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For a nonzero integer $n$, a set of $m$ distinct nonzero integers$\{a_1,a_2,...,a_m\}$ such that $a_ia_j+n$ is a perfect square for all $1 \leqi < j \leq m$, is called a $D(n)$-$m$-tuple. In this paper, we show that thereinfinitely many essentially different quadruples which are simultaneously$D(n_1)$-quadruples and $D(n_2)$-quadruples with $n_1\neq n_2$.

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