Extensions of Some Parametric Families of D(16)-Triples

Let n be an integer. A set ofm positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. In this paper, we consider extensions of some parametric families of D(16)-triples. We prove that if {k − 4,k + 4,4k,d}, for k ≥ 5, is a D(16)-quadruple, then d = k3 − 4k. Furthermore, if {k− 4,4k,9k− 12}, for k > 5, is a D(16)-quadruple, then d = 9k3− 48k2 + 76k− 32. But for k = 5, this statement is not valid. Namely, the D(16)-triple {1,20,33} has exactly two extensions to a D(16)quadruple: {1,20,33,105} and {1,20,33,273}.