A generalization of Gauss' triangular theorem

TL;DR

This paper proves that specific quadratic polynomials are universal, meaning they can represent all non-negative integers as solutions to their Diophantine equations, extending Gauss's classical results.

Contribution

The paper confirms Sun's conjectures by proving universality of three particular quadratic polynomials, expanding the understanding of quadratic forms and their representational capabilities.

Findings

Proved universality for (a,b,c) = (2,2,6), (2,3,5), (2,3,7)

Extended Gauss's triangular theorem to new quadratic forms

Validated conjectures posed by Sun

Abstract

A quadratic polynomial $\Phi_{a,b,c}(x,y,z)=x(ax+1)+y(by+1)+z(cz+1)$ is called universal if the diophantine equation $\Phi_{a,b,c}(x,y,z)=n$ has an integer solution $x,y,z$ for any non negative integer $n$. In this article, we show that if $(a,b,c)=(2,2,6), (2,3,5)$ or $(2,3,7)$, then $\Phi_{a,b,c}( x,y,z)$ is universal. These were conjectured by Sun in \cite {Sun}.