Diophantine quadruples with the properties $D(n_1)$ and $D(n_2)$

Abstract

For a nonzero integer $n$, a set of $m$ 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 $D(n)$-$m$-tuple. In this paper, we show that there infinitely many essentially different quadruples which are simultaneously $D(n_1)$-quadruples and $D(n_2)$-quadruples with $n_1\ne n_2$.

8 pages