\(D(-1)\) tuples in imaginary quadratic fields
type: Item
instance of: scholarly article
title: \(D(-1)\) tuples in imaginary quadratic fields
zbMATH Open document ID: 1488.11077
author: Yanyan Li
published in: Acta Mathematica Hungarica
publication date: 2021-08-03T00:00:00Z
review text: This paper deals with a generalization of classical Diophantine \(m\)-tuples in the ring of integers of a number field. Let \(K\) be a number field, and let \(O_K\) be its ring of integers. A set \(\{a_1, \ldots, a_m\} \subset O_K \setminus \{0\}\) is said to be a Diophantine \(m\)-tuple in \(O_K\) with the property \(D(n)\), where \(n \in O_K\), if there exists \(x_{ij} \in O_K\) such that \(a_ia_j + n = x^2_{ij}\), for all \(1 \leq i < j \leq m\). In this paper, it is proved that in case where \(K\) is an imaginary quadratic number field, for any integer \(m > 36\) there does not exist any Diophantine \(m\)-tuple in \(O_K\) with the property \(D(-1)\).
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11J68
zbMATH DE Number: 7377607
zbMATH Keywords: Diophantine tuples
zbMATH Keywords: imaginary quadratic field
zbMATH Keywords: Pell equation
zbMATH Keywords: simultaneous approximation
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
full work available at URL: s10474-021-01150-w
OpenAlex ID: W3171508718
cites work: On the size of Diophantine m-tuples in imaginary quadratic number rings
cites work: On \(D(-1)\)-quadruples
cites work: THE EQUATIONS 3x2−2 = y2 AND 8x2−7 = z2
cites work: On the number of solutions of simultaneous Pell equations
cites work: COMPLETE SOLUTION OF A PROBLEM OF DIOPHANTUS AND EULER
cites work: Q4347183
cites work: There are only finitely many Diophantine quintuples
cites work: Effective solution of the D(-1)-quadruple conjecture
cites work: On Diophantine quintuples and \(D(-1)\)-quadruples
cites work: Q3069437
cites work: There is no Diophantine quintuple
cites work: A SYSTEM OF RELATIVE PELLIAN EQUATIONS AND A RELATED FAMILY OF RELATIVE THUE EQUATIONS
cites work: Explicit upper bound for an average number of divisors of quadratic polynomials
cites work: Correction to: ``Explicit upper bound for the average number of divisors of irreducible quadratic polynomials''
cites work: Bounds on the number of Diophantine quintuples
DOI: 10.1007/S10474-021-01150-W
description: scientific article; zbMATH DE number 7377607