Tight universal octagonal forms

TL;DR

This paper classifies all tight universal octagonal forms that represent all integers above a certain threshold without representing smaller ones, and extends the 15-Theorem to these forms with an effective criterion.

Contribution

It provides a complete classification of tight $\\mathcal T(n)$-universal octagonal forms for all $n\ge 2$ and generalizes the 15-Theorem for these forms.

Findings

All tight $\\mathcal T(n)$-universal octagonal forms are identified.

An effective criterion for determining tight $\\mathcal T(n)$-universality is established.

The results extend the classical 15-Theorem to octagonal forms.

Abstract

Let $P_8(x)=3x^2-2x$. For positive integers $a_1,a_2,\dots,a_k$, a polynomial of the form $a_1P_8(x_1)+a_2P_8(x_2)+\cdots+a_kP_8(x_k)$ is called an octagonal form. For a positive integer $n$, an octagonal form is called tight $\mathcal T(n)$-universal if it represents (over $\mathbb{z}$) every positive integer greater than or equal to $n$ and does not represent any positive integer less than $n$. In this article, we find all tight $\mathcal T(n)$-universal octagonal forms for every $n\ge 2$. Furthermore, we provide an effective criterion on tight $\mathcal T(n)$-universality of an arbirary octagonal form, which is a generalization of "15-Theorem" of Conway and Schneeberger.