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

Abstract

For a nonzero integer $n$, a set of distinct nonzero integers $\{a_1,a_2,\dots ,a_m\}$ such that $a_i{a}_j+n$ is a perfect square for all $1\le i<j\le m$, 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 $n\ne 1$. 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 $n\ne 1$. 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 $n\ne 1$, as well as some Diophantine triples which are also $D(n)$-sets for three distinct $n$'s with $n\ne 1$. We further consider some related questions.