Polignac Numbers, Conjectures of Erdos on Gaps Between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture
In the present work we prove a number of results about gaps between consecutive primes. The proofs need the method of Y. Zhang which led to the proof of infinitely many bounded gaps between primes. Several of the results refer to the so called Polignac numbers which we define as those even integers which can be written in infinitely many ways as the difference of two consecutive primes. Others refer to several 60 70 years old conjecture of Paul Erdos about the distribution of the normalized gaps between consecutive primes and about the distribution of the ratio of consecutive primegaps. The methods involve an extended version of Zhangs method, a property of the GPY weights proved by the author a few years ago and other ideas as well.