The \(abc\)-conjecture
type: Item
instance of: scholarly article
title: The \(abc\)-conjecture
zbMATH Open document ID: 0794.11015
author: Serge Lang
published in: Elemente der Mathematik
publication date: 1994-09-08T00:00:00Z
full work available at URL: https://eudml.org/doc/141545
review text: The Masser-Oesterlé \(abc\) conjecture asserts that, given \(\varepsilon>0\), for all triples \(a\), \(b\), \(c\) of integers with \(a+b+c=0\) and \((a,b,c)=1\), we have \(\max( | a|,| b|,| c|)\ll_ \varepsilon \prod_{p\mid abc} p^{1+\varepsilon}\). First, Lang motivates this conjecture by stating and proving the analogue, Mason's theorem, in the function field case. He then shows how Mason's theorem and the \(abc\) conjecture (would) imply, respectively, an analogue of the Fermat conjecture over function fields, and the Fermat conjecture over number fields for sufficiently large exponent (``asymptotic Fermat''). The paper also discusses how the \(abc\) conjecture relates to a conjecture of M. Hall on \(| y^ 2- x^ 3|\) and to the generalized Szpiro conjecture.
reviewed by: Paul Vojta
Mathematics Subject Classification ID: 11D75
Mathematics Subject Classification ID: 11D04
zbMATH DE Number: 509048
zbMATH Keywords: Hall conjecture
zbMATH Keywords: Masser-Oesterlé \(abc\) conjecture
zbMATH Keywords: Mason's theorem
zbMATH Keywords: function field
zbMATH Keywords: Fermat conjecture over function fields
zbMATH Keywords: Fermat conjecture over number fields
zbMATH Keywords: Szpiro conjecture
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 509048