Rational Diophantine sextuples containing two regular quadruples and one regular quintuple

Dujella, Andrej; Kazalicki, Matija; Petričević, Vinko

doi:10.32817/ams.1.1.2

Cited by 16 publications

(16 citation statements)

References 14 publications

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“…To increase the rank, we will use the parametrization of rational Diophantine triples due to Lasić ([23]) (see also [13]):…”

Section: Construction Of An Elliptic Curve With Rank 12mentioning

confidence: 99%

“…In 2019, Stoll ([28]) proved that extension of Fermat's set to a rational quintuple with the same property is unique. Gibbs ([20]), while Dujella, Kazalicki, Mikić and Szikszai ( [11]) recently proved that there are infinitely many rational Diophantine sextuples (see also [10,12,13]). For an overview of results on Diophantine m-tuples and its generalizations see [8].…”

Section: Introductionmentioning

confidence: 99%

“…where a63 = −13122w 16 3 − 7348320w 15 3 − 1570137696w 14 3 − 206172584064w 13 3 − 19541430237312w 12 3 − 1402008391816704w 11 3 − 77606011598363136w 10 3…”

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High rank elliptic curves induced by rational Diophantine triples

Dujella¹,

Peral²

2020

Glas. Mat. Ser. III

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A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over ℚ with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank ≥ 7, which are both the current records for that kind of curves.

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“…To increase the rank, we will use the parametrization of rational Diophantine triples due to Lasić ([23]) (see also [13]):…”

Section: Construction Of An Elliptic Curve With Rank 12mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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High rank elliptic curves induced by rational Diophantine triples

Dujella¹,

Peral²

2020

Glas. Mat. Ser. III

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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%

On elliptic curves induced by rational Diophantine quadruples

Dujella,

Soydan

2021

Preprint

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In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups Z/2Z × Z/kZ for k = 2, 4, 6, 8, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.

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“…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%

Rank zero elliptic curves induced by rational Diophantine triples

Dujella

Mikić

2020

Rad Hrvatske akademije znanosti i umjetnosti

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Rational Diophantine triples, i.e. rationals a, b, c with the property that ab+1, ac+1, bc+1 are perfect squares, are often used in construction of elliptic curves with high rank. In this paper, we consider the opposite problem and ask how small can be the rank of elliptic curves induced by rational Diophantine triples. It is easy to find rational Diophantine triples with elements with mixed signs which induce elliptic curves with rank 0. However, the problem of finding such examples of rational Diophantine triples with positive elements is much more challenging, and we will provide the first such known example.

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Rational Diophantine sextuples containing two regular quadruples and one regular quintuple

Cited by 16 publications

References 14 publications

High rank elliptic curves induced by rational Diophantine triples

High rank elliptic curves induced by rational Diophantine triples

On elliptic curves induced by rational Diophantine quadruples

Rank zero elliptic curves induced by rational Diophantine triples

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