Determining universality of $m$-gonal form with first five coefficients

TL;DR

This paper classifies the first five coefficients of $m$-gonal forms to determine when such forms are universal, meaning they represent all positive integers beyond a certain point, with applications in number theory.

Contribution

It provides a classification of the initial coefficients of $m$-gonal forms that guarantees universality based on representability of integers up to $m-4$.

Findings

Characterization of coefficient sets for universality

Representation of integers up to $m-4$ as a criterion

Applications in number theory and form classification

Abstract

In this paper, we classify the $(a_1,a_2,a_3,a_4,a_5)$ for which the universality of an $m$-gonal form $F_m(\mathbf x)$ having its first five coefficients as $(a_1,a_2,a_3,a_4,a_5)$ is characterized as the representability of positive integers up to $m-4$ and see its applications.