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

Janos Pintz

Abstract

In the present work we prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent fantastic achievement of Yitang Zhang about the existence of bounded gaps between consecutive primes. Most of these results would have belonged to the category of science fiction a decade ago. However, the presented results are far from being immediate consequences of Zhang's famous theorem: they require various new ideas, other important properties of the applied sieve function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, and Zhang, respectively.

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

Abstract

In the present work we prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent fantastic achievement of Yitang Zhang about the existence of bounded gaps between consecutive primes. Most of these results would have belonged to the category of science fiction a decade ago. However, the presented results are far from being immediate consequences of Zhang's famous theorem: they require various new ideas, other important properties of the applied sieve function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, and Zhang, respectively.

Paper Structure

This paper contains 13 sections, 16 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. History of the problems. Formulation of the results
  3. Approximations to the Twin Prime Conjecture
  4. Polignac numbers
  5. The normalized value distribution of $d_n$
  6. Comparison of two consecutive values of $d_n$
  7. Arithmetic progressions in the sequence of generalized twin primes
  8. Preparation for the proofs
  9. Polignac numbers
  10. The normalized value distribution of $d_n$
  11. Comparison of two consecutive values of $d_n$
  12. Arbitrarily long arithmetic progressions of generalized twin primes
  13. How to make Zhang's theorem effective?

Key Result

Theorem A

$\liminf\limits_{n \to \infty} d_n \leq 7\cdot 10^7$.

Theorems & Definitions (24)

  • Theorem A: Y. Zhang Zhang2013
  • Theorem B: GPY2009
  • Theorem C: GPY2009
  • Theorem D: GPY2010
  • Theorem E: Pin2013
  • Definition 1
  • Definition 2
  • Proposition
  • Theorem 1
  • Theorem 2
  • ...and 14 more