On lattice triangles satisfying $\boldsymbol{B(T)=3}$ with collinear interior lattice points

TL;DR

The paper classifies integers $k$ for which every lattice triangle with $B(T)=3$ and $I(T)=k$ has its $k$ interior lattice points collinear (2-collinear integers). It develops a framework using integer unimodular shear transformations to reduce triangles to a standard form, then analyzes specific base cases and employs the Schemmel totient function to bound possible $k$ and rule out non-2-collinear cases. The main result is that the only 2-collinear integers are $1$, $2$, $4$, and $7$, with detailed proofs for these values and a general non-2-collinearity criterion for other $k$. The methods connect lattice-point geometry with number-theoretic tools, suggesting extensions to higher-dimensional lattice polygons and polytopes.

Abstract

A lattice point in $\mathbb{R}^2$ is a point $(x,y)$ with $x,y\in\mathbb{Z}$, and a lattice triangle is a triangle whose three vertices are all lattice points. We investigate the integers $k$ with the property that if $T$ is a lattice triangle with $3$ boundary points and $k$ points in the interior, then all $k$ boundary points must be collinear.

Theorems & Definitions (25)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Theorem 3.1
• proof