$D(4)$-数对$\{a, ka\}$的扩展，其中$k\in\{7,8,10,11,12,13\}$
Extensions of $D(4)$-pairs $\{a, ka\}$ with $k\in \{7,8,10,11,12,13\}$
AI总结：
该研究探讨了当$k\in\{7,8,10,11,12,13\}$时$D(4)$-数对$\{a,ka\}$的扩展问题，证明其可扩展为含依赖$a$的正整数族元素$c$的$D(4)$-三元组，且此类三元组可唯一扩展为$D(4)$-四元组。
摘要：
我们研究$D(4)$-数对$\{a,b\}$的可扩展性，其中$b=ka$且$k\in\{7,8,10,11,12,13\}$。首先，我们证明其可扩展为含元素$c$的$D(4)$-三元组，$c$是依赖于$a$的正整数族中的成员；接着，我们证明此类三元组可唯一扩展为$D(4)$-四元组。
英文摘要：
We study the extensibility of $D(4)$-pairs $\{a,b\}$, where $b = ka$ and $k \in \{7,8,10,11,12,13\}$. Firstly, we show that it can be extended to a $D(4)$-triple with an element c, which is a member of a family of positive integers depending on a. Then, we prove that such a triple has a unique extension to a $D(4)$-quadruple.
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。
