论文标题
原始的通用Quaternary二次形式
Primitively universal quaternary quadratic forms
论文作者
论文摘要
如果代表所有正整数，则（正定和积分的）二次表格$ f $如果代表所有积极的整数，则为$ \ textit {undivil} $，如果代表所有正整数，则据说为$ \ textit {prinistife comminity unisival} $原始的整数。我们还说，$ f $是$ \ textit {原始} $几乎代表所有积极整数。 Conway和Schneeberger证明了（见[1]），恰好有$ 204 $等效类的Quaternary二次形式。最近，Earnest and Gunawardana在[4]中证明，在$ 204 $的Quaternary二次形式的$ 204 $等效类中，正好有152美元的$ 152 $等效类别的原始Quaternary Quartnary Quadratic形式。在本文中，我们证明，原始的五二次二次形式的恰好有$ 107 $等效类别。我们还确定了所有剩下的$ 152-107 = 45 $等效类别的原始Quaternary Quadratic形式的等价类别中的每个正整数的集合。
A (positive definite and integral) quadratic form $f$ is said to be $\textit{universal}$ if it represents all positive integers, and is said to be $\textit{primitively universal}$ if it represents all positive integers primitively. We also say $f$ is $\textit{primitively almost universal}$ if it represents almost all positive integers primitively. Conway and Schneeberger proved (see [1]) that there are exactly $204$ equivalence classes of universal quaternary quadratic forms. Recently, Earnest and Gunawardana proved in [4] that among $204$ equivalence classes of universal quaternary quadratic forms, there are exactly $152$ equivalence classes of primitively almost universal quaternary quadratic forms. In this article, we prove that there are exactly $107$ equivalence classes of primitively universal quaternary quadratic forms. We also determine the set of all positive integers that are not primitively represented by each of the remaining $152-107=45$ equivalence classes of primitively almost universal quaternary quadratic forms.
