Sums of four polygonal numbers with coefficients

Xiang-Zi Meng ¹   ; Zhi-Wei Sun ¹

¹ Department of Mathematics Nanjing University Nanjing 210093, People’s Republic of China

Acta Arithmetica, Tome 180 (2017) no. 3, pp. 229-249

Résumé

Let $m\ge3$ be an integer. The polygonal numbers of order $m+2$ are given by $p_{m+2}(n)=m\binom n2+n$ $(n=0,1,2,\ldots)$. A famous claim of Fermat proved by Cauchy states that each nonnegative integer is the sum of $m+2$ polygonal numbers of order $m+2$. For $(a,b)=(1,1),(2,2),(1,3),(2,4)$, we study whether any sufficiently large integer can be expressed as $$ p_{m+2}(x_1)+p_{m+2}(x_2)+ap_{m+2}(x_3)+bp_{m+2}(x_4) $$ with $x_1,x_2,x_3,x_4$ nonnegative integers. We show that the answer is positive if $(a,b)\in\{(1,3),(2,4)\}$, or $(a,b)=(1,1)\ \ 4 \,|\, m$, or $(a,b)=(2,2)\ \ m\not\equiv2\pmod4$. In particular, we confirm a conjecture of Z.-W. Sun that any natural number can be written as $p_6(x_1)+p_6(x_2)+2p_6(x_3)+4p_6(x_4)$ with $x_1,x_2,x_3,x_4$ nonnegative integers.

Détail

DOI : 10.4064/aa8630-4-2017

Keywords: integer polygonal numbers order given binom ldots famous claim fermat proved cauchy states each nonnegative integer sum polygonal numbers order study whether sufficiently large integer expressed nonnegative integers answer positive amp amp equiv pmod particular confirm conjecture w sun natural number written nonnegative integers