满足$B(T)\in\{4,5\}$的三角形中共线内部格点
Collinear Interior Lattice Points in Triangles Satisfying $B(T)\in\{4,5\}$
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中文总结 AI 辅助
本文基于前期$B(T)=3$的研究，结合Alder广义欧拉函数$g(k)$等，分类$B4$-、$B5$-共线整数，确定$B4$-共线整数为1、2、5，无$B5$-共线整数，揭示边界格点数量对共线性模式的影响。
中文摘要 AI 辅助
正整数$k$称为$Bn$-共线的，当存在至少一个具有$n$个边界点（$B(T)=n$）和$k$个内部格点的格点三角形，且所有此类三角形的内部点均共线。基于$B(T)=3$的前期研究，本文完全分类$B4$-和$B5$-共线整数。结合典范格点分类与Alder广义欧拉函数$g(k)$的算术性质，证明唯一的$B4$-共线整数为$k\in\{1,2,5\}$；进一步表明不存在$B5$-共线整数，确立结构对比：3、4个边界格点存在共线性约束，而5个边界点打破该模式。
英文摘要
A positive integer $k$ is called $Bn$-collinear if at least one lattice triangle with $n$ boundary points ($B(T)=n$) and $k$ interior lattice points exists, and every such triangle has all of its interior points collinear. Building on prior work on $B(T)=3$, we completely classify the $B4$- and $B5$-collinear integers. Using canonical lattice classifications together with arithmetic properties of Alder's generalized totient function $g(k)$, we prove that the only $B4$-collinear integers are $k\in\{1,2,5\}$. Furthermore, we show that no integer is $B5$-collinear. This establishes a structural contrast: while three and four boundary lattice points exhibit some collinearity constraints, five boundary points disrupt the pattern.
