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A New Lower Bound in the $abc$ Conjecture

Curtis Bright

TL;DR

This work strengthens the known lower bounds in the abc conjecture by constructing infinitely many abc triples with a+b=c and c substantially larger than rad(abc). The authors develop a lattice-based framework on the odd-prime lattice and a full-rank kernel sublattice, translating short lattice vectors into extremal abc triples via Rankin-type bounds. By optimizing key parameters, they obtain a new exponent κ ≈ 6.563 in the lower bound c > rad(abc) exp(κ sqrt(log c)/log log c), improving on the previous κ=6.068 and highlighting how the lattice geometry of prime factors governs extremal abc behavior. The result depends on the δ-constant from Rankin (δ ≈ 3.65931) and underscores how advances in lattice theory could further tighten these lower bounds, with potential future improvements if δ is refined further.

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 $4\sqrt{2δ/e}$ where $δ$ 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 $n/δ$.

A New Lower Bound in the $abc$ Conjecture

TL;DR

This work strengthens the known lower bounds in the abc conjecture by constructing infinitely many abc triples with a+b=c and c substantially larger than rad(abc). The authors develop a lattice-based framework on the odd-prime lattice and a full-rank kernel sublattice, translating short lattice vectors into extremal abc triples via Rankin-type bounds. By optimizing key parameters, they obtain a new exponent κ ≈ 6.563 in the lower bound c > rad(abc) exp(κ sqrt(log c)/log log c), improving on the previous κ=6.068 and highlighting how the lattice geometry of prime factors governs extremal abc behavior. The result depends on the δ-constant from Rankin (δ ≈ 3.65931) and underscores how advances in lattice theory could further tighten these lower bounds, with potential future improvements if δ is refined further.

Abstract

We prove that there exist infinitely many coprime numbers , , with and . These are the most extremal examples currently known in the 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 in place of . We show that the constant may be replaced by where is a constant such that all full-rank unimodular lattices of sufficiently large dimension contain a nonzero vector with norm at most .
Paper Structure (9 sections, 13 theorems, 44 equations, 1 figure)

This paper contains 9 sections, 13 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The odd prime number lattice
  4. The kernel sublattice
  5. Hermite's constant
  6. A full-rank kernel sublattice
  7. Asymptotic formulae
  8. Exceptional $abc$ triples
  9. Optimal choice of $m$

Key Result

Lemma 2.1

$\lVert{\mathbold{x}}\rVert_1 = 2 \log h\lparen p/q\rparen$ where ${\mathbold{x}}=\sum_{i=1}^n e_i{\mathbold{b}}_i$ and $p/q=\prod_{i=1}^n p_i^{e_i}$ is expressed in lowest terms.

Figures (1)

  • Figure 1: Plots of $\lbrace[\rbrace-NoValue-]{\,(x,y){}:{}(x,y,z)\in L_{2,m}\,}$ for $1\leq m\leq 8$.

Theorems & Definitions (22)

  • Lemma 2.1
  • Proof 1
  • Lemma 2.2
  • Proof 2
  • Lemma 2.3
  • Corollary 2.4
  • Proof 3
  • Lemma 2.5
  • Proof 4
  • Lemma 2.6
  • ...and 12 more