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

Abstract

Let n be an integer. A set of m 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}.