Almost Universal Weighted Ternary Sums of Polygonal Numbers

TL;DR

This paper characterizes when weighted sums of generalized m-gonal numbers are almost universal, providing criteria and examples of forms that do or do not represent all integers or classes.

Contribution

It establishes a criterion for almost universality of weighted ternary sums of polygonal numbers and analyzes specific cases where forms are not almost universal.

Findings

Derived a criterion for almost universality based on parameters a,b,c,m.

Identified forms that are not almost universal despite representing all congruence classes.

Provided examples illustrating the boundary between universal and non-universal forms.

Abstract

For a natural number $m$, generalized $m$-gonal numbers are defined by the formula $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb Z$. In this paper, we determine a criterion on $a,b,c,m$ for which the weighted ternary sum $P_{a,b,c,m}:=ap_m(x)+bp_m(y)+cp_m(z)$ is almost universal. We also prove for some $a,b,c,m$ that the form $P_{a,b,c,m}$ is not almost universal, while it represents all possible congruence classes.