Patterns of primes in arithmetic progressions

TL;DR

This paper proves that for any number of consecutive primes, there exists a bounded pattern of such primes forming arbitrarily long arithmetic progressions, extending previous foundational results in prime number theory.

Contribution

It generalizes Green-Tao and Maynard/Tao results by showing bounded prime patterns contain arbitrarily long arithmetic progressions and constitute a positive proportion of all such patterns.

Findings

Existence of bounded prime patterns with arbitrarily long arithmetic progressions

Positive proportion of m-tuples form such prime patterns

Extension of Green-Tao and Maynard/Tao results

Abstract

We show that there exists a bounded pattern of m consecutive primes for any m>0, that means a tuple H_m of m distinct non-negative integers h_i (i=1,2,...m) such that its translations contain arbitrarily long (finite) arithmetic progressions. More precisely, the set of natural numbers n for which all components n+h_i (i=1,2,...m) are consecutive primes contains arbitrarily long (finite) arithmetic progressions. Moreover, the set of m-tuples that satisfy this property represents a positive proportion of all m-tuples. The present result is the generalization of the results of Green-Tao (about the existence of arbitrarily long arithmetic progressions) and of Maynard/Tao (about the existence of infinitely many bounded blocks of m primes, where m is an arbitrary natural number). It also generalizes the author's work which first showed the existence of infinitely many Polignac numbers and they contain arbitrarily long (finite) arithmetic progressions (arXiv: 1305.6289v1, 27 May 2013) which was a common generalization of the above mentioned result of Green-Tao and that of Zhang (about the exisatence of infinitely many bounded gaps between primes).