Doubly regular Diophantine quadruples
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Doubly regular Diophantine quadruples Andrej Dujella1
· Vinko Petriˇcevi´c1
Received: 4 February 2020 / Accepted: 12 August 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract For a nonzero integer n, a set of m distinct nonzero integers {a1 , a2 , . . . , am } such that ai a j +n is a perfect square for all 1 ≤ i < j ≤ m, is called a D(n)-m-tuple. In this paper, by using properties of so-called regular Diophantine m-tuples and certain family of elliptic curves, we show that there are infinitely many essentially different sets consisting of perfect squares which are simultaneously D(n 1 )-quadruples and D(n 2 )-quadruples with distinct nonzero squares n 1 and n 2 . Keywords Diophantine quadruples · Regular quadruples · Elliptic curves Mathematics Subject Classification Primary 11D09 · Secondary 11G05
1 Introduction For a nonzero integer n, a set of distinct nonzero integers {a1 , a2 , . . . , am } such that ai a j +n is a perfect square for all 1 ≤ i < j ≤ m, is called a Diophantine m-tuple with the property D(n) or D(n)-m-tuple. Sometimes it is convenient to allow that n = 0 in this definition. The D(1)m-tuples are called simply Diophantine m-tuples, and sets of nonzero rationals with the same property are called rational Diophantine m-tuples. The first rational Diophantine quadruple, 1 33 17 105 the set 16 , 16 , 4 , 16 , was found by Diophantus of Alexandria. By multiplying elements of this set by 16 we obtain the D(256)-quadruple {1, 33, 68, 105}. The first Diophantine quadruple, the set {1, 3, 8, 120}, was found by Fermat. In 1969, Baker and Davenport [2], proved that Fermat’s set cannot be extended to a Diophantine quintuple. Recently, He, Togbé and Ziegler proved that there are no Diophantine quintuples [14]. Euler proved that there are infinitely many rational Diophantine quintuples. The first example of a rational Diophantine sextuple, the set {11/192, 35/192, 155/27, 512/27, 1235/48, 180873/16}, was found by Gibbs [13], while Dujella, Kazalicki, Miki´c and Szikszai [7] recently proved that there are infinitely many rational Diophantine sextuples (see also [6,8,9]). It is not known whether there
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Andrej Dujella [email protected] Vinko Petriˇcevi´c [email protected]
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Department of Mathematics, Faculty of Science, University of Zagreb, Bijeniˇcka cesta 30, 10000 Zagreb, Croatia 0123456789().: V,-vol
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exists a rational Diophantine septuple. Gibbs’ example shows that there exists a D(2985984)sextuple. It is not known whether there exist a D(n)-septuple for some n = 0. Moreover, it is not known whether there exist a D(n)-sextuple for any n which is not a perfect square. For an overview of results on Diophantine m-tuples and its generalizations see [5]. In [15], Kihel and Kihel asked if there are Diophantine triples {a, b, c} which are D(n)triples for several distinct n’s. In [1], several infinite families of Diophantine triples were presented which are also D(n)-sets for two additional n’s. Furthermore, there are examples of Diophantine trip
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