Landau’s problems on primes

János Pintz[1]

  • [1] Rényi Mathematical Institute of the Hungarian Academy of Sciences Budapest Reáltanoda u. 13–15 H-1053, Hungary

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 2, page 357-404
  • ISSN: 1246-7405

Abstract

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At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following :(1)Are there infinitely many primes of the form n 2 + 1 ?(2)The (Binary) Goldbach Conjecture, that every even number exceeding 2 can be written as the sum of two primes.(3)The Twin Prime Conjecture.(4)Does there exist always at least one prime between neighbouring squares?All these problems are still open. In the present work a survey will be given about partial results in Problems (2)–(4), with special emphasis on the recent results of D. Goldston, C. Yıldırım and the author on small gaps between primes.

How to cite

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Pintz, János. "Landau’s problems on primes." Journal de Théorie des Nombres de Bordeaux 21.2 (2009): 357-404. <http://eudml.org/doc/10886>.

@article{Pintz2009,
abstract = {At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following :(1)Are there infinitely many primes of the form ?(2)The (Binary) Goldbach Conjecture, that every even number exceeding 2 can be written as the sum of two primes.(3)The Twin Prime Conjecture.(4)Does there exist always at least one prime between neighbouring squares?All these problems are still open. In the present work a survey will be given about partial results in Problems (2)–(4), with special emphasis on the recent results of D. Goldston, C. Yıldırım and the author on small gaps between primes.},
affiliation = {Rényi Mathematical Institute of the Hungarian Academy of Sciences Budapest Reáltanoda u. 13–15 H-1053, Hungary},
author = {Pintz, János},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {primes; twin primes; Goldbach},
language = {eng},
number = {2},
pages = {357-404},
publisher = {Université Bordeaux 1},
title = {Landau’s problems on primes},
url = {http://eudml.org/doc/10886},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Pintz, János
TI - Landau’s problems on primes
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 2
SP - 357
EP - 404
AB - At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following :(1)Are there infinitely many primes of the form ?(2)The (Binary) Goldbach Conjecture, that every even number exceeding 2 can be written as the sum of two primes.(3)The Twin Prime Conjecture.(4)Does there exist always at least one prime between neighbouring squares?All these problems are still open. In the present work a survey will be given about partial results in Problems (2)–(4), with special emphasis on the recent results of D. Goldston, C. Yıldırım and the author on small gaps between primes.
LA - eng
KW - primes; twin primes; Goldbach
UR - http://eudml.org/doc/10886
ER -

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