There Are Infinitely Many Rational Diophantine Sextuples

Abstract

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].

On the size of Diophantine m-tuples in imaginary quadratic number rings

“…A long-standing conjecture, motivated by work of Baker and Davenport [1], that there is no Diophantine quintuple, was recently proven [7]. At a similar time, in [4], it was found that there are infinitely many Diophantine sextuples in rational numbers. Generalizing results from integers to rationals can be very hard.…”

A polynomial variant of a problem of Diophantus and its consequences

“…Let us mention that there exist a polynomial D(n)-sextuple in Z[X], where n is not a constant polynomial (see [19,20]). Moreover, in all those examples n is a square in Z[X], while for n ∈ Z[X] non-square, there exist examples of D(n)quintuples in Z[X] (see [14]).…”

Elliptic curves induced by Diophantine triples

“…Third curve. In a recent paper [9] it is shown the existence of infinitely many rational Diophantine sextuples. In the proof elliptic curves are used in order to extend a family of rational Diophantine triples to a family of sextuples.…”