On the largest element in [formula omitted]-quadruples

Abstract

Let n be a nonzero integer. A set of nonzero integers {a1,…,am} such that aiaj+n is a perfect square for all 1≤i<j≤m is called a D(n)-m-tuple. In this paper, we consider the question, for a given integer n which is not a perfect square, how large and how small can be the largest element in a D(n)-quadruple. We construct families of D(n)-quadruples in which the largest element is of order of magnitude |n|3, resp. |n|2∕5.