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

arXiv eprints
 Pub Date:
 October 2016
 arXiv:
 arXiv:1610.04415
 Bibcode:
 2016arXiv161004415A
 Keywords:

 Mathematics  Number Theory;
 11D09;
 11A99
 EPrint:
 partially presented at ELAZ 2016 conference (Strobl am Wolfgangsee, Austria)