Ternary universal sums of generalized polygonal numbers

An integer of the form $p_m(x)= \frac{(m-2)x^2-(m-4)x}{2} \ (m\ge 3)$, for some integer $x$ is called a generalized polygonal number of order $m$. A ternary sum $\Phi_{i,j,k}^{a,b,c}(x,y,z)=ap_{i+2}(x)+bp_{j+2}(y)+cp_{k+2}(z)$ of generalized polygonal numbers, for some positive integers $a,b,c$ and some integers $1\leq i\leq j \leq k$, is said to be universal over $\mathbb{Z}$ if the equation $\Phi_{i,j,k}^{a,b,c}(x,y,z)=n$ has an integer solution $x,y,z$ for any nonnegative integer $n$. In this article, we prove the universalities of $17$ ternary sums of generalized polygonal numbers, which was conjectured by Sun.