关于Polignac猜想

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，这被称为星座。随着递归的进行，较长星座内的相邻间隙被加在一起以产生相同和的较短星座。这些加法或闭包对应于去除对于埃拉托斯特尼筛的该阶段可被素数整除的合数。虽然我们不知道在间隙的循环中，闭合将发生在哪里，但我们可以精确地列举出每个阶段有多少不同星座的副本将幸存下来。在本文中，我们扩大了我们的研究这些系统的星座的一个固定的总和。通过推广我们的方法，我们能够证明，对于每一个偶数$2n$的间隙$g=2n$发生无限经常通过埃拉托斯特尼筛的阶段。此外，我们还证明了在Eratosthenes筛的每个阶段，空隙数$g=2n$与空隙数$g=2$之比渐近收敛于Hardy和Littlewood在其1923年论文的猜想B中对素数空隙所作的估计。

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 are known as constellations. As the recursion proceeds, adjacent gaps within longer constellations are added together to produce shorter constellations of the same sum. These additions or closures correspond to removing composite numbers that are divisible by the prime for that stage of Eratosthenes sieve. Although we don't know where in the cycle of gaps a closure will occur, we can enumerate exactly how many copies of various constellations will survive each stage. In this paper, we broaden our study of these systems of constellations of a fixed sum. By generalizing our methods, we are able to demonstrate that for every even number $2n$ the gap $g=2n$ occurs infinitely often through the stages of Eratosthenes sieve. Moreover, we show that asymptotically the ratio of the number of gaps $g=2n$ to the number of gaps $g=2$ at each stage of Eratosthenes sieve converges to the estimates made for gaps among primes by Hardy and Littlewood in Conjecture B of their 1923 paper.

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