Diophantine triples with largest two elements in common
type: Item
instance of: scholarly article
title: Diophantine triples with largest two elements in common
zbMATH Open document ID: 1474.11078
author: Mihai Cipu
author: Andrej Dujella
author: Yasutsugu Fujita
published in: Periodica Mathematica Hungarica
publication date: 2021-06-29T00:00:00Z
review text: A set of \(m\) positive integers \(\{a_{1},a_{2},\cdots, a_{m}\}\) is called a Diophantine \(m\)-tuple if \(a_{i}a_{j}+1\) is a perfect square for all \(i\) and \(j\) with \(1 \leq i <j \leq m\). \textit{A. Dujella} [J. Reine Angew. Math. 566, 183--214 (2004; Zbl 1037.11019)] proved that that there exist only finitely many Diophantine quintuples whereas there is no Diophantine sextuple. \textit{B. He} et al. [Trans. Am. Math. Soc. 371, No. 9, 6665--6709 (2019; Zbl 1430.11044)] proved that there does not exist a Diophantine quintuple. A strong conjecture asserting that all Diophantine quadruples are regular is still open. For Diophantine quadruples \(\{a,b,c,d\}\) with \(a<b<c<d\) containing various pairs \(\{a,b\}\) or triples \(\{a,b,c\},\) such as\par (i) \(\{k-1,k+1\}\) with \(k\geq 2\) an integer \par (ii) \(\{k,4k\pm4\}\) with \(k\) a positive integer \par (iii) \(\{k,a^{2}k\pm2a,(a+1)^{2}k\pm2(a+1)\}\) with \(a,k\) positive integers \par (iv) \(\{a,b,c\}\) with \(c \geq 200b^{4}\) \par it is known that \(d\) must be equal to \[a+b+c+2abc+2(\sqrt{ab+1})(\sqrt{ac+1})(\sqrt{bc+1}).\] In proving each of these results, firstly the conditions that \(ad+1=x^{2}, bd+1=y^{2}, cd+1=z^{2}\) for some positive integers \(x,y, z\) are to be transformed into a system of Pellian equations \(az^{2}-cx^{2}=a-c, bz^{2}-cy^{2}=b-c, \) and an upper bound for \(z\) is deduced by using Baker's method or hypergeometric method. In this paper, it is proved that if \(\{a,b,c\}\) is a Diophantine triple then \(\{a+1,b,c\}\) is not a Diophantine triple. It is also proved that if \(c<16b^{3}\) and \(\{a_{1},b,c\}\) and \(\{a_{2},b,c\}\) are Diophantine triples with \(a_{1}<a_{2}<b<c\), then \(\{a_{1},a_{2},b,c\}\) is a Diophantine quadruple. It is proved using the properties of regular Diophantine quadruples.
reviewed by: Ranjeet Sehmi
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11J68
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 7364769
zbMATH Keywords: diophantine \(m\)-tuples
zbMATH Keywords: Pellian equations
zbMATH Keywords: hypergeometric method
zbMATH Keywords: linear forms in logarithms
MaRDI profile type: Publication
full work available at URL: s10998-020-00331-4
OpenAlex ID: W3013920479
cites work: Q3208737
cites work: THE EQUATIONS 3x2−2 = y2 AND 8x2−7 = z2
cites work: On the number of solutions of simultaneous Pell equations
cites work: ON THE FAMILY OF DIOPHANTINE TRIPLES {k− 1,k+ 1, 16k3− 4k}
cites work: Bounds for Diophantine quintuples
cites work: Two-parameter families of uniquely extendable Diophantine triples
cites work: On the number of extensions of a Diophantine triple
cites work: An absolute bound for the size of Diophantine \(m\)-tuples
cites work: There are only finitely many Diophantine quintuples
cites work: Q4215652
cites work: The number of Diophantine quintuples II
cites work: The extendibility of Diophantine pairs. II: Examples
cites work: The extensibility of Diophantine pairs \(\{k - 1,k+1\}\)
cites work: A note on the regularity of the Diophantine pair \protect \lbrace k,4k\pm 4\protect \rbrace
cites work: On a family of Diophantine triples \(\{k,A^2k+2A,(A+1)^2k+2(A+1)\}\) with two parameters
cites work: On a family of Diophantine triples \(\{K,A^2K + 2A, (A + 1)^2 K + 2(A + 1)\}\) with two parameters. II
cites work: There is no Diophantine quintuple
cites work: Linear forms in two logarithms and interpolation determinants
cites work: Q5804241
cites work: Simultaneous rational approximations and related diophantine equations
DOI: 10.1007/S10998-020-00331-4
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description: scientific article; zbMATH DE number 7364769