A weak effective \(abc\)-conjecture
type: Item
instance of: scholarly article
title: A weak effective \(abc\)-conjecture
zbMATH Open document ID: 1241.11033
author: Jerzy Browkin
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2009-11-18T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1229696557
review text: For a triple \((a,b,c)\) of relatively prime positive integers satisfying \(a+b=c\) define \[ L=L(a,b,c)={\log c \over \log \text{rad}(a,b,c)}, \] where the radical \(\text{rad} (m)\) of a positive integer is the product of the distinct prime factors of \(m\). The \(abc\)--conjecture states that \(\limsup L(a,b,c)=1\), where \((a,b,c)\) runs over all triples as above. The \textit{weak effective \(abc(r)\)-conjecture} for a given real \(r\leq 1.5\) says that all triples \((a,b,c)\) satisfying \(L(a,b,c)>r\) are the presently calculated triples (given in a table of the paper). Applications are given for several classical Diophantine equations, for example for the Nagell-Ljunggren equation \((x^n-1)/(x-1)=y^q\) the author shows that the above conjecture for \(r=145/63)=2.3015\ldots\) implies that this equation has only the three presently known solutions, \((x,n,y,q)=(3,5,11,2)=(7,4,20,2)=(18,3,7,3)\).
reviewed by: Maurice Mignotte
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 5635540
zbMATH Keywords: Ljunggren-Nagell equation
zbMATH Keywords: Catalan equation
zbMATH Keywords: Fermat equation
zbMATH Keywords: Brocard-Ramanujan equation
zbMATH Keywords: Wieferich primes
MaRDI profile type: Publication
OpenAlex ID: W1972085732
Wikidata QID: Q123279299
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DOI: 10.7169/FACM/1229696557
description: scientific article; zbMATH DE number 5635540