A set of m distinct nonzero rationals {a1,a2,…,am} such that aiaj+1 is a perfect square for all 1 ≤ i < j ≤ m, is called a rational Diophantine m-tuple. It is proved recently that there are infinitely many rational Diophantine sextuples. In this paper, we construct infinite families of rational Diophantine sextuples with special structure, namely the sextuples containing quadruples and quintuples of certain type.

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, and in 2016 Dujella, Kazalicki, Mikić and Szikszai proved that there are infinitely many of them. In this paper, we prove that there exist infinitely many rational Diophantine sextuples such that the denominators of all the elements in the sextuples are perfect squares.

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We prove that there exist infinitely many rationals a, b and c with the property that a 2 − 1, b 2 − 1, c 2 − 1, ab − 1, ac − 1 and bc − 1 are all perfect squares. This provides a solution to a variant of the problem studied by Diophantus and Euler.