Almost universal ternary sums of polygonal numbers

TL;DR

This paper investigates conditions under which the sum of three generalized m-gonal numbers is almost universal, meaning it represents all sufficiently large integers, extending the understanding of polygonal number representations.

Contribution

The paper establishes new criteria on m for which the ternary sum of generalized m-gonal numbers is almost universal.

Findings

Identifies specific values of m where the sum is almost universal.

Provides a characterization of the representability of integers by these sums.

Extends classical results on polygonal numbers to a broader class.

Abstract

For a natural number $m$, generalized $m$-gonal numbers are those numbers of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb Z$. In this paper we establish conditions on $m$ for which the ternary sum $p_m(x)+p_m(y)+p_m(z)$ is almost universal.