Universal sums of $m$-gonal numbers

TL;DR

This paper investigates universal quadratic polynomials formed by sums of polygonal numbers, establishing bounds on the minimal sets needed to ensure all natural numbers are represented.

Contribution

It provides an asymptotic upper bound on the size of sets that guarantee universality of sums of m-gonal numbers.

Findings

Derived an asymptotic upper bound for the minimal representing set size.

Characterized conditions under which sums of m-gonal numbers are universal.

Extended classical results on polygonal number representations.

Abstract

In this paper we study universal quadratic polynomials which arise as sums of polygonal numbers. Specifically, we determine an asymptotic upper bound (as a function of $m$) on the size of the set $S_m\subset\mathbb{N}$ such that if a sum of $m$-gonal numbers represents $S_m$, then it represents $\mathbb{N}$.