2019
DOI: 10.1016/j.jnt.2019.06.006
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Rational Diophantine sextuples with square denominators

Abstract: A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple, and in 2016 Dujella, Kazalicki, Mikić and Szikszai proved that there are infinitely many of them. In this paper, we prove that there exist infinitely m… Show more

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Cited by 19 publications
(9 citation statements)
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“…Recently, Stoll [31] proved that Euler's extension of Fermat's quadruple to the rational quintuple is unique. In 1999, Gibbs (see [25]) found the first rational sextuple { 11 192 , 35 192 , 155 27 , 512 27 , 1235 48 , 180873 16 }, while in 2017, Dujella, Kazalicki, Mikić and Szikszai [15] proved that there exist infinitely many rational Diophantine sextuples (see also [14,16,17]). For surveys of results and conjectures concerning Diophantine m-tuples and their generalizations see [10] and [11,Sections 14.6,16.7].…”
Section: Introductionmentioning
confidence: 99%
“…Recently, Stoll [31] proved that Euler's extension of Fermat's quadruple to the rational quintuple is unique. In 1999, Gibbs (see [25]) found the first rational sextuple { 11 192 , 35 192 , 155 27 , 512 27 , 1235 48 , 180873 16 }, while in 2017, Dujella, Kazalicki, Mikić and Szikszai [15] proved that there exist infinitely many rational Diophantine sextuples (see also [14,16,17]). For surveys of results and conjectures concerning Diophantine m-tuples and their generalizations see [10] and [11,Sections 14.6,16.7].…”
Section: Introductionmentioning
confidence: 99%
“…Recently, He, Togbé and Ziegler [HTZ19] building upon the work of Dujella [Duj04] proved that there does not exist an integer Diophantine quintuple. On the other hand, it was shown in [DKMS17] that there are infinitely many rational Diophantine sextuples (for more constructions see [DK17], [DKP20] and [DKP19]), and it is not known if there are rational Diophantine septuples. For a short survey on Diophantine m-tuples see [Duj16].…”
Section: Introductionmentioning
confidence: 99%
“…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 [23], while Dujella, Kazalicki, Mikić and Szikszai [14] recently proved that there are infinitely many rational Diophantine sextuples (see also [13,15,16]). It is not known whether there exists any rational Diophantine septuple.…”
Section: Introductionmentioning
confidence: 99%