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 ?Π n? and some of arithmetic density zero such as primes of the form ? n log log n ?.
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机译：格林陶氏定理是现代数论中最著名的定理之一，它指出素数存在任意长的算术级数。在一个相关但又不同的方向上，Shiu的一个新定理证明，存在任意长的连续素数字符串，这些字符串位于任何包含无限多个素数的算术级数中。本文使用Shiu和Maier的技术，将Shiu定理推广到素数的某些子集，例如形式为的素数。还有一些算术密度零，如形式的素数？ n log log n？。
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