Primitively universal quaternary quadratic forms

TL;DR

This paper classifies primitive universality in quaternary quadratic forms, establishing the exact number of such classes and identifying integers not primitively represented by nearly universal forms.

Contribution

It proves there are exactly 107 equivalence classes of primitively universal quaternary quadratic forms and details the integers not primitively represented by the remaining nearly universal classes.

Findings

107 equivalence classes of primitively universal forms

Identification of integers not primitively represented by 45 classes

Extension of previous classifications of universal forms

Abstract

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.