$D(-1)$-triples of triangular numbers

TL;DR

This paper investigates special triples of triangular numbers where the product of any two minus one is a perfect square, establishing conditions for membership and proving infinite such triples exist.

Contribution

It provides a necessary condition for triangular numbers to be part of these triples and proves the infinitude of such triples involving any given triangular number.

Findings

Necessary condition for T_n to be in such a pair

Existence of infinitely many D(-1)-triples containing T_n

Abstract

We study pairs and triples consisting of triangular numbers such that the product of any two distinct elements decreased by 1 is a perfect square. For a positive integer $n$, we establish a necessary condition for the $n$-th triangular number $T_n$ to be a member of such a pair, and we prove that any such $T_n$ is also a member of infinitely many $D(-1)$-triples.