Bulletin of the
Korean Mathematical Society BKMS

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 nonnegative 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}.

Keywords: triangular theorem, universal polynomials

MSC numbers: Primary 11E12, 11E20