Universal Sums of m-Gonal Numbers

Abstract

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 ${\mathbb{N}}$.