Diophantine triples with the property $D(n)$ for distinct $n$

TL;DR

This paper demonstrates the existence of infinitely many Diophantine triples with property D(n) for various n, including complex integers, showing rich multiplicity and diversity in such number sets.

Contribution

It proves the existence of infinitely many D(n)-triples for each integer n and extends results to Gaussian integers with multiple n-values.

Findings

Infinitely many D(n)-triples exist for each integer n.

Existence of infinitely many D(-1)-triples in Gaussian integers with multiple n-values.

Such triples are not equivalent to any with D(1) property.

Abstract

We prove that for every integer $n$, there exist infinitely many $D(n)$-triples which are also $D(t)$-triples for $t\in\mathbb{Z}$ with $n\ne t$. We also prove that there are infinitely many triples with the property $D(-1)$ in $\mathbb{Z}[i]$ which are also $D(n)$-triple in $\mathbb{Z}[i]$ for two distinct $n$'s other than $n = -1$ and these triples are not equivalent to any triple with the property $D(1)$.