichtman, and Teräväinen, who prove a refined estimate O(N^33/50), giving the first power-savings since 1962.

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