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].

Let u = (u n ) ∞ n=0 be a Lucas sequence, that is a binary linear recurrence sequence of integers with initial terms u 0 = 0 and u 1 = 1. We show that if k is large enough then one can find k consecutive terms of u such that none of them is relatively prime to all the others. We even give the exact values g u and G u for each u such that the above property first holds with k = g u ; and that it holds for all k G u , respectively. We prove similar results for Lehmer sequences as well, and also a generalization for linear recurrence divisibility sequences of arbitrarily large order. On our way to prove our main results, we provide a positive answer to a question of Beukers from 1980, concerning the sums of the multiplicities of 1 and −1 values in non-degenerate Lucas sequences. Our results yield an extension of a problem of Pillai from integers to recurrence sequences, as well.

Abstract. Let f ∈ Z[X] be quadratic or cubic polynomial. We prove that there exists an integer G f ≥ 2 such that for every integer k ≥ G f one can find infinitely many integers n ≥ 0 with the property that none of f (n + 1), f (n + 2), . . . , f (n + k) is coprime to all the others. This extends previous results on linear polynomials and, in particular, on consecutive integers.

Abstract. We prove that for any elliptic divisibility sequence and any sufficiently large integer k, one can find k consecutive terms of the sequence such that none of these terms is coprime to all the others. In other words, elliptic divisibility sequences are Pillai sequences, named for a problem posed originally by Pillai for the sequence of integers. In fact we give an upper bound for the smallest value k0 past which this property is valid. We also provide a more general theorem where the coprimality condition is severely relaxed. In case of some particular sequences we give the values of k0, as well.