Cannonball Polygons with Multiplicities

TL;DR

This paper introduces a new class of polygons called cannonball polygons, generalizing the Cannonball Problem by associating integer-valued functions with these polygons and analyzing their properties for various multiplicities.

Contribution

It extends the Cannonball Problem to a broader setting using arithmetic functions and provides asymptotic formulas for the count of such polygons with higher multiplicities.

Findings

Existence of cannonball polygons with multiplicity 8 for any largest side length Z.

Asymptotic formulas for the number of classes of cannonball polygons with multiplicity s > 8.

Generalization of the Cannonball Problem through arithmetic functions and polygon association.

Abstract

We generalize the Cannonball Problem by introducing integer-valued and non-increasing arithmetic functions $w$. We associate these functions $w$ with certain polygons, which we call cannonball polygons. Through this correspondence, we show that for any $Z\in\mathbb{N}$, there exists a cannonball polygon with multiplicity 8 and largest side of length $Z$. Moreover, for any multiplicity $s$ greater than 8, we provide an asymptotic formula for the number of distinct classes of cannonball polygons with multiplicity $s$.