On the extension of $D(-8k^2)$-pair $\{8k^2, 8k^2+1\}$

TL;DR

This paper investigates the extension limits of a specific $D(-8k^2)$-pair, showing it can be extended to at most a quadruple, and proposes future research on related $D(-k^2)$-triples.

Contribution

The paper proves that the $D(-8k^2)$-pair $ extstyleigrace{8k^2, 8k^2+1}$ can be extended to at most a quadruple and introduces a new potential research direction.

Findings

The $D(-8k^2)$-pair can be extended to at most four elements.

The third and fourth elements in the extension are limited to specific values.

A new $D(-k^2)$-triple is proposed for future study.

Abstract

Let $n$ be a nonzero integer. A set of $m$ positive integers is called a $D(n)$-$m$-tuple if the product of any two of its distinct elements increased by $n$ is a perfect square. Let $k$ be a positive integer. By elementary means, we show that the $D(-8k^2)$-pair $\{8k^2, 8k^2+1\}$ can be extended to at most a quadruple (the third and fourth element can only be $1$ and $32k^2+1$). At the end, we suggest considering a $D(-k^2)$-triple $\{ 1, 2k^2, 2k^2+2k+1\}$ as possible future research direction.