近哥德巴赫图中顶点的度数计数
Counting degrees of vertices in near Goldbach graphs
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中文总结 AI 辅助
本文针对近哥德巴赫图G(n)，推导了正偶数度数的精确公式，给出了度数近似函数，还引入近独立事件集概念并关联哥德巴赫猜想相关结论。
中文摘要 AI 辅助
近哥德巴赫图是一种简单无向图，其顶点集由所有正偶数组成，当且仅当(a+b)/2和|a−b|/2均为奇素数或1时，顶点a和b之间存在边。有限近哥德巴赫图G(n)的顶点集为{x∈2ℕ:x≤2n}，邻接规则与上述一致。本文中，我们得到了正偶数x在G(x/2)中的度数的两个精确公式；计算了函数η(x)=∏_{p|x,p>2}[(p−1)/(p−2)]·[xe^−0.183407/(log x)^2]，用于近似大正偶数x在G(x/2)中的度数；最后引入近独立事件集的概念，并证明若大正整数x的可除性事件集是近独立的，则x可表示为两个奇素数之和。
英文摘要
A near Goldbach graph is a simple undirected graph whose vertex set consists of all positive even integers and there is an edge between two vertices $a,b$ if and only if $\frac{a+b}{2}, \frac{|a-b|}{2}$ are either odd primes or $1$. A finite near Goldbach graph $G(n)$ has the vertex set $\{x\in 2\mathbb{N}\, :\, x\leq 2n\}$ with the same adjacency rule. In this paper, we obtain two exact formulas for the degree of the even positive integer $x$ in $G(x/2)$. We compute a function $η(x)=\prod\limits_{p\mid x,\, p>2} \frac{p-1}{p-2}\, \frac{xe^{-0.183407}}{(\log\, x)^2}$ that approximates the degree of $x$ in $G(x/2)$ for a large even positive integer $x$. Finally, we introduce the concept of a nearly independent set of events and show that if the set of divisibility events for a large even integer $x$ is nearly independent, then $x$ can be expressed as the sum of two odd primes.
