Primitively universal quaternary quadratic forms

Abstract

A (positive definite and integral) quadratic form f is said to be universal if it represents all positive integers, and is said to be primitively universal if it represents all positive integers primitively. We also say f is almost primitively 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 almost primitively 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 almost primitively universal quaternary quadratic forms.