On certain quaternary quadratic forms

TL;DR

This paper characterizes when certain quaternary quadratic forms can represent all nonnegative integers, identifying specific conditions on parameters and initial representability that guarantee universality.

Contribution

It determines all parameter sets for which the quadratic form is universal and establishes initial representability conditions for universality.

Findings

Identified all parameter combinations for universality.

Proved that representing specific small integers ensures universality.

Provided a complete characterization of the quadratic forms' representational capacity.

Abstract

In this paper, we determine all the positive integers $a, b$ and $c$ such that every nonnegative integer can be represented as $$ f^{a,b}_c(x,y,z,w)=ax^2+by^2+c(z^2+zw+w^2) \,\, \textrm{with} \,\,x,y,z,w\in\mathbb{Z}. $$ Furthermore, we prove that $f^{a,b}_c$ can represent all the nonnegative integers if it represents $n=1,2,3,5,6,10.$