Rational D(q)-quintuples

A PARAMETRIZATION OF RATIONAL D(q)-TRIPLES

“…Remark 4.2: Many constructions of D(q)-m-tuples started from a D(q)-pair and expanded it to a D(q)-triple using regularity. Examples are the construction of rational D(q)-quintuples by Dujella [5] and Draºi¢ [3], as well as the construction of strong D(q)-triples by Dujella, Paganin and Sadek [14]. The parametrizations we found may be used as a better starting point in similar constructions.…”

“…Regarding rational D(q)-n-tuples, Dujella [6] has shown that there are innitely many rational D(q)-quadruples for any q ∈ Q. Draºi¢ and Kazalicki [4] parametrized rational D(q)-quadruples (a, b, c, d) with a xed product of elements m = abcd, using triples of points on the elliptic curve E m : y 2 = x 3 +(4q 2 −2m)x 2 + m 2 x, and for each q ∈ Q they found all m ∈ Q such that there exists a rational D(q)-quadruple with product of elements equal to m. Dujella and Fuchs in [9] have shown that, assuming the Parity Conjecture, for innitely squarefree integers q = 1 there exist innitely many rational D(q)-quintuples, Draºi¢ improved their results in [3] by proving the same statement for a larger class of numbers q. There is no known rational D(q)-sextuple for q = a 2 , a ∈ Q.…”

Divisibility by 2 on quartic models of elliptic curves and rational Diophantine $D(q)$-quintuples

“…It was also used to show that there are infinitely many rational Diophantine sextuples, see [20]. In [8], it was proved that assuming the Parity Conjecture for the twists of several explicitly given elliptic curves, the density of rational numbers q for which there exist infinitely many rational D(q)-quintuples is at least 295026/296010 ≈ 99.5%.…”