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].

A D(4)-m-tuple is a set of m integers such that the product of any two of them increased by 4 is a perfect square. A problem of extendibility of D(4)-m-tuples is closely connected with the properties of elliptic curves associated with them. In this paper we prove that the torsion group of an elliptic curve associated with a D(4)-triple can be either Z/2Z × Z/2Z or Z/2Z × Z/6Z, except for the D(4)-triple {−1, 3, 4} when the torsion group is Z/2Z × Z/4Z.

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.

The problem of the extendibility of Diophantine triples is closely connected with the Mordell-Weil group of the associated elliptic curve. In this paper, we examine Diophantine triples {k − 1, k + 1, c l (k)} and prove that the torsion group of the associated curves is Z/2Z × Z/2Z for l = 3, 4 and l ≡ 1 or 2 (mod 4). Additionally, we prove that the rank is greater than or equal to 2 for all l ≥ 2. This represents an improvement of previous results by Dujella, Pethő and Najman, where cases k = 2 and l ≤ 3 were considered. 2010 AMS Mathematics subject classification. Primary 11G05. Keywords and phrases. Diophantine triples, elliptic curves, Mordell-Weil group.

Abstract. We find the number of elliptic curves with a cyclic isogeny of degree n over various number fields by studying the modular curves X 0 (n). We show that for n = 14, 15, 20, 21, 49 there exist infinitely many quartic fields K such that #Y 0 (n)(Q) = #Y 0 (n)(K) < ∞. In the case n = 27 we prove that there are infinitely many sextic fields such that #Y 0 (n)(Q) = #Y 0 (n)(K) < ∞.