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On Nearly Linear Recurrence Sequences Read chapter Read first chapter

2017 | OriginalPaper | Chapter

Patterns of Primes in Arithmetic Progressions

Author : János Pintz

Published in: Number Theory – Diophantine Problems, Uniform Distribution and Applications

Publisher: Springer International Publishing

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Abstract

After the proof of Zhang about the existence of infinitely many bounded gaps between consecutive primes the author showed the existence of a bounded d such that there are arbitrarily long arithmetic progressions of primes with the property that p ′ = p + d is the prime following p for each element of the progression. This was a common generalization of the results of Zhang and Green-Tao. In the present work it is shown that for every m we have a bounded m-tuple of primes such that this configuration (i.e. the integer translates of this m-tuple) appear as arbitrarily long arithmetic progressions in the sequence of all primes. In fact we show that this is true for a positive proportion of all m-tuples. This is a common generalization of the celebrated works of Green-Tao and Maynard/Tao.

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previous chapter Orbits of Algebraic Dynamical Systems in Subgroups and Subfields
next chapter On Simple Linear Recurrences
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Metadata
Title
Patterns of Primes in Arithmetic Progressions
Author
János Pintz
Copyright Year
2017
Book
Number Theory – Diophantine Problems, Uniform Distribution and Applications

Print ISBN: 978-3-319-55356-6

Electronic ISBN: 978-3-319-55357-3

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-55357-3_19

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