For an integer n, a set of distinct nonzero integers \(\{a_1, a_2, ... , a_m\}\) such that \(a_i a_j + n\) is a perfect square for all \(1 \le i < j \le m\), is called a Diophantine m-tuple with the property D(n) or simply a D(n)-set. D(1)-sets are also called Diophantine m-tuples. The first Diophantine quadruple, the set \(\{1,3,8,120\}\) was found by Fermat. He, Togbé and Ziegler proved in 2019 that there does not exist a Diophantine quintuple. On the other hand, it is known that there exist infinitely many rational Diophantine sextuples. When considering D(n)-sets, usually an integer n is fixed in advance. However, we may ask if a set can have the property D(n) for several different n’s. For example, \(\{8,21,55\}\) is a D(1)-triple and D(4321)-triple. In joint work with Adžaga, Kreso and Tadić, we presented several families of Diophantine triples, which are D(n)-sets for two distinct n’s with \(n \ne 1\). In joint work with Petričević we proved that there are infinitely many (essentially different) quadruples which are simultaneously \(D(n_1)\)-quadruples and \(D(n_2)\)-quadruples with \(n_1 \ne n_2\). Moreover, the elements in some of these quadruples are squares, so they are also D(0)-quadruples. E.g. \(\{54^2, 100^2, 168^2, 364^2\}\) is a \(D(8190^2)\), \(D(40320^2)\) and D(0)-quadruple. In recent joint work with Kazalicki and Petričević, we considered D(n)-quintuples with square elements (so they are also D(0)-quintuples). We proved that there are infinitely many such quintuples. One example is a \(D(480480^2)\)-quintuple \(\{225^2, 286^2, 819^2, 1408^2, 2548^2\}\). In this survey paper, we describe methods used in constructions of mentioned triples, quadruples and quintuples.
