On Diophantine triples containing a triangular number

TL;DR

This paper introduces a method to generate infinitely many families of Diophantine triples composed of triangular numbers, all sharing a common triangular number, expanding understanding of their structure.

Contribution

The paper presents a new construction technique for generating infinite families of Diophantine triples of triangular numbers with a shared element.

Findings

Constructed infinitely many $D(m^2)$-triples of triangular numbers.

All triples from the construction contain the same triangular number.

Provides a framework for further exploration of Diophantine triples with special properties.

Abstract

A general construction yielding infinitely many families of $D(m^2)$-triples of triangular numbers is presented. Moreover, each triple obtained from this construction contains the same triangular number $T_n$.