To count clean triangles we count on $imph(n)$

TL;DR

This paper introduces a new function, imph(n), to count clean lattice triangles of a given area up to unimodular equivalence, advancing combinatorial geometry methods.

Contribution

It develops the imph(n) function and applies it to enumerate clean lattice triangles, providing a novel approach in lattice geometry.

Findings

Derived formulas for counting clean triangles using imph(n)

Established connections between imph(n) and Euler's phi function

Provided enumeration results for specific areas

Abstract

A clean lattice triangle in ${\mathbb R}^2$ is a triangle that does not contain any lattice points on its sides other than its vertices. The central goal of this paper is to count the number of clean triangles of a given area up to unimodular equivalence. In doing so we use a variant of the Euler phi function which we call $imph(n)$ (imitation phi).