On some universal sums of generalized polygonal numbers
Volume 145 / 2016
Abstract
For $m=3,4,\ldots $ those $p_m(x)=(m-2)x(x-1)/2+x$ with $x\in {\mathbb Z}$ are called generalized $m$-gonal numbers. Sun (2015) studied for what values of positive integers $a,b,c$ the sum $ap_5+bp_5+cp_5$ is universal over $\mathbb Z$ (i.e., any $n\in {\mathbb N}=\{0,1,2,\ldots \}$ has the form $ap_5(x)+bp_5(y)+cp_5(z)$ with $x,y,z\in {\mathbb Z}$). We prove that $p_5+bp_5+3p_5$ $(b=1,2,3,4,9)$ and $p_5+2p_5+6p_5$ are universal over $\mathbb Z$, as conjectured by Sun. Sun also conjectured that any $n\in {\mathbb N}$ can be written as $p_3(x)+p_5(y)+p_{11}(z)$ and $3p_3(x)+p_5(y)+p_7(z)$ with $x,y,z\in {\mathbb N}$; in contrast, we show that $p_3+p_5+p_{11}$ and $3p_3+p_5+p_7$ are universal over $\mathbb Z$. Our proofs are essentially elementary and hence suitable for general readers.
