Ternary universal sums of generalized polygonal numbers

An integer of the form P m (x) = (m - 2)(x2)- ( m - 4 )x/2 (m >= 3), for some integer x, is called ,b a generalized polygonal number of order m. A ternary sum Phi(a,b,c)(i,j,k)(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 <= i <= j <= k, is said to be universal over Z if for any nonnegative integer n, the equation Phi(a,b,c)(i,j,k)(x,y,z) = n has an integer solution x, y, z. In this paper, we prove the universalities of 17 ternary sums of generalized polygonal numbers, which was conjectured by Sun.