Uniqueness of the 2-universality Criterion

TL;DR

This paper proves that the minimal criterion for 2-universality of positive-definite integer-matrix quadratic forms is unique, similar to the uniqueness of the Fifteen Theorem for universal quadratic forms.

Contribution

It establishes the uniqueness of the 2-universality criterion, extending the concept of the Fifteen Theorem to this specific class of quadratic forms.

Findings

The 2-universality criterion is unique.

The criterion parallels the uniqueness of the Fifteen Theorem.

This result clarifies the structure of 2-universal quadratic forms.

Abstract

Kim, Kim, and Oh gave a minimal criterion for the 2-universality of positive-definite integer-matrix quadratic forms. We show that this 2-universality criterion is unique in the sense of the uniqueness of the Conway-Schneeberger Fifteen Theorem.