Lattice Equable Quadrilaterals I -- Parallelograms

TL;DR

This paper classifies lattice equable parallelograms, revealing their side length constraints and infinite families, especially focusing on Pythagorean cases linked to Pell-like equations.

Contribution

It establishes the possible gcds of side lengths and characterizes infinite families of Pythagorean equable parallelograms using Pell-like equations.

Findings

gcd of side lengths can only be 3, 4, or 5

shortest side can only be 3, 4, 5, 6, or 10

five infinite families of Pythagorean equable parallelograms

Abstract

This paper studies equable parallelograms whose vertices lie on the integer lattice. Using Rosenberger's Theorem on generalised Markov equations, we show that the g.c.d. of the side lengths of such parallelograms can only be 3, 4 or 5, and in each of these cases the set of parallelograms naturally forms an infinite tree all of whose vertices have degree 4, bar the root. The paper then focuses on what we call Pythagorean equable parallelograms. These are lattice equable parallelograms whose complement in a circumscribing rectangle consists of two Pythagorean triangles. We prove that for these parallelograms the shortest side can only be 3, 4, 5, 6 or 10, and there are five infinite families of such parallelograms, given by solutions to corresponding Pell-like equations.