Universal sums of generalized octagonal numbers

An integer of the form P8(x)=3x2−2x for some integer x is called a generalized octagonal number. A quaternary sum Φa,b,c,d(x,y,z,t)=aP8(x)+bP8(y)+cP8(z)+dP8(t) of generalized octagonal numbers is called universal if Φa,b,c,d(x,y,z,t)=n has an integer solution x,y,z,t for any positive integer n. In this article, we show that if a=1 and (b,c,d)=(1,3,3),(1,3,6),(2,3,6),(2,3,7) or (2,3,9), then Φa,b,c,d(x,y,z,t) is universal. These were conjectured by Sun in [10]. We also give an effective criterion on the universality of an arbitrary sum a1P8(x1)+a2P8(x2)+⋯+akP8(xk) of generalized octagonal numbers, which is a generalization of “15-theorem” of Conway and Schneeberger.