On the largest element in D(n)-quadruples

Abstract

Let $n$ be a nonzero integer. A set of nonzero integers $\{a_1,\ldots,a_m\}$ such that $a_ia_j+n$ is a perfect square for all $1\leq i<j\leq m$ is called a $D(n)$-$m$-tuple. In this paper, we consider the question, for 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. $|n|^{2/5}$.