Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Forbidden Integer Ratios of Consecutive Power Sums Read chapter Read first chapter

2016 | OriginalPaper | Chapter

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

Author : János Pintz

Published in: From Arithmetic to Zeta-Functions

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Selberg Sums: A New Perspective
next chapter Idempotents and Congruence
Literature
1.
2.
A. de Polignac, Six propositions arithmologiques déduites de crible d’Ératosthene. Nouv. Ann. Math. 8, 423–429 (1849)
3.
L.E. Dickson, A new extension of Dirichlet’s theorem on prime numbers. Messenger Math. 33 (2), 155–161 (1904)
4.
P.D.T.A. Elliott, H. Halberstam, A conjecture in prime number theory, in Symposia Mathematica, vol. 4 (INDAM, Rome, 1968/1969) (Academic, London, 1970), pp. 59–72
5.
6.
7.
P. Erdős, Some problems on the distribution of prime numbers. Teoria dei Numeri, Math. Congr. Varenna, 8 pp. (1954/1955).
8.
9.
B. Farkas, J. Pintz, Sz. Gy. Révész, On the optimal weight function in the Goldston–Pintz–Yıldırım method for finding small gaps between consecutive primes, in Turán Memorial, Number Theory, Combinatorics and Analysis (de Gruyter, Berlin, 2014), pp. 75–104
10.
11.
D.A. Goldston, J. Pintz, C. Yıldırım, Primes in tuples. I. Ann. Math. (2) 170 (2), 819–862 (2009)
12.
13.
14.
B. Green, T. Tao, The primes contain arbitrarily long arithmetic progressions. Ann. Math. (2) 167 (2), 481–547 (2008)
15.
H. Halberstam, H.-E. Richert, Sieve Methods (Academic Press, London, 1974)MATH
16.
G.H. Hardy, J.E. Littlewood, Some problems of ‘Partitio Numerorum’, III: on the expression of a number as a sum of primes. Acta Math. 44, 1–70 (1923)MathSciNetMATH
17.
18.
J. Maynard, Small gaps between primes, Ann. Math. (2) 181 (1), 383–413 (2015)
19.
20.
J. Pintz, Are there arbitrarily long arithmetic progressions in the sequence of twin primes?, in An Irregular Mind. Szemerédi is 70. Bolyai Society Mathematical Studies, vol. 21 (Springer, Berlin, 2010), pp. 525–559
21.
J. Pintz, On the singular series in the prime k-tuple conjecture (2010). arXiv: 1004.1084v1 [math.NT]
22.
J. Pintz, Some new results on gaps between consecutive primes, in Turán Memorial, Number Theory, Combinatorics and Analysis (de Gruyter, Berlin, 2014), pp. 231–247
23.
24.
G. Ricci, Sull’andamento della differenza di numeri primi consecutivi. Riv. Mat. Univ. Parma 5, 3–54 (1954)MathSciNetMATH
25.
26.
I.M. Vinogradov, Representation of an odd number as a sum of three prime numbers. Dokl. Akad. Nauk. SSSR 15, 291–294 (1937) (Russian)
27.
E. Westzynthius, Über die Verteilung der Zahlen, die zu der n ersten Primzahlen teilerfremd sind. Commun. Phys. Math. Helsingfors (5) 25, 1–37 (1931)
28.
Y. Zhang, Bounded gaps between primes. Ann. Math. (2) 179 (3), 1121–1174 (2014)
Metadata
Title
Polignac Numbers, Conjectures of Erdős on Gaps Between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture
Author
János Pintz
Copyright Year
2016
Book
From Arithmetic to Zeta-Functions

Print ISBN: 978-3-319-28202-2

Electronic ISBN: 978-3-319-28203-9

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-28203-9_22

Premium Partner

    Image Credits