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对于m=3,4,...,m角数如下给出对正整数a,b,c和i,j,k≥3,三元组(api,bpj,cpk)在Z上通用,指每个自然数n可表成api(x)＋bpi(y)+cpk(z)(其中x,y,z为整数)。孙智伟老师已找出33对通用三元组(ap3,bp3,cp5)和45对通用三元组(ap3,bp5,cp5),并且认为这些组都是通用三元组。本文利用三元二次型理论证明下述39对三元组(P3,p3,3p5), (p3,2p3,2p5), (p3,2p3,3p5), (p3,3p3,2p5), (p3,6p3,2p5), (p3,9p3,p5), 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,p5,2p5), (6p3,p5,3p5),(9p3,p5,p5),(9p3,p5,3p5).的确都是通用的。