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

## TL;DR

This paper proves the existence of infinitely many quadruples of integers that simultaneously satisfy the $D(n_1)$ and $D(n_2)$ properties for distinct nonzero integers $n_1$ and $n_2$, expanding the understanding of Diophantine tuples.

## Contribution

It demonstrates the infinite existence of quadruples that are simultaneously $D(n_1)$ and $D(n_2)$-quadruples with different $n$ values, a new result in Diophantine tuple theory.

## Key findings

- Existence of infinitely many such quadruples.
- Construction method for these quadruples.
- Distinctness of the $n$ values involved.

## Abstract

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 $1 \leq i < j \leq m$, 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 $n_1\neq n_2$.