On certain consequences of the \((abc)\) conjecture in arithmetic and in logic
type: Item
instance of: scholarly article
title: On certain consequences of the \((abc)\) conjecture in arithmetic and in logic
zbMATH Open document ID: 0877.11004
author: Michel Langevin
published in: Rocky Mountain Journal of Mathematics
publication date: 1997-06-24T00:00:00Z
full work available at URL: contents.htm
review text: Assuming the correctness of the \(abc\)-conjecture of Oesterlé-Masser, the author shows how to deduce the following result: Let \(x,y,d,d'\) be positive integers with \((x,d) =(y,d') =1\), and \((x,d) \neq(y,d')\), and such that \(x+id\) and \(y+id'\) have the same prime factors for \(1\leq i\leq 5\). Then \((x,y,d,d')\) belongs to a finite set. This conditional result can be improved when \(d\) and \(d'\) are fixed integers. It follows that the Erdös-Woods question as to the existence of an integer \(k>2\) such that for given positive integers \(x\) and \(y\), if \(x+i\) and \(y+i\) have the same prime factors for \(1\leq i\leq 5\), then \(x=y\), can be answered in the affirmative under the assumption of the \(abc\)-conjecture.
Mathematics Subject Classification ID: 11A99
Mathematics Subject Classification ID: 11D99
Mathematics Subject Classification ID: 11U99
zbMATH DE Number: 1026401
zbMATH Keywords: \(abc\)-conjecture
zbMATH Keywords: integers
zbMATH Keywords: prime factors
reviewed by: Roelof J. Stroeker
MaRDI profile type: Publication
OpenAlex ID: W1997907093
Wikidata QID: Q123008494
DOI: 10.1216/RMJM/1181072035
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description: scientific article; zbMATH DE number 1026401