Fermat's polygonal number theorem for repeated generalized polygonal numbers

Sums of four polygonal numbers with coefficients

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 asserts 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 $(a,b)=(1,1)\ \&\ 4\mid m$, or $(a,b)=(2,2)\ \&\ m\not\equiv…

一种高效的素数计数估计算法

An incremental approximation to the prime-counting function via polygonal numbers

提出一种基于广义三角数结构化非均匀划分的素数计数函数π(x)近似算法，具有O(1)均摊更新复杂度和O(√x)总复杂度，通过数值实验修正项达到与经典解析近似相当的精度。