On Wilson's Theorem and Polignac Conjecture

On Polignac's conjecture and arithmetic progressions

In this paper we investigate the recent advances by Zhang, Maynard and Pintz towards Polignac's conjecture and give some new results concerning the relationship between Polignac numbers and arithmetic progressions.

About de Polignac's conjecture

In this note we present a method to bound gaps between primes via the divergence of the series of reciprocals of the prime numbers, a consequence of a version of the Bertrand's test for convergence of series of positive numbers and a suitable series of positive terms related to the logarithmic integral function. Using this proposed method we discuss the Polignac's conjecture and present some consequences.

Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture

In the present work we prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent fantastic achievement of Yitang Zhang about the existence of bounded gaps between consecutive primes. Most of these results would have belonged to the category of science fiction a decade ago. However, the presented results are far from being immediate consequences of Zhang's famous theorem: they require various new ideas, other important properties of the applied sieve functi…

关于 de Polignac 猜想中形如 $p + 2^k$ 的数的一些结果

Some results on a conjecture of de Polignac about numbers of the form $p + 2^k$

本文针对形如 $p + 2^k$ 的数，构造等差数列反驳 Erdős 猜想，验证 Chen 关于公差下界的猜想，并将可表示数的密度上界改进至 0.490341088858244。

On Polignac's Conjecture

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 wil…

10 conjectures in additive number theory

Following an idea of Rowland we give a conjectural way to generate increasing sequences of primes using algorithms involving the gcd. These algorithms seem not so useless for searching primes since it appears we found sometime primes much more greater than the number of required iterations. In an other hand we propose new formulations of famous conjectures from the additive theory of numbers (the weak twin prime conjecture, the Polignac conjecture, the Goldbach conjecture or the very general Schinzel's hypothesis H). For the moment these are experimental results obtained using pari-gp.