$(k,l)$-universality of ternary quadratic forms $ax^2 + by^2+ cz^2$

Abstract

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

Type
Publication
In INTEGERS, Volume 18, A20