# Triples which are $D(n)$-sets for several $n$'s

## TL;DR

This paper explores Diophantine triples that are simultaneously $D(n)$-sets for multiple distinct values of n, providing infinite families and analyzing their properties, thereby addressing a longstanding conjecture.

## Contribution

It constructs infinite families of Diophantine triples that are $D(n)$-sets for multiple distinct n values, challenging previous conjectures and expanding understanding of such sets.

## Key findings

- Existence of triples that are $D(n)$-sets for two or three distinct n values
- Construction methods for infinite families of such triples
- Counterexamples to earlier conjectures about $D(n)$-sets

## Abstract

For a nonzero integer $n$, a set of distinct nonzero integers $\{a_1,a_2,\ldots,a_m\}$ such that $a_ia_j+n$ is a perfect square for all $1\leq i<j\leq m$, is called a Diophantine $m$-tuple with the property $D(n)$ or simply $D(n)$-set. $D(1)$-sets are known as simply Diophantine $m$-tuples. Such sets were first studied by Diophantus of Alexandria, and since then by many authors. It is natural to ask if there exists a Diophantine $m$-tuple ($D(1)$-set) which is also a $D(n)$-set for some $n\neq 1$. This question was raised by Kihel and Kihel in 2001. They conjectured that there are no Diophantine triples which are also $D(n)$-sets for some $n\neq 1$. However, the conjecture does not hold, since, for example, $\{8, 21, 55\}$ is a $D(1)$ and $D(4321)$-triple, while $\{1, 8, 120\}$ is a $D(1)$ and $D(721)$-triple. We present several infinite families of Diophantine triples $\{a, b, c\}$ which are also $D(n)$-sets for two distinct $n$'s with $n\neq 1$, as well as some Diophantine triples which are also $D(n)$-sets for three distinct $n$'s with $n\neq 1$. We further consider some related questions.