Tight universal 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 $a_1{P}_8(x_1)+a_2{P}_8(x_2)+\cdots +a_k{P}_8(x_k)$ is called an octagonal form. For a positive integer $n$, an octagonal form is called tight $\mathcal{T}(n)$-universal if it represents (over $z$) 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.