Primitively $2$-universal senary integral quadratic forms

TL;DR

This paper classifies primitively 2-universal positive definite integral quadratic forms of rank six, establishing the minimal rank and enumerating all equivalence classes of such forms.

Contribution

It proves the minimal rank for primitively 2-universal forms is six and classifies all 201 equivalence classes of primitively 2-universal senary quadratic forms.

Findings

Minimal rank of primitively 2-universal forms is six.

Exactly 201 equivalence classes of such forms exist.

Classification extends understanding of universal quadratic forms.

Abstract

For a positive integer $m$, a (positive definite integral) quadratic form is called primitively $m$-universal if it primitively represents all quadratic forms of rank $m$. It was proved in arXiv:2202.13573 that there are exactly $107$ equivalence classes of primitively $1$-universal quaternary quadratic forms. In this article, we prove that the minimal rank of primitively $2$-universal quadratic forms is six, and there are exactly $201$ equivalence classes of primitively $2$-universal senary quadratic forms.