Diophantine quadruples in \(\mathbb{Z}[i][X]\)
type: Item
instance of: scholarly article
title: Diophantine quadruples in \(\mathbb{Z}[i][X]\)
zbMATH Open document ID: 1488.11076
author: Alan Filipin
author: Ana Jurasić
published in: Periodica Mathematica Hungarica
publication date: 2021-08-03T00:00:00Z
review text: ``A set consisting of \(m\) positive integers such that the product of any two of its distinct elements increased by \(1\) is a perfect square is called a Diophantine \(m\)-tuple. There is long history of finding such sets. One of the questions of interest, which various mathematicians try to solve, is how large those sets can be. Very recently, \textit{B. He} et al. [Trans. Am. Math. Soc. 371, No. 9, 6665--6709 (2019; Zbl 1430.11044)] proved the folklore conjecture that there cannot be \(5\) elements in Diophantine \(m\)-tuple, i.e. \(m < 5\).'' \par ``There are also many generalizations of the original problem, for example we can add a fixed integer \(n\) instead of \(1\) or consider the problem over domains other than \(\mathbb{Z}\) or \(\mathbb{Q}\).'' \par ``Definition. Let \(m \geq 2\) and let \(R\) be a commutative ring with unity. Let \(n \in R\) be a non-zero element and let \(\{a_1,\dots, a_m\}\) be a set of \(m\) distinct non-zero elements from \(R\) such that \(a_ia_j + n\) is a square of an element of \(R\) for \(1 \leq i < j \leq m\). The set \(\{a_1,\dots, a_m\}\) is called a Diophantine \(m\)-tuple with the property \(D(n)\) or simply a \(D(n)-m\)-tuple in \(R\).'' \par The authors consider a polynomial variant of the problem, with \(R = \mathbb{Z}[i][X]\) and \(n = 1\). The authors prove that every Diophantine quadruple in \(\mathbb{Z}[i][X]\) is regular (Theorem 1.5). More precisely, they prove that if \(\{a,b,c,d\}\) is a set of non-zero polynomials from \(\mathbb{Z}[i][X]\), not all constant, such that the product of any of its distinct elements increased by \(1\) is a square of a polynomial from \(\mathbb{Z}[i][X]\), then \((a+b-c-d)^2 = 4(ab+1)(cd+1)\). \par ``Since we do not have the relation ``\(<\)'' between the elements of \(\mathbb{Z}[i][X]\), we use the relation ``\(\leq\)'' between their degrees. In order to prove Theorem 1.5, we partially follow the strategies used in \textit{A. Dujella} and \textit{C. Fuchs} [J. Number Theory 106, No. …
reviewed by: Andrzej Dąbrowski
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11D45
zbMATH DE Number: 7377563
zbMATH Keywords: Diophantine \(m\)-tuples
zbMATH Keywords: polynomials
zbMATH Keywords: regular quadruples
MaRDI profile type: Publication
full work available at URL: s10998-020-00353-y
OpenAlex ID: W3043994161
cites work: On the size of Diophantine m-tuples in imaginary quadratic number rings
cites work: Q3208737
cites work: On the polynomial quadruples with the property \(D(-1;1)\)
cites work: Sets in Which xy + k is Always a Square
cites work: On the number of extensions of a Diophantine triple
cites work: Diophantine \(m\)-tuples and elliptic curves
cites work: Generalization of a problem of Diophantus
cites work: On the size of Diophantine \(m\)-tuples
cites work: Bounds for the size of sets with the property D(n)
cites work: There are only finitely many Diophantine quintuples
cites work: A polynomial variant of a problem of Diophantus and Euler
cites work: Complete solution of the polynomial version of a problem of Diophantus
cites work: A POLYNOMIAL VARIANT OF A PROBLEM OF DIOPHANTUS FOR PURE POWERS
cites work: Diophantine \(m\)-tuples for linear polynomials
cites work: Diophantine \(m\)-tuples for linear polynomials. II: Equal degrees
cites work: ON THE SIZE OF SETS IN A POLYNOMIAL VARIANT OF A PROBLEM OF DIOPHANTUS
cites work: On a problem of Diophantus with polynomials
cites work: Q5350067
cites work: A polynomial variant of a problem of Diophantus and its consequences
cites work: Some rational Diophantine sextuples
cites work: There is no Diophantine quintuple
cites work: A VARIATION ON A PROBLEM OF DAVENPORT AND DIOPHANTUS
DOI: 10.1007/S10998-020-00353-Y
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description: scientific article; zbMATH DE number 7377563