A lower bound in the \(abc\) conjecture
type: Item
instance of: scholarly article
title: A lower bound in the \(abc\) conjecture
zbMATH Open document ID: 0998.11033
published in: Journal of Number Theory
publication date: 2002-10-07T00:00:00Z
review text: The \(abc\) conjecture is the following claim: Let \(a\), \(b\) and \(c\) be relatively prime rational integers such that \(a+b=c\), then, for all fixed \(c>0\), there exists a constant \(K\) such that \[ h(a,b,c) \leq r(a,b,c)+c h(a,b,c)+K, \] where \(h(a,b,c)=\max\{\log |a|, \log |b|, \log |c|\}\) and \(r(a,b,c) =\sum_{p \mid abc} \log p\), \(p\) prime. The author shows that there exist infinitely many \((a,b,c)\) as above such that \[ h(a,b,c) \geq r(a,b,c) + 4K_1 {\sqrt{h(a,b,c)}\over \log h(a,b,c)}, \] where \(K_1=1{.}517\). The proof uses sphere packing.
reviewed by: Maurice Mignotte
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11H31
zbMATH DE Number: 1538255
zbMATH Keywords: \(abc\) conjecture
zbMATH Keywords: lower bounds
zbMATH Keywords: sphere packing methods
Wikidata QID: Q123336390
author: Machiel van Frankenhuijsen
MaRDI profile type: Publication
full work available at URL: jnth.1999.2484
OpenAlex ID: W2082736755
cites work: Q4023809
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DOI: 10.1006/JNTH.1999.2484
description: scientific article; zbMATH DE number 1538255