On the Diophantine pair \(\{a,3a\}\)
type: Item
instance of: scholarly article
title: On the Diophantine pair \(\{a,3a\}\)
zbMATH Open document ID: 1475.11045
author: Kouèssi Norbert Adédji
author: Bo He
author: Ákos Pintér
author: Alain Togbé
published in: Journal of Number Theory
publication date: 2021-07-05T00:00:00Z
review text: A set of positive integers is called Diophatine if the product of any two distinct elements is one less than a perfect square. It is known that if the set \(\{a,b,c,d\}\) is Diophantine and \(a<b\le 8a\), \(b<c<d\), \(d>4c(ab+1)\), then \(c\) must be one of four values given explicitly in terms of \(a\) and \(b\). By using standard tools in this kind of problems, the authors prove that if \(b=3a\) or \(b=8a\), then none of the admissible values for \(c\) can be attained. Therefore, any Diophantine quadruple that contains either \(\{a,3a\}\) or \(\{a,8a\}\) must be regular.
reviewed by: Mihai Cipu
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 7367590
zbMATH Keywords: Diophantine \(m\)-tuples
zbMATH Keywords: Pell equations
zbMATH Keywords: Baker's method
zbMATH Keywords: reduction method
Wikidata QID: Q114157061
MaRDI profile type: Publication
full work available at URL: j.jnt.2021.03.011
OpenAlex ID: W3153431523
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DOI: 10.1016/J.JNT.2021.03.011
description: scientific article; zbMATH DE number 7367590