Collinear Interior Lattice Points in Triangles Satisfying $B(T)\in \{4,5\}$

Abstract

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.