Some universal quadratic sums over the integers

Abstract

Let $a,b,c,d,e,f\in \mathbb{N}$ with $a\ge c\ge e>0$, $b\le a$ and $b\equiv a(\mathrm{mod}2)$, $d\le c$ and $d\equiv c(\mathrm{mod}2)$, $f\le e$ and $f\equiv e(\mathrm{mod}2)$. 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 $x^2+y^2+32z^2$.

19 pages, final published version

References in corpus (2)

• Universal sums of three quadratic polynomials
• Ternary universal sums of generalized polygonal numbers

Cited by in corpus (1)

• Arithmetic progressions represented by diagonal ternary quadratic forms