On some universal sums of generalized polygonal numbers

Fan Ge¹ ; Zhi-Wei Sun²

¹Department of Mathematics University of Rochester Rochester, NY 14627, U.S.A.
²Department of Mathematics Nanjing University Nanjing 210093, People’s Republic of China

Colloquium Mathematicum, Tome 145 (2016) no. 1, pp. 149-155

Résumé

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.

Détail

DOI : 10.4064/cm6742-3-2016

Keywords: ldots those m x mathbb called generalized m gonal numbers sun studied what values positive integers sum universal mathbb mathbb ldots has form mathbb prove nbsp universal mathbb conjectured sun sun conjectured mathbb written mathbb contrast universal mathbb proofs essentially elementary hence suitable general readers