Quadratic forms representing all integers coprime to 3

TL;DR

This paper proves a universality theorem for positive-definite integer-valued quadratic forms representing all positive integers coprime to 3, extending the 290-theorem to this specific coprimality condition.

Contribution

It establishes a new universality criterion for quadratic forms based on their representation of integers coprime to 3, generalizing previous results like Bhargava and Hanke's theorem.

Findings

Quadratic forms representing all positive integers coprime to 3 are universal if they represent all such integers up to 290.

The paper extends the 290-theorem to integers coprime to 3.

Under GRH, forms representing all odd integers up to 451 represent all odd integers.

Abstract

Following Bhargava and Hanke's celebrated 290-theorem, we prove a universality theorem for all positive-definite integer-valued quadratic forms that represent all positive integers coprime to $3$. In particular, if a positive-definite quadratic form represents all positive integers coprime to $3$ and $\leq 290$, then it represents all positive integers coprime to $3$. We use similar methods to those used by Rouse to prove (assuming GRH) that a positive-definite quadratic form representing every odd integer between $1$ and $451$ represents all positive odd integers.