On Universal Representations Of The Forms AT_x+bT_y+cp₅（z） And AT_x+ Bp₅（y）+cp₅（z）

For m=3,4..., the m-gonal numbers are given by For positive integers a,b,c and i,j, k≥ 3, we call the triple (api,bpj,cpk) universal over Z if for any n∈N, there are integers x, y, z such that n= api(x)+bpj(y)+cpk(z). Recently, Sun has found that there are 33 candidates for universal triples (ap3, bp3, cp5) and 45 candidates for universal triples (ap3,bp5, cps), he thought that they are all universal triples. In this paper, by using the theory of ternary quadratic forms, we show that the following 39 triples (p3,p3,3p5), (p3,2p3,2p5), (P3,2p3,3p5), (p3,3p3,2p5), (P3,6p3,2p5), (p3,9p3,p5), (2p3,2p3,p5), (2p3,4p3,p5), (2p3,9p3,p5), (3p3,3p3,p5), (3p3,4p3,p5), (p3,p5,2p5), (p3,p5,3p5), (p3,P5,4p5), (p3,p5,6p5), (p3,p5,7p5), (p3,p5,9p5), (p3,2p5,2p5), (p3,2p5,3p5), (p3,2p5,4p5), (p3,2p5,6p5), (2p3,p5,p5), (2p3,p5,2p5), (2p3,p5,3p5), (2p3,p5,4p5), (3p3,p5,p5), (3p3,p5,2p5), (3p3,P5,3p5), (3p3,P5,4p5), (3p3,P5,5p5), (3p3,p5,6p5), (4p3,p5,p5), (4p3,p5,2p5), (4p3,p5,3p5), (6p3,p5,p5),(6p3,p5,p5) (6p3,p5,3p5),(9p3,p5,p5),(9P3,p5,3p5). are indeed universal.