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Polignac Numbers, Conjectures of Erdős on Gaps Between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture

In: From Arithmetic to Zeta-Functions

Author

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  • János Pintz

    (Rényi Mathematical Institute of the Hungarian Academy of Sciences)

Abstract

In the present work we prove a number of results about gaps between consecutive primes. The proofs need the method of Y. Zhang which led to the proof of infinitely many bounded gaps between primes. Several of the results refer to the so-called Polignac numbers which we define as those even integers which can be written in infinitely many ways as the difference of two consecutive primes. Others refer to several 60–70 years old conjecture of Paul Erdős about the distribution of the normalized gaps between consecutive primes and about the distribution of the ratio of consecutive primegaps. The methods involve an extended version of Zhangs method, a property of the GPY weights proved by the author a few years ago and other ideas as well.

Suggested Citation

  • János Pintz, 2016. "Polignac Numbers, Conjectures of Erdős on Gaps Between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture," Springer Books, in: Jürgen Sander & Jörn Steuding & Rasa Steuding (ed.), From Arithmetic to Zeta-Functions, pages 367-384, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-28203-9_22
    DOI: 10.1007/978-3-319-28203-9_22
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