Sums of four polygonal numbers with coefficients
Volume 180 / 2017
Abstract
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.
