paper

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

arXiv:1610.04415

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 $D (- 8 k^{2})$ -pair ${8 k^{2}, 8 k^{2} + 1}$ can be extended to at most a quadruple (the third and fourth element can only be $1$ and $32 k^{2} + 1$ ). At the end, we suggest considering a $D (- k^{2})$ -triple ${1, 2 k^{2}, 2 k^{2} + 2 k + 1}$ as possible future research direction.

partially presented at ELAZ 2016 conference (Strobl am Wolfgangsee, Austria)

On the extension of D(−8k2)-pair {8k2,8k2+1}

Abstract

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