Polignac Numbers, Conjectures of Erd\"os on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture

TL;DR

This paper builds on Zhang's breakthrough on bounded prime gaps to prove new results about prime gaps, arithmetic progressions, and generalized twin primes, employing novel ideas and advanced sieve methods.

Contribution

It introduces new techniques and insights to extend Zhang's theorem, establishing results on prime gaps and progressions that were previously thought impossible.

Findings

Proved new bounds on gaps between consecutive primes.

Established existence of arithmetic progressions within generalized twin primes.

Extended the implications of Zhang's bounded gap theorem.

Abstract

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 function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, and Zhang, respectively.