2022/02/28 by Jangwon Ju, Ju, Jangwon, Daejun Kim +7
Mathematics · #11E12 #11E20 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2202.13573
openalex publication_date 2022/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.