$(k,l)$-universality of ternary quadratic forms $a{x}^2+b{y}^2+c{z}^2$

Abstract

Let $\mathbb{N}$ denote the set of positive integers and ${\mathbb{N}}_0=\mathbb{N}\cup \{0\}$. The positive diagonal integral ternary quadratic form $a{x}^2+b{y}^2+c{z}^2,,(a,b,c\in \mathbb{N})$ is said to be $(k,l)$-universal if it represents every integer in the arithmetic progression $\{kn+l,,|,,n\in {\mathbb{N}}_0\}$, where $k,l\in \mathbb{N}$ are such that $l\le k.$ We show that there are only finitely many $(k,l)$-universal positive integral ternary quadratic forms $a{x}^2+b{y}^2+c{z}^2$ for a fixed pair $(k,l)\in {\mathbb{N}}^2$ with $l\le k$. We also prove the existence of a finite set $S=S(k,l)$ of positive integers $\equiv l(\mathrm{mod}k)$ such that if $a{x}^2+b{y}^2+c{z}^2$ represents every integer in $S$ then $a{x}^2+b{y}^2+c{z}^2$ is $(k,l)$-universal. Assuming that certain ternaries are $(k,l)$-universal we determine all the $(k,l)$-universal ternaries $a{x}^2+b{y}^2+c{z}^2$ $(a,b,c\in \mathbb{N}),$ as well as the sets $S(k,l),$ for all $k,l\in \mathbb{N}$ with $1\le l\le k\le 11$.