关于$D(-8k^2)$ -对$\{8k^2, 8k^2+1\}$的扩张
On the extension of $D(-8k^2)$-pair $\{8k^2, 8k^2+1\}$
摘要：
令$n$为非零整数。一组$m$正整数称为$D(n)$ - $m$ -元组，如果它的任何两个不同元素加上$n$的乘积是完全平方。令$k$为正整数。通过初等方法，我们证明了$D(-8k^2)$对$\{8k^2, 8k^2+1\}$最多可以扩展为四元组（第三和第四个元素只能是$1$和$32k^2+1$）.最后，我们建议考虑$D(-k^2)$ -三重$\{ 1, 2k^2, 2k^2+2k+1\}$作为未来可能的研究方向。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
