紧通用八边形

摘要：

令$P_8(x)=3x^2-2x$。对于正整数$a_1,a_2,\dots,a_k$，形式为$a_1P_8(x_1)+a_2P_8(x_2)+\cdots+a_kP_8(x_k)$的多项式称为八边形。对于正整数$n$，八边形形式称为紧$\mathcal T(n)$ -泛，如果它表示（在$\mathbb{z}$上）大于或等于$n$的每个正整数，并且不表示小于$n$的任何正整数。在这篇文章中，我们发现所有紧$\mathcal T(n)$ -泛八边形的形式为每个$n\ge 2$。进一步，我们给出了任意八边形的紧$\mathcal T(n)$的一个有效判据--普适性，它是Conway和Schneeberger的“15定理”的推广.

英文摘要：

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.

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