A New Lower Bound in the $abc$ Conjecture

TL;DR

This paper establishes a new lower bound in the $abc$ conjecture by proving the existence of infinitely many extremal coprime triples with a larger bound than previously known, refining the understanding of the conjecture's limits.

Contribution

The paper introduces a new lower bound constant in the $abc$ conjecture, improving upon previous results and linking it to lattice theory and unimodular lattices.

Findings

Existence of infinitely many coprime triples with $a+b=c$ exceeding previous bounds.

New lower bound constant $6.563$ in the $abc$ conjecture.

Connection between the bound and properties of unimodular lattices.

Abstract

We prove that there exist infinitely many coprime numbers $a$, $b$, $c$ with $a+b=c$ and $c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$. These are the most extremal examples currently known in the $abc$ conjecture, thereby providing a new lower bound on the tightest possible form of the conjecture. This builds on work of van Frankenhuysen (1999) who proved the existence of examples satisfying the above bound with the constant $6.068$ in place of $6.563$. We show that the constant $6.563$ may be replaced by $4\sqrt{2\delta/e}$ where $\delta$ is a constant such that all full-rank unimodular lattices of sufficiently large dimension $n$ contain a nonzero vector with $\ell_1$ norm at most $n/\delta$.