Arithmetic progressions represented by diagonal ternary quadratic forms

TL;DR

This paper investigates which diagonal ternary quadratic forms can represent all members of certain arithmetic progressions, expanding understanding of their universality properties through the theory of quadratic forms.

Contribution

The paper proves new results on the $(d,r)$-universality of specific diagonal ternary quadratic forms conjectured by Pehlivan, Williams, and Sun, using quadratic form theory.

Findings

2x^2+3y^2+10z^2 is (8,5)-universal

x^2+3y^2+8z^2 and x^2+2y^2+12z^2 are (10,1)- and (10,9)-universal

3x^2+5y^2+15z^2 is (15,8)-universal

Abstract

Let $d>r\ge 0$ be integers. For positive integers $a,b,c$, if any term of the arithmetic progression $\{r+dn:\ n=0,1,2,\ldots\}$ can be written as $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb{Z}$, then the form $ax^2+by^2+cz^2$ is called $(d,r)$-universal. In this paper, via the theory of ternary quadratic forms we study the $(d,r)$-universality of some diagonal ternary quadratic forms conjectured by L. Pehlivan and K. S. Williams, and Z.-W. Sun. For example, we prove that $2x^2+3y^2+10z^2$ is $(8,5)$-universal, $x^2+3y^2+8z^2$ and $x^2+2y^2+12z^2$ are $(10,1)$-universal and $(10,9)$-universal, and $3x^2+5y^2+15z^2$ is $(15,8)$-universal.