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 A generalization of Gauss' triangular theorem Bull. Korean Math. Soc. 2018 Vol. 55, No. 4, 1149-1159 https://doi.org/10.4134/BKMS.b170633Published online May 2, 2018Printed July 31, 2018 Jangwon Ju, Byeong-Kweon Oh Seoul National University, Seoul National University 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 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 Downloads: Full-text PDF