On the sums of three generalized polygonal numbers

TL;DR

This paper investigates conditions under which sums of three generalized polygonal numbers can represent all sufficiently large positive integers, using congruence theta functions to establish these criteria.

Contribution

It provides new criteria for when sums of three generalized polygonal numbers are almost universal, advancing understanding of their representational capabilities.

Findings

Identifies specific conditions on parameters for almost universality.

Uses congruence theta functions to analyze representation problems.

Establishes finiteness of exceptions in representation of positive integers.

Abstract

For each natural number $m\ge 3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. In this paper, with the help of the congruence theta function, we establish conditions on $a$, $b$, $c$ for which the sum $P_a(x)+P_b(y)+P_c(z)$ represents all but finitely many positive integers.