论文标题
在四个五边形数字的总和上
On sums of four pentagonal numbers with coefficients
论文作者
论文摘要
五边形数字是$ p_5（n）= n（3n-1）/2 \（n = 0,1,2，\ ldots）$给出的整数。令$（b，c，d）$是三元组$（1,1,2），（1,2,3），（1,2,6）$和$（2,3,4）$之一。我们表明，每个$ n = 0,1,2，\ ldots $可以写为$ W+bx+cy+cy+dz $，带有$ w，x，y，z $五角形数字，该号码首先由Z.-W。 2016年的太阳。特别是，任何非负整数都是五个五边形数的总和，其中两个相等。这是Fermat声称的Cauchy的经典结果。
The pentagonal numbers are the integers given by $p_5(n)=n(3n-1)/2\ (n=0,1,2,\ldots)$. Let $(b,c,d)$ be one of the triples $(1,1,2),(1,2,3),(1,2,6)$ and $(2,3,4)$. We show that each $n=0,1,2,\ldots$ can be written as $w+bx+cy+dz$ with $w,x,y,z$ pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integer is a sum of five pentagonal numbers two of which are equal; this refines a classical result of Cauchy claimed by Fermat.
