论文标题
在更严格的双素素中，一般而言，在Polignac的猜想上
On a Stricter Twin Primes Conjecture, and on the Polignac's Conjecture in general
论文作者
论文摘要
Polignac的猜想是由Alphonse de Polignac于1849年首次提出的，他断言，对于任何偶数M，都有许多质数P，P+M的夫妻无限地存在。当M = 2时，这将减少到双素素的猜想。尽管有数值证据和许多理论进展，但自那以后，猜想还是抵制了正式的证据。在本文的第一部分中，我们研究了猜想的更严格的版本，如下所示：“让$ p_ {n} $为n-th-prime。然后，在$（p_ {n} -2）^{2} $和$ p_ {n}^{2} $'''''''''''''''之间总是存在双素。为了证明这一猜想是合理的，我们针对此范围内的双素对数制定了一个预测（基于双向方法），并将预测与$ p_ {n} $值为6500000的值的实际结果进行比较。我们还分析了$ p_ {n} $的较高值应该发生的情况。在第二部分中，我们研究了普通Polignac猜想的有效性。我们预测了M = 2的M = 2的任何值的任何值的溶液数量的比率，并解释了该比率如何取决于M的分解。我们将预测与M最高3000（以及特殊情况30030）的实际值（从1000000-th的范围）进行比较到210000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000。
The Polignac's Conjecture, first formulated by Alphonse de Polignac in 1849, asserts that, for any even number M, there exist infinitely many couples of prime numbers P, P+M. When M = 2, this reduces to the Twin Primes Conjecture. Despite numerical evidence, and many theoretical progresses, the conjecture has resisted a formal proof since. In the first part of this paper, we investigate a stricter version of the conjecture, expressed as follows: ''Let $p_{n}$ be the n-th prime. Then, there always exist twin primes between $(p_{n}-2)^{2}$ and $p_{n}^{2}$ ''. To justify this conjecture, we formulate a prediction (based on a double-sieve method) for the number of twin prime pairs in this range, and compare the prediction with the real results for values of $p_{n}$ up to 6500000. We also analyse what should happen for higher values of $p_{n}$. In the second part, we investigate the validity of the general Polignac's Conjecture. We predict the ratio of the number of solutions for any value of M divided by the number of solutions for M = 2, and explain how this ratio depends on the factorization of M. We compare the predictions with the real values for M up to 3000 (and for the special case 30030) in the range of from the 1000000-th prime to the 21000000-th prime.
