On $x(ax+1)+y(by+1)+z(cz+1)$ and $x(ax+b)+y(ay+c)+z(az+d)$

TL;DR

This paper classifies integer triples and quadruples for which all nonnegative integers can be represented by specific quadratic forms, extending known results and proposing conjectures for further cases.

Contribution

It provides a complete classification of triples and quadruples of positive integers for which all nonnegative integers are representable by the given forms, and conjectures about additional cases.

Findings

Seven triples $(a,b,c)$ allow representation of all nonnegative integers.

Five quadruples $(a,b,c,d)$ allow representation of all nonnegative integers.

Conjectures are proposed for additional triples $(a,b,c)$ with $a>2$.

Abstract

In this paper we first investigate for what positive integers $a,b,c$ every nonnegative integer $n$ can be represented as $x(ax+1)+y(by+1)+z(cz+1)$ with $x,y,z$ integers. We show that $(a,b,c)$ can be either of the following seven triples: $$(1,2,3),\ (1,2,4),\ (1,2,5),\ (2,2,4),\ (2,2,5),\ (2,3,3),\ (2,3,4),$$ and conjecture that any triple $(a,b,c)$ among $$(2,2,6),\ (2,3,5),\ (2,3,7),\ (2,3,8),\ (2,3,9),\ (2,3,10)$$ also has the desired property. For integers $0\le b\le c\le d\le a$ with $a>2$, we prove that any nonnegative integer can be represented as $x(ax+b)+y(ay+c)+z(az+d)$ with $x,y,z$ integers, if and only if the quadruple $(a,b,c,d)$ is among $$(3,0,1,2),\ (3,1,1,2),\ (3,1,2,2),\ (3,1,2,3),\ (4,1,2,3).$$