A generalization of Gauss' triangular theorem

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}.

8 pages