埃拉托色尼筛隙星座
Constellations of gaps in Eratosthenes sieve
摘要：
几年前，我们发现了一种直接处理埃拉托斯特尼筛选每个阶段中生成器之间间隙的递归。这种递归提供了生成器之间的间隙序列的显式枚举，该序列被称为星座。在过去的一年里，我们确定了一个离散的线性系统，它精确地模拟了筛子所有阶段的任何间隙的种群。在2014年8月，我们总结了我们的结果，从分析这个离散模型的单一差距的人口。本文将离散系统推广到林隙星群的种群模型。最显著的结果是关于等差数列的一个强Polignac结果。我们以前已经建立了波利尼亚克猜想的等价物适用于埃拉托塞尼筛-每个偶数都是筛中的一个间隙，其人口朝着哈代和利特尔伍德猜想B所暗示的比率收敛。将该工作扩展到星座，我们在这里建立，对于任何偶数间隙$g$，如果$p$是最大素数，使得$p\# \; | g$和$P$是下一个素数大于$p$，那么对于每个$2 \le j_1 < P-1$，星座$g,g,\ldots,g$的长度为$j_1$在埃拉托斯特尼筛中出现。该星座对应于$j_1+1$连续候选素数的等差数列。
英文摘要：
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations. Over the last year we identified a discrete linear system that exactly models the population of any gap across all stages of the sieve. In August 2014 we summarized our results from analyzing this discrete model on populations of single gaps. This paper extends the discrete system to model the populations of constellations of gaps. The most remarkable result is a strong Polignac result on arithmetic progressions. We had previously established that the equivalent of Polignac's conjecture holds for Eratosthenes sieve -- that every even number arises as a gap in the sieve, and its population converges toward the ratio implied by Hardy and Littlewood's Conjecture B. Extending that work to constellations, we here establish that for any even gap $g$, if $p$ is the maximum prime such that $p\# \; | g$ and $P$ is the next prime larger than $p$, then for every $2 \le j_1 < P-1$, the constellation $g,g,\ldots,g$ of length $j_1$ arises in Eratosthenes sieve. This constellation corresponds to an arithmetic progression of $j_1+1$ consecutive candidate primes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
