Universal sums of generalized octagonal numbers

Abstract

An integer of the form $P_8(x)=3x^2-2x$ for some integer $x$ is called a generalized octagonal number. A quaternary sum ${\Phi }_{a,b,c,d}(x,y,z,t)=a{P}_8(x)+b{P}_8(y)+c{P}_8(z)+d{P}_8(t)$ of generalized octagonal numbers is called {\it universal} if ${\Phi }_{a,b,c,d}(x,y,z,t)=n$ 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 ${\Phi }_{a,b,c,d}(x,y,z,t)$ is universal. These were conjectured by Sun in \cite {sun}. We also give an effective criterion on the universality of an arbitrary sum $a_1{P}_8(x_1)+a_2{P}_8(x_2)+\cdots +a_k{P}_8(x_k)$ of generalized octagonal numbers, which is a generalization of "$15$-theorem" of Conway and Schneeberger.

10 pages