There are only finitely many \(D(4)\)-quintuples
type: Item
instance of: scholarly article
title: There are only finitely many \(D(4)\)-quintuples
zbMATH Open document ID: 1237.11014
DOI: 10.1216/RMJ-2011-41-6-1847
author: Alan Filipin
published in: Rocky Mountain Journal of Mathematics
publication date: 2012-01-12T00:00:00Z
review text: A \(D(n)\)-\(m\)-tuple is an \(m\)-tuple \((a_1,a_2,\dots,a_m)\in\mathbb Z^m\) such that \(a_ia_j+n=\square\) for all \(1\leq i < j\leq n\). The \(D(1)\)-\(m\)-tuples which are better known as Diophantine \(m\)-tuples are well studied and Dujella showed that there do not exist Diophantine sex-tuples and at most finitely many quintuples [\textit{A. Dujella}, J. Reine Angew. Math. 566, 183--214 (2004; Zbl 1037.11019)]. In this paper the author finds the explicit bound \[ \max\{a,b,c,d,e\}<10^{10^{28}} \] for a \(D(4)\)-quintuple \(\{a,b,c,d,e\}\). This result goes along the same lines as Fujita's bound for the number of \(D(1)\)-quintuples [\textit{Y. Fujita}, Glasn. Mat., III. Ser. 45, No. 1, 15--29 (2010; Zbl 1211.11040)]. In particular, the result is achieved by considering simultaneous Pell equations.
reviewed by: Volker Ziegler
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11D45
zbMATH DE Number: 5996727
zbMATH Keywords: Diophantine m-tuples
zbMATH Keywords: simultaneous Pell equations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5996727