Strings of special primes in arithmetic progressions

TL;DR

This paper extends results on prime strings in arithmetic progressions to special subsets of primes, demonstrating the existence of arbitrarily long strings within these subsets using advanced number theory techniques.

Contribution

It generalizes Shiu's theorem to primes of specific forms and subsets with zero density, broadening understanding of prime distributions in arithmetic progressions.

Findings

Existence of arbitrarily long strings of primes in certain subsets.

Generalization of Shiu's theorem to primes of the form ⌊πn⌋.

Application to primes of the form ⌊n log log n⌋.

Abstract

The Green-Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves that there exist arbitrarily long strings of consecutive primes that lie in any arithmetic progression that contains infinitely many primes. Using the techniques of Shiu and Maier, this paper generalizes Shiu's Theorem to certain subsets of the primes such as primes of the form $\lfloor \pi n\rfloor$ and some of arithmetic density zero such as primes of the form $\lfloor n\log\log n\rfloor$.