Some universal quadratic sums over the integers

TL;DR

This paper proves that 44 specific quadratic sum tuples are universal over integers, meaning they can represent all nonnegative integers, using the theory of ternary quadratic forms.

Contribution

It confirms the universality of 44 candidate tuples from Sun's list through the application of ternary quadratic form theory.

Findings

44 tuples are proven universal over integers

The tuple (16,4,2,0,1,1) is universal, related to x^2+y^2+32z^2

The paper validates Sun's candidate list using quadratic forms

Abstract

Let $a,b,c,d,e,f\in\mathbb N$ with $a\ge c\ge e>0$, $b\le a$ and $b\equiv a\pmod2$, $d\le c$ and $d\equiv c\pmod2$, $f\le e$ and $f\equiv e\pmod2$. If any nonnegative integer can be written as $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ with $x,y,z\in\mathbb Z$, then the ordered tuple $(a,b,c,d,e,f)$ is said to be universal over $\mathbb Z$. Recently, Z.-W. Sun found all candidates for such universal tuples over $\mathbb Z$. In this paper, we use the theory of ternary quadratic forms to show that 44 concrete tuples $(a,b,c,d,e,f)$ in Sun's list of candidates are indeed universal over $\mathbb Z$. For example, we prove the universality of $(16,4,2,0,1,1)$ over $\mathbb Z$ which is related to the form $x^2+y^2+32z^2$.