Integer polygons of given perimeter

TL;DR

This paper extends classical results on integer triangles to polygons with fixed and arbitrary numbers of sides, providing exact formulas and asymptotic estimates for the count of such polygons with a given perimeter.

Contribution

It generalizes Honsberger's classical triangle perimeter problem to polygons with any fixed number of sides and arbitrary polygons, offering exact and asymptotic formulas.

Findings

Number of integer m-gons with perimeter n approximates (2^{m-1}-m)/(2^m m!) * n^{m-1}

Number of polygons with arbitrary sides asymptotic to 2^{n-1}/n

Exact formulas for counting integer polygons of fixed side number

Abstract

A classical result of Honsberger states that the number of incongruent triangles with integer sides and perimeter $n$ is the nearest integer to $\frac{n^2}{48}$ ($n$ even) or $\frac{(n+3)^2}{48}$ ($n$ odd). We solve the analogous problem for $m$-gons (for arbitrary but fixed $m\geq3$), and for polygons (with arbitrary number of sides). We also show that the solution to the latter is asymptotic to $\frac{2^{n-1}}n$, and the former (for fixed $m$) to $\frac{2^{m-1}-m}{2^mm!}n^{m-1}$.