Primitively universal quaternary quadratic 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.