On the torsion group of elliptic curves induced by <i>D</i>(4)-triples

Dujella, Andrej; Mikić, Miljen

doi:10.2478/auom-2014-0033

Cited by 10 publications

(4 citation statements)

References 14 publications

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“…It remains an open question whether or not there exists a Diophantine triple with all positive elements inducing an elliptic curve with torsion Z/2Z × Z/8Z and rank 0. A candidate for such triple is given in [10,22]:…”

Section: Torsion Z/2z × Z/8zmentioning

confidence: 99%

Elliptic curves induced by Diophantine triples

Dujella

Peral

2018

RACSAM

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Given a Diophantine triple {c 1 (t), c 2 (t), c 3 (t)}, the elliptic curve over Q(t) induced by this triple, i.e.can have as torsion group one of the non cyclic groups in Mazur's theorem, i.e. Z/2Z × Z/2Z, Z/2Z × Z/4Z, Z/2Z × Z/6Z or Z/2Z × Z/8Z. In this paper we present results concerning the rank over Q(t) of these curves improving in some of the cases the previously known results.2010 Mathematics Subject Classification. 11G05.

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Section: Torsion Z/2z × Z/8zmentioning

confidence: 99%

Elliptic curves induced by Diophantine triples

Dujella

Peral

2018

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“…It is an open problem whether this torsion group is possible for elliptic curves induced by an integer Diophantine triple (see e.g. [13,24]). On the other hand, examples of elliptic curves, induced by rational Diophantine triples, with torsion group Z/2Z × Z/6Z and rank equal to 1, 2, 3 and 4 can be found in [9] (for examples of elliptic curves with torsion groups Z/2Z×Z/4Z and Z/2Z×Z/8Z…”

Section: Explicit Formulasmentioning

confidence: 99%

There Are Infinitely Many Rational Diophantine Sextuples

Dujella

Kazalicki

Mikić

et al. 2016

Int Math Res Notices

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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. In this paper, we prove that there exist infinitely many rational Diophantine sextuples. set 1 16 , 33 16 , 17 4 , 105 16 found by Diophantus (see [4]). Euler found infinitely many rational Diophantine quintuples (see [20]), e.g. he was able to extend the integer Diophantine quadruple {1, 3, 8, 120} found by Fermat, to the rational quintuple 1, 3, 8, 120, 777480 8288641 . Let us note that Baker and Davenport [2] proved that Fermat's set cannot be extended to an integer Diophantine quintuple, while Dujella and Pethő [15] showed that there is no integer Diophantine quintuple which contains the pair {1, 3}. For results on the existence of infinitely many rational D(q)-quintuples, i.e. sets in which xy + q is always a square, for q = 1 see [12].

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“…But 4a > b > 10 5 , so a > 25000, which leads to a contradiction. The authors in [8,Lemma 1] show that c = a + b + 2r or c > ab in a D(4)-triple {a, b, c} with a < b < c. As in [3], to get the better bound on the number on quintuples, we will also consider the subcases ab < c ≤ a 2 b 2 and c > a Proof. The statement follows from [3] and the previous considerations.…”

Section: The Lower Bound On Bmentioning

confidence: 99%