Polymath Wiki - User contributions [en]

ABC conjecture

2015-02-16T20:09:45Z

Hannes:

The '''abc conjecture''' asserts, roughly speaking, that if a+b=c and a,b,c are coprime, then a,b,c cannot all be too smooth; in particular, the product of all the primes dividing a, b, or c has to exceed <math>c^{1-\varepsilon}</math> for any fixed <math>\varepsilon > 0</math> (if a,b,c are smooth).

This shows for instance that <math>(1-\varepsilon) \log N / 3</math>-smooth a,b,c of size N which are coprime cannot sum to form a+b=c. This unfortunately seems to be too weak to be of much use for the [[finding primes]] project.

A probabilistic heuristic justification for the ABC conjecture can be found at [http://terrytao.wordpress.com/2012/09/18/the-probabilistic-heuristic-justification-of-the-abc-conjecture/ this blog post].

* [[wikipedia:Abc_conjecture|Wikipedia page for the ABC conjecture]]
* [http://ncatlab.org/nlab/show/abc%20conjecture nLab page for the ABC conjecture]
* [http://www.ams.org/notices/200002/fea-mazur.pdf Questions about Powers of Numbers], Notices of the AMS, February 2000.
* [http://www.ams.org/notices/200210/fea-granville.pdf It's As Easy As abc], Andrew Granville and Thomas J. Tucker, Notices of the AMS, November 2002.

==Mochizuki's proof==

=== Papers ===
Mochizuki's claimed proof of the abc conjecture is conducted primarily through the following series of papers:

# (IUTT-I) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf Inter-universal Teichmuller Theory I: Construction of Hodge Theaters], Shinichi Mochizuki
# (IUTT-II) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20II.pdf Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic Evaluation], Shinichi Mochizuki
# (IUTT-III) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice], Shinichi Mochizuki
# (IUTT-IV) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations], Shinichi Mochizuki
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf A Panoramic Overview of Inter-universal Teichmuller Theory], Shinichi Mochizuki

Progress reports:
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202013-12.pdf On the Verification of Inter-Universal Teichmüller theory: A process report (as of december 2013)], Shinichi Mochizuki
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202014-12.pdf On the Verification of Inter-Universal Teichmüller theory: A process report (as of december 2014)], Shinichi Mochizuki

See also [http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Brief%20Introduction%20to%20Inter-universal%20Geometry%20(Tokyo%202004-01).pdf these earlier slides] of Mochizuki on inter-universal Teichmuller theory. The answers to [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture this MathOverflow post] (and in particular [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/106658#106658 Minhyong Kim's answer]) describe the philosophy behind Mochizuki's proof strategy. Go Yamashita has a [http://www.kurims.kyoto-u.ac.jp/~motizuki/FAQ%20on%20Inter-Universality.pdf short FAQ on inter-universality], which is a concept that appears in Mochizuki's arguments, though it does not appear to be the central ingredient in these papers.

The argument also relies heavily on Mochizuki's previous work on the Hodge-Arakelov theory of elliptic curves, including the following references:

* (HAT) [http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves.pdf The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories], Shinichi Mochizuki
* (GTKS) [http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Galois-Theoretic%20Kodaira-Spencer%20Morphism%20of%20an%20Elliptic%20Curve.pdf The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve], Shinichi Mochizuki
* (HAT-Survey-I) [http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Survey%20of%20the%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves%20I.pdf A Survey of the Hodge-Arakelov Theory of Elliptic Curves I], Shinichi Mochizuki
* (HAT-Survey-II) [http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Survey%20of%20the%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves%20II.pdf A Survey of the Hodge-Arakelov Theory of Elliptic Curves II], Shinichi Mochizuki
* (AbsTopIII) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Topics%20in%20Absolute%20Anabelian%20Geometry%20III.pdf Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms], Shinichi Mochizuki, RIMS Preprint 1626 (March 2008).
* (EtTh) [http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Etale%20Theta%20Function%20and%20its%20Frobenioid-theoretic%20Manifestations.pdf The Etale Theta Function and its Frobenioid-theoretic Manifestations], S. Mochizuki, Publ. Res. Inst. Math. Sci. 45 (2009), pp. 227-349. (See also [http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Etale%20Theta%20Function%20and%20its%20Frobenioid-theoretic%20Manifestations%20(comments).pdf this list] of errata for the paper.)

Anyone seeking to get a thorough "bottom-up" understanding of Mochizuki's argument will probably be best advised to start with these latter papers first. The papers (AbsTopIII), (EtTh) are directly cited heavily by the IUTT series of papers; the earlier papers (HAT), (GTKS) cover thematically related material but serve more as inspiration than as a source of mathematical results in the IUTT series.

The theory of (IUTT I-IV) is used to establish a Szpiro-type inequality, which is similar to [http://en.wikipedia.org/wiki/Szpiro's_conjecture Szpiro's conjecture] but with an additional genericity hypothesis on a certain parameter <math>\ell</math>. In order to then deduce the true Szpiro's conjecture (which is essentially equivalent to the abc conjecture), the results from the paper

* (GenEll) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Arithmetic%20Elliptic%20Curves%20in%20General%20Position.pdf Arithmetic Elliptic Curves in General Position], S. Mochizuki, Arithmetic Elliptic Curves in General Position,Math. J. Okayama Univ. 52 (2010), pp. 1-28.

are used. (Note that the published version of this paper requires some small corrections, listed [http://www.kurims.kyoto-u.ac.jp/~motizuki/Arithmetic%20Elliptic%20Curves%20in%20General%20Position%20(comments).pdf here].) See [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107386#107386 this MathOverflow post of Vesselin Dimitrov] for more discussion.

Here are the remainder of [http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html Shinichi Mochizuki's papers], and here is the [http://en.wikipedia.org/wiki/Shinichi_Mochizuki Wikipedia page for Shinichi Mochizuki].

===Specific topics===

* The last part of (IUTT-IV) explores the use of different models of ZFC set theory in order to more fully develop inter-universal Teichmuller theory (this part is not needed for the applications to the abc conjecture). There appears to be an inaccuracy in a remark in Section 3, page 43 of that paper regarding the conservative nature of the extension of ZFC by the addition of the Grothendieck universe axiom; see [http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/#comment-10605 this blog comment]. However, this remark was purely for motivational purposes and does not impact the proof of the abc conjecture.

* There is some discussion at [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107279#107279 this MathOverflow post] as to whether the explicit bounds for the abc conjecture are too strong to be consistent with known or conjectured lower bounds on abc. In particular, there appears to be a serious issue with the main Diophantine inequality (Theorem 1.10 of IUTT-IV), in that it appears to be inconsistent with commonly accepted conjectures, namely the abc conjecture and the uniform Serre open image conjecture. Mochizuki has written [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV%20(comments).pdf comments] in October 2012 to say that he hopes to post a revised version of Theorem 1.10 and its proof in the not too distant future.

* The question of whether the results in this paper can be made completely effective (which would be of importance for several applications) is discussed in some of the comments to [http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ this blog post].

* The category and topos theory viewpoint is discussed at the [http://nforum.mathforge.org/discussion/4260/abc-conjecture nForum page for the abc conjecture].
===Lectures===
* announced Lecture Series by [http://www.kurims.kyoto-u.ac.jp/~gokun/myworks.html Go Yamashita] at Kyushu University ([http://www.math.kyushu-u.ac.jp/seminars/view/1373 announcement in japanese]), three weeks (86,5 hours in total):
** 16.-19.09.2014 (18,5h)
** 09.-13.03.2015 (33,5h)
** 16.-20.03.2015 (35h)

The lectures in March will be part of a two-weeks workshop at RIMS: [http://www.kurims.kyoto-u.ac.jp/~motizuki/2015-03%20IUTeich%20Program%20(English).pdf program]

===Survey articles===
*Ivan Fesenko, [https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki]

===Blogs===
*[http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/ ABC conjecture rumor], Secret Blogging Seminar, 12 June, 2012
*[http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ Mochizuki on ABC], Quomodocumque, Jordan Ellenberg, 3 Sept, 2012
*[http://richardelwes.co.uk/2012/09/04/as-easy-as-123/ As easy as 123…], Simple City, Richard Elwes' Blog, 4 Sept, 2012
*[https://plus.google.com/103703080789076472131/posts/j1sEGnPyiRu Timothy Gowers Google+], 4 Sept, 2012
*[https://plus.google.com/117663015413546257905/posts/Npu7xDniXMS John Baez Google+], 4 Sept 2012, see also a [https://plus.google.com/117663015413546257905/posts/2vTzJJSueRb repost]
**[https://plus.google.com/117663015413546257905/posts/hzqBCeujWEg John Baez Google+], 5 Sept, 2012
**[https://plus.google.com/117663015413546257905/posts/d1RsN4KnCUs John Baez Google+], 12 Sept, 2012, by Minhyong Kim.
*[https://plus.google.com/114134834346472219368/posts/c7LkaWV69KL Terence Tao Google+], 4 Sept, 2012
*[http://www.math.columbia.edu/~woit/wordpress/?p=5104 Proof of the abc Conjecture?], Not Even Wrong, 4 Sept, 2012
*[http://gaussianos.com/posible-demostracion-de-la-veracidad-de-la-conjetura-abc/ Posible demostración de la veracidad de la conjetura ABC], Gaussianos, 5 Sept, 2012.
*[http://bit-player.org/2012/the-abc-game The abc game], bit-player, 7 Sept, 2012
*[http://oumathclub.wordpress.com/2012/09/09/the-abc-conjecture/ The abc Conjecture], U. Oklahoma math club, 9 Sept, 2012
*[http://golem.ph.utexas.edu/category/2012/09/the_axgrothendieck_theorem_acc.html The Ax-Grothendieck Theorem According to Category Theory], The n-Category Café, 10 Sept, 2012
*[http://www.oblomovka.com/wp/2012/09/11/touch-of-the-galois/ touch of the galois], Oblomovka, 11 Sept, 2012
*[http://rjlipton.wordpress.com/2012/09/12/the-abc-conjecture-and-cryptography/ The ABC Conjecture And Cryptography], Gödel’s Lost Letter and P=NP, 12 Sept, 2012
*[http://mochizukidenial.wordpress.com/ Mochizuki Denial], 14 Sept 2012
*[http://leisureguy.wordpress.com/2012/09/16/abc-proof-opens-new-vistas-in-math/ “ABC” proof opens new vistas in math], Later On, 16 Sept, 2012
*[http://mathbabe.org/2012/11/14/the-abc-conjecture-has-not-been-proved The ABC Conjecture has not been proved], Mathbabe, 14 Nov, 2012.
*[https://plus.google.com/u/0/115831511988650789490/posts/hJQoYM2FS6g in IUTeich the theta function corresponds to the gaze of the little girl into the “small house”], lieven lebruyn Google+, 27 May 2013
*[https://plus.google.com/u/0/115831511988650789490/posts/FWU8YD6xnNY MochizukiDenial], lieven lebruyn Google+, 28 May 2013
*[http://www.quora.com/Joseph-Heavner/Posts/An-overview-of-Inter-universal-Teichm%C3%BCller-Theory-and-Shinichi-Mochizukis-proof-of-the-ABC-Conjecture-along-with-th An overview of Inter-universal Teichmüller Theory and Shinichi Mochizuki's proof of the ABC Conjecture, along with the current situation and how we can begin to understand this theory], Joseph Heavner, Quora, Aug 18 2013
*[http://www.math.columbia.edu/~woit/wordpress/?p=6514 Latest on abc], Not Even Wrong, 19 Dec 2013
*[https://plus.google.com/+RichardElwes/posts/jMVfRcnRaoV Richard Elwes, Google+], 20 Dec 2013
*[http://www.math.columbia.edu/~woit/wordpress/?p=7451 Peter Woit on Progress-Report 2014], 13 Jan 2015

===2013 study of Geometry of Frobenioids===
*[https://plus.google.com/u/0/115831511988650789490/posts/Y1XVCDLWRP5 a baby Arithmetic Frobenioid], lieven lebruyn Google+, 29 May 2013
*[https://plus.google.com/u/0/115831511988650789490/posts/dx5vuxVewzN MinuteMochizuki 2 : a quadratic arithmetic Frobenioid], lieven lebruyn Google+, 31 May 2013
*[http://matrix.cmi.ua.ac.be/content/minutemochizuki-1 MinuteMochizuki 1], the bourbaki code, lieven's blog, 1 June 2013
*[http://matrix.cmi.ua.ac.be/content/minutemochizuki-2 MinuteMochizuki 2], the bourbaki code, lieven's blog, 1 June 2013
*[https://plus.google.com/115831511988650789490/posts/Y7okWptRtEW Mochizuki's menagerie of morphisms], lieven lebruyn Google+, 4 June 2013
*[https://plus.google.com/115831511988650789490/posts/aYDv916LeEi Mochizuki's categorical prime number sieve], lieven lebruyn Google+, 5 June 2013
*[https://plus.google.com/115831511988650789490/posts/4qxuDqXPgug Mochizuki's Frobenioids for the Working Category Theorist], lieven lebruyn Google+, 7 June 2013
*[http://matrix.cmi.ua.ac.be/content/minutemochizuki-3 MinuteMochizuki 3], lieven lebruyn Google+, 9 June 2013
*[https://plus.google.com/115831511988650789490/posts/SGM3gcvyoP1 my problem with Mochizuki's Frobenioid1] lieven lebruyn Google+, 11 June 2013
*[https://plus.google.com/115831511988650789490/posts/hK66h2artZc Mochizuki's Frobenioid reconstruction: the final bit] lieven lebruyn Google+, 12 July 2013

===Q & A===
*[http://mathoverflow.net/questions/852/what-is-inter-universal-geometry What is inter-universal geometry?], Mathoverflow, 17 Oct, 2009
*[http://mathoverflow.net/questions/106321/mochizukis-proof-and-siegel-zeros Mochizuki’s proof and Siegel zeros], Mathoverflow, 4 Sept, 2012
*[http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture Philosophy behind Mochizuki's work on the ABC conjecture], Mathoverflow, 7 Sept, 2012 (see also [http://meta.mathoverflow.net/discussion/1438/mochizuki-proof-of-abc the metapost] for this question)
*[http://cstheory.stackexchange.com/questions/12504/implications-of-proof-of-abc-conjecture-for-cs-theory Implications of proof of abc conjecture for cs theory], Theoretical Computer Science Stackexchange, 11 Sept, 2012
*[http://mathoverflow.net/questions/107379/model-theoretic-content-of-mochizukis-teichmuller-theory-papers Model-theoretic content of Mochizuki’s Teichmüller theory papers], Mathoverflow, 17 Sept 2012
*[http://math.stackexchange.com/questions/199609/groupification-and-perfection-of-a-commutative-monoid Groupification and perfection of a commutative monoid], Mathematics Stackexchange, 20 Sept 2012
*[http://www.quora.com/As-of-September-2014-what-is-the-mathematical-communitys-current-understanding-of-Mochizukis-proof-of-the-abc-conjecture As of September 2014, what is the mathematical community's current understanding of Mochizuki's proof of the abc conjecture?] Quora, September 2014.
*[http://mathoverflow.net/questions/195353/what-is-a-frobenioid What is a Frobenioid?], Mathoverflow, 31 January 2015.
*[http://mathoverflow.net/questions/195841/what-is-an-%C3%A9tale-theta-function What is an étale theta function?], Mathoverflow, 07 February 2015.

Note that Mathoverflow has a number of policies and guidelines regarding appropriate questions and answers to post on that site; see [http://mathoverflow.net/faq this FAQ for details].

===News Media===
*[http://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378 Proof claimed for deep connection between primes], Nature News, 10 September 2012, reprinted by Scientific American
*[http://www.newscientist.com/article/dn22256-fiendish-abc-proof-heralds-new-mathematical-universe.html Fiendish 'ABC proof' heralds new mathematical universe], New Scientist, 10 September 2012
*[http://news.yahoo.com/mathematician-claims-proof-connection-between-prime-numbers-131737044.html Mathematician Claims Proof of Connection between Prime Numbers], Yahoo News, 11 Sept 2012, reprinted by Huffington Post and MSNBC
*[http://news.sciencemag.org/sciencenow/2012/09/abc-conjecture.html ABC Proof Could Be Mathematical Jackpot], Science, 12 Sept 2012
*[http://www.nytimes.com/2012/09/18/science/possible-breakthrough-in-maths-abc-conjecture.html A Possible Breakthrough in Explaining a Mathematical Riddle], The New York Times, 17 Sept 2012
*[http://www.telegraph.co.uk/news/worldnews/asia/japan/9552155/Worlds-most-complex-mathematical-theory-cracked.html World's most complex mathematical theory 'cracked'], The Telegraph, 19 Sept 2012, reprinted by several other news outlets
*[http://www.dailyprincetonian.com/2012/09/20/31183/ U.-educated mathematician offers proof of pivotal number theory conjecture], The Daily Princetonian, 20 Sept 2012
*[http://bostonglobe.com/ideas/2012/11/03/abc-proof-too-tough-even-for-mathematicians/o9bja4kwPuXhDeDb2Ana2K/story.html An ABC proof too tough even for mathematicians], Kevin Hartnett, 3 Nov 2012.
*[http://projectwordsworth.com/the-paradox-of-the-proof/ The Paradox of the Proof], Caroline Chen, 10 May 2013.
*[http://www.newscientist.com/article/dn26753-mathematicians-anger-over-his-unread-500page-proof.html Mathematician's anger over his unread 500-page proof], Jacob Aron, 02 Jan 2015.

===Crowd News===
*[http://news.ycombinator.com/item?id=4476367 Shin Mochizuki has released his long-rumored proof of the ABC conjecture ], Hacker News, 5 Sept 2012
**[http://news.ycombinator.com/item?id=4502856 Proof Claimed for Deep Connection between Prime Numbers], Hacker News, 11 Sept 212
*[http://science.slashdot.org/story/12/09/10/226217/possible-proof-of-abc-conjecture Possible Proof of ABC Conjecture], Slashdot, September 10, 2012
*[http://www.metafilter.com/119847/Mathematics-world-abuzz-with-a-proof-of-the-ABC-Conjecture Mathematics world abuzz with a proof of the ABC Conjecture], MetaFilter, 11 Sept 2012
*[http://theconversation.edu.au/the-abc-conjecture-as-easy-as-1-2-3-or-not-10836 The abc conjecture, as easy as 1, 2, 3 ... or not ], The Conversation, Alexandru Ghitza, 26 Nov 2012.
*[http://www.sciencenews.org/view/generic/id/349199/description/A_theorem_in_limbo_shows_that_QED_is_not_the_last_word_in_a_mathematical_proof A theorem in limbo shows that QED is not the last word in a mathematical proof], March 25, 2013.

ABC conjecture

2015-02-16T17:45:55Z

Hannes: added link, please move to correct place

The '''abc conjecture''' asserts, roughly speaking, that if a+b=c and a,b,c are coprime, then a,b,c cannot all be too smooth; in particular, the product of all the primes dividing a, b, or c has to exceed <math>c^{1-\varepsilon}</math> for any fixed <math>\varepsilon > 0</math> (if a,b,c are smooth).

This shows for instance that <math>(1-\varepsilon) \log N / 3</math>-smooth a,b,c of size N which are coprime cannot sum to form a+b=c. This unfortunately seems to be too weak to be of much use for the [[finding primes]] project.

A probabilistic heuristic justification for the ABC conjecture can be found at [http://terrytao.wordpress.com/2012/09/18/the-probabilistic-heuristic-justification-of-the-abc-conjecture/ this blog post].

* [[wikipedia:Abc_conjecture|Wikipedia page for the ABC conjecture]]
* [http://ncatlab.org/nlab/show/abc%20conjecture nLab page for the ABC conjecture]
* [http://www.ams.org/notices/200002/fea-mazur.pdf Questions about Powers of Numbers], Notices of the AMS, February 2000.
* [http://www.ams.org/notices/200210/fea-granville.pdf It's As Easy As abc], Andrew Granville and Thomas J. Tucker, Notices of the AMS, November 2002.

==Mochizuki's proof==

=== Papers ===
Mochizuki's claimed proof of the abc conjecture is conducted primarily through the following series of papers:

# (IUTT-I) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf Inter-universal Teichmuller Theory I: Construction of Hodge Theaters], Shinichi Mochizuki
# (IUTT-II) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20II.pdf Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic Evaluation], Shinichi Mochizuki
# (IUTT-III) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice], Shinichi Mochizuki
# (IUTT-IV) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations], Shinichi Mochizuki
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf A Panoramic Overview of Inter-universal Teichmuller Theory], Shinichi Mochizuki

Progress reports:
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202013-12.pdf On the Verification of Inter-Universal Teichmüller theory: A process report (as of december 2013)], Shinichi Mochizuki
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202014-12.pdf On the Verification of Inter-Universal Teichmüller theory: A process report (as of december 2014)], Shinichi Mochizuki

See also [http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Brief%20Introduction%20to%20Inter-universal%20Geometry%20(Tokyo%202004-01).pdf these earlier slides] of Mochizuki on inter-universal Teichmuller theory. The answers to [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture this MathOverflow post] (and in particular [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/106658#106658 Minhyong Kim's answer]) describe the philosophy behind Mochizuki's proof strategy. Go Yamashita has a [http://www.kurims.kyoto-u.ac.jp/~motizuki/FAQ%20on%20Inter-Universality.pdf short FAQ on inter-universality], which is a concept that appears in Mochizuki's arguments, though it does not appear to be the central ingredient in these papers.

The argument also relies heavily on Mochizuki's previous work on the Hodge-Arakelov theory of elliptic curves, including the following references:

* (HAT) [http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves.pdf The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories], Shinichi Mochizuki
* (GTKS) [http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Galois-Theoretic%20Kodaira-Spencer%20Morphism%20of%20an%20Elliptic%20Curve.pdf The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve], Shinichi Mochizuki
* (HAT-Survey-I) [http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Survey%20of%20the%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves%20I.pdf A Survey of the Hodge-Arakelov Theory of Elliptic Curves I], Shinichi Mochizuki
* (HAT-Survey-II) [http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Survey%20of%20the%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves%20II.pdf A Survey of the Hodge-Arakelov Theory of Elliptic Curves II], Shinichi Mochizuki
* (AbsTopIII) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Topics%20in%20Absolute%20Anabelian%20Geometry%20III.pdf Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms], Shinichi Mochizuki, RIMS Preprint 1626 (March 2008).
* (EtTh) [http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Etale%20Theta%20Function%20and%20its%20Frobenioid-theoretic%20Manifestations.pdf The Etale Theta Function and its Frobenioid-theoretic Manifestations], S. Mochizuki, Publ. Res. Inst. Math. Sci. 45 (2009), pp. 227-349. (See also [http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Etale%20Theta%20Function%20and%20its%20Frobenioid-theoretic%20Manifestations%20(comments).pdf this list] of errata for the paper.)

Anyone seeking to get a thorough "bottom-up" understanding of Mochizuki's argument will probably be best advised to start with these latter papers first. The papers (AbsTopIII), (EtTh) are directly cited heavily by the IUTT series of papers; the earlier papers (HAT), (GTKS) cover thematically related material but serve more as inspiration than as a source of mathematical results in the IUTT series.

The theory of (IUTT I-IV) is used to establish a Szpiro-type inequality, which is similar to [http://en.wikipedia.org/wiki/Szpiro's_conjecture Szpiro's conjecture] but with an additional genericity hypothesis on a certain parameter <math>\ell</math>. In order to then deduce the true Szpiro's conjecture (which is essentially equivalent to the abc conjecture), the results from the paper

* (GenEll) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Arithmetic%20Elliptic%20Curves%20in%20General%20Position.pdf Arithmetic Elliptic Curves in General Position], S. Mochizuki, Arithmetic Elliptic Curves in General Position,Math. J. Okayama Univ. 52 (2010), pp. 1-28.

are used. (Note that the published version of this paper requires some small corrections, listed [http://www.kurims.kyoto-u.ac.jp/~motizuki/Arithmetic%20Elliptic%20Curves%20in%20General%20Position%20(comments).pdf here].) See [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107386#107386 this MathOverflow post of Vesselin Dimitrov] for more discussion.

Here are the remainder of [http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html Shinichi Mochizuki's papers], and here is the [http://en.wikipedia.org/wiki/Shinichi_Mochizuki Wikipedia page for Shinichi Mochizuki].

===Specific topics===

* The last part of (IUTT-IV) explores the use of different models of ZFC set theory in order to more fully develop inter-universal Teichmuller theory (this part is not needed for the applications to the abc conjecture). There appears to be an inaccuracy in a remark in Section 3, page 43 of that paper regarding the conservative nature of the extension of ZFC by the addition of the Grothendieck universe axiom; see [http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/#comment-10605 this blog comment]. However, this remark was purely for motivational purposes and does not impact the proof of the abc conjecture.

* There is some discussion at [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107279#107279 this MathOverflow post] as to whether the explicit bounds for the abc conjecture are too strong to be consistent with known or conjectured lower bounds on abc. In particular, there appears to be a serious issue with the main Diophantine inequality (Theorem 1.10 of IUTT-IV), in that it appears to be inconsistent with commonly accepted conjectures, namely the abc conjecture and the uniform Serre open image conjecture. Mochizuki has written [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV%20(comments).pdf comments] in October 2012 to say that he hopes to post a revised version of Theorem 1.10 and its proof in the not too distant future.

* The question of whether the results in this paper can be made completely effective (which would be of importance for several applications) is discussed in some of the comments to [http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ this blog post].

* The category and topos theory viewpoint is discussed at the [http://nforum.mathforge.org/discussion/4260/abc-conjecture nForum page for the abc conjecture].
===Lectures===
* announced Lecture Series by [http://www.kurims.kyoto-u.ac.jp/~gokun/myworks.html Go Yamashita] at Kyushu University ([http://www.math.kyushu-u.ac.jp/seminars/view/1373 announcement in japanese]), three weeks (86,5 hours in total):
** 16.-19.09.2014 (18,5h)
** 09.-13.03.2015 (33,5h)
** 16.-20.03.2015 (35h)

The lectures in March will be part of a two-weeks workshop at RIMS: [http://www.kurims.kyoto-u.ac.jp/~motizuki/2015-03%20IUTeich%20Program%20(English).pdf program]

===Blogs===
*[http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/ ABC conjecture rumor], Secret Blogging Seminar, 12 June, 2012
*[http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ Mochizuki on ABC], Quomodocumque, Jordan Ellenberg, 3 Sept, 2012
*[http://richardelwes.co.uk/2012/09/04/as-easy-as-123/ As easy as 123…], Simple City, Richard Elwes' Blog, 4 Sept, 2012
*[https://plus.google.com/103703080789076472131/posts/j1sEGnPyiRu Timothy Gowers Google+], 4 Sept, 2012
*[https://plus.google.com/117663015413546257905/posts/Npu7xDniXMS John Baez Google+], 4 Sept 2012, see also a [https://plus.google.com/117663015413546257905/posts/2vTzJJSueRb repost]
**[https://plus.google.com/117663015413546257905/posts/hzqBCeujWEg John Baez Google+], 5 Sept, 2012
**[https://plus.google.com/117663015413546257905/posts/d1RsN4KnCUs John Baez Google+], 12 Sept, 2012, by Minhyong Kim.
*[https://plus.google.com/114134834346472219368/posts/c7LkaWV69KL Terence Tao Google+], 4 Sept, 2012
*[http://www.math.columbia.edu/~woit/wordpress/?p=5104 Proof of the abc Conjecture?], Not Even Wrong, 4 Sept, 2012
*[http://gaussianos.com/posible-demostracion-de-la-veracidad-de-la-conjetura-abc/ Posible demostración de la veracidad de la conjetura ABC], Gaussianos, 5 Sept, 2012.
*[http://bit-player.org/2012/the-abc-game The abc game], bit-player, 7 Sept, 2012
*[http://oumathclub.wordpress.com/2012/09/09/the-abc-conjecture/ The abc Conjecture], U. Oklahoma math club, 9 Sept, 2012
*[http://golem.ph.utexas.edu/category/2012/09/the_axgrothendieck_theorem_acc.html The Ax-Grothendieck Theorem According to Category Theory], The n-Category Café, 10 Sept, 2012
*[http://www.oblomovka.com/wp/2012/09/11/touch-of-the-galois/ touch of the galois], Oblomovka, 11 Sept, 2012
*[http://rjlipton.wordpress.com/2012/09/12/the-abc-conjecture-and-cryptography/ The ABC Conjecture And Cryptography], Gödel’s Lost Letter and P=NP, 12 Sept, 2012
*[http://mochizukidenial.wordpress.com/ Mochizuki Denial], 14 Sept 2012
*[http://leisureguy.wordpress.com/2012/09/16/abc-proof-opens-new-vistas-in-math/ “ABC” proof opens new vistas in math], Later On, 16 Sept, 2012
*[http://mathbabe.org/2012/11/14/the-abc-conjecture-has-not-been-proved The ABC Conjecture has not been proved], Mathbabe, 14 Nov, 2012.
*[https://plus.google.com/u/0/115831511988650789490/posts/hJQoYM2FS6g in IUTeich the theta function corresponds to the gaze of the little girl into the “small house”], lieven lebruyn Google+, 27 May 2013
*[https://plus.google.com/u/0/115831511988650789490/posts/FWU8YD6xnNY MochizukiDenial], lieven lebruyn Google+, 28 May 2013
*[http://www.quora.com/Joseph-Heavner/Posts/An-overview-of-Inter-universal-Teichm%C3%BCller-Theory-and-Shinichi-Mochizukis-proof-of-the-ABC-Conjecture-along-with-th An overview of Inter-universal Teichmüller Theory and Shinichi Mochizuki's proof of the ABC Conjecture, along with the current situation and how we can begin to understand this theory], Joseph Heavner, Quora, Aug 18 2013
*[http://www.math.columbia.edu/~woit/wordpress/?p=6514 Latest on abc], Not Even Wrong, 19 Dec 2013
*[https://plus.google.com/+RichardElwes/posts/jMVfRcnRaoV Richard Elwes, Google+], 20 Dec 2013
*[http://www.math.columbia.edu/~woit/wordpress/?p=7451 Peter Woit on Progress-Report 2014], 13 Jan 2015
*[https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf Ivan Fesenko, Arithmetic Deformation Theory via Arithmetic Fundamental Groups and Nonarchimedean Theta-Functions, Notes on the Work of Shinichi Mochizuki]

===2013 study of Geometry of Frobenioids===
*[https://plus.google.com/u/0/115831511988650789490/posts/Y1XVCDLWRP5 a baby Arithmetic Frobenioid], lieven lebruyn Google+, 29 May 2013
*[https://plus.google.com/u/0/115831511988650789490/posts/dx5vuxVewzN MinuteMochizuki 2 : a quadratic arithmetic Frobenioid], lieven lebruyn Google+, 31 May 2013
*[http://matrix.cmi.ua.ac.be/content/minutemochizuki-1 MinuteMochizuki 1], the bourbaki code, lieven's blog, 1 June 2013
*[http://matrix.cmi.ua.ac.be/content/minutemochizuki-2 MinuteMochizuki 2], the bourbaki code, lieven's blog, 1 June 2013
*[https://plus.google.com/115831511988650789490/posts/Y7okWptRtEW Mochizuki's menagerie of morphisms], lieven lebruyn Google+, 4 June 2013
*[https://plus.google.com/115831511988650789490/posts/aYDv916LeEi Mochizuki's categorical prime number sieve], lieven lebruyn Google+, 5 June 2013
*[https://plus.google.com/115831511988650789490/posts/4qxuDqXPgug Mochizuki's Frobenioids for the Working Category Theorist], lieven lebruyn Google+, 7 June 2013
*[http://matrix.cmi.ua.ac.be/content/minutemochizuki-3 MinuteMochizuki 3], lieven lebruyn Google+, 9 June 2013
*[https://plus.google.com/115831511988650789490/posts/SGM3gcvyoP1 my problem with Mochizuki's Frobenioid1] lieven lebruyn Google+, 11 June 2013
*[https://plus.google.com/115831511988650789490/posts/hK66h2artZc Mochizuki's Frobenioid reconstruction: the final bit] lieven lebruyn Google+, 12 July 2013

===Q & A===
*[http://mathoverflow.net/questions/852/what-is-inter-universal-geometry What is inter-universal geometry?], Mathoverflow, 17 Oct, 2009
*[http://mathoverflow.net/questions/106321/mochizukis-proof-and-siegel-zeros Mochizuki’s proof and Siegel zeros], Mathoverflow, 4 Sept, 2012
*[http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture Philosophy behind Mochizuki's work on the ABC conjecture], Mathoverflow, 7 Sept, 2012 (see also [http://meta.mathoverflow.net/discussion/1438/mochizuki-proof-of-abc the metapost] for this question)
*[http://cstheory.stackexchange.com/questions/12504/implications-of-proof-of-abc-conjecture-for-cs-theory Implications of proof of abc conjecture for cs theory], Theoretical Computer Science Stackexchange, 11 Sept, 2012
*[http://mathoverflow.net/questions/107379/model-theoretic-content-of-mochizukis-teichmuller-theory-papers Model-theoretic content of Mochizuki’s Teichmüller theory papers], Mathoverflow, 17 Sept 2012
*[http://math.stackexchange.com/questions/199609/groupification-and-perfection-of-a-commutative-monoid Groupification and perfection of a commutative monoid], Mathematics Stackexchange, 20 Sept 2012
*[http://www.quora.com/As-of-September-2014-what-is-the-mathematical-communitys-current-understanding-of-Mochizukis-proof-of-the-abc-conjecture As of September 2014, what is the mathematical community's current understanding of Mochizuki's proof of the abc conjecture?] Quora, September 2014.
*[http://mathoverflow.net/questions/195353/what-is-a-frobenioid What is a Frobenioid?], Mathoverflow, 31 January 2015.
*[http://mathoverflow.net/questions/195841/what-is-an-%C3%A9tale-theta-function What is an étale theta function?], Mathoverflow, 07 February 2015.

Note that Mathoverflow has a number of policies and guidelines regarding appropriate questions and answers to post on that site; see [http://mathoverflow.net/faq this FAQ for details].

===News Media===
*[http://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378 Proof claimed for deep connection between primes], Nature News, 10 September 2012, reprinted by Scientific American
*[http://www.newscientist.com/article/dn22256-fiendish-abc-proof-heralds-new-mathematical-universe.html Fiendish 'ABC proof' heralds new mathematical universe], New Scientist, 10 September 2012
*[http://news.yahoo.com/mathematician-claims-proof-connection-between-prime-numbers-131737044.html Mathematician Claims Proof of Connection between Prime Numbers], Yahoo News, 11 Sept 2012, reprinted by Huffington Post and MSNBC
*[http://news.sciencemag.org/sciencenow/2012/09/abc-conjecture.html ABC Proof Could Be Mathematical Jackpot], Science, 12 Sept 2012
*[http://www.nytimes.com/2012/09/18/science/possible-breakthrough-in-maths-abc-conjecture.html A Possible Breakthrough in Explaining a Mathematical Riddle], The New York Times, 17 Sept 2012
*[http://www.telegraph.co.uk/news/worldnews/asia/japan/9552155/Worlds-most-complex-mathematical-theory-cracked.html World's most complex mathematical theory 'cracked'], The Telegraph, 19 Sept 2012, reprinted by several other news outlets
*[http://www.dailyprincetonian.com/2012/09/20/31183/ U.-educated mathematician offers proof of pivotal number theory conjecture], The Daily Princetonian, 20 Sept 2012
*[http://bostonglobe.com/ideas/2012/11/03/abc-proof-too-tough-even-for-mathematicians/o9bja4kwPuXhDeDb2Ana2K/story.html An ABC proof too tough even for mathematicians], Kevin Hartnett, 3 Nov 2012.
*[http://projectwordsworth.com/the-paradox-of-the-proof/ The Paradox of the Proof], Caroline Chen, 10 May 2013.
*[http://www.newscientist.com/article/dn26753-mathematicians-anger-over-his-unread-500page-proof.html Mathematician's anger over his unread 500-page proof], Jacob Aron, 02 Jan 2015.

===Crowd News===
*[http://news.ycombinator.com/item?id=4476367 Shin Mochizuki has released his long-rumored proof of the ABC conjecture ], Hacker News, 5 Sept 2012
**[http://news.ycombinator.com/item?id=4502856 Proof Claimed for Deep Connection between Prime Numbers], Hacker News, 11 Sept 212
*[http://science.slashdot.org/story/12/09/10/226217/possible-proof-of-abc-conjecture Possible Proof of ABC Conjecture], Slashdot, September 10, 2012
*[http://www.metafilter.com/119847/Mathematics-world-abuzz-with-a-proof-of-the-ABC-Conjecture Mathematics world abuzz with a proof of the ABC Conjecture], MetaFilter, 11 Sept 2012
*[http://theconversation.edu.au/the-abc-conjecture-as-easy-as-1-2-3-or-not-10836 The abc conjecture, as easy as 1, 2, 3 ... or not ], The Conversation, Alexandru Ghitza, 26 Nov 2012.
*[http://www.sciencenews.org/view/generic/id/349199/description/A_theorem_in_limbo_shows_that_QED_is_not_the_last_word_in_a_mathematical_proof A theorem in limbo shows that QED is not the last word in a mathematical proof], March 25, 2013.

Bounded gaps between primes

2013-06-28T07:29:14Z

Hannes: /* World records */ added + to avoid confusion

This is the home page for the Polymath8 project "bounded gaps between primes".

== World records ==

* <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
* <math>k_0</math> is a quantity such that every admissible <math>k_0</math>-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in <math>k_0</math> lead to improvements in <math>H</math>. (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].)
* <math>\varpi</math> is a technical parameter related to a specialized form of the [http://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture Elliott-Halberstam conjecture]. Would like to be as large as possible. Improvements in <math>\varpi</math> lead to improvements in <math>k_0</math>, as described in the page on [[Dickson-Hardy-Littlewood theorems]]. In more recent work, the single parameter <math>\varpi</math> is replaced by a pair <math>(\varpi,\delta)</math> (in previous work we had <math>\delta=\varpi</math>). These estimates are obtained in turn from Type I, Type II, and Type III estimates, as described at the page on [[distribution of primes in smooth moduli]].

In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored.

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| 14 May
| 1/1,168 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| All subsequent work is based on Zhang's breakthrough paper.
|-
| 21 May
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| 28 May
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| 30 May
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| 31 May
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| 1 Jun
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| 2 Jun
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| 3 Jun
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| 4 Jun
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| 5 Jun
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| 6 Jun
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| 7 Jun
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>828 \varpi + 172\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| Jun 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])

[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])

[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
|-
| Jun 12
|
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])

[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
|-
| Jun 13
|
|
|

[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
|
|-
| Jun 14
|
|
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
|
|-
| Jun 15
| <math>348\varpi+68\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])

6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
|-
| Jun 16
| <math>348\varpi+68\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])

<strike>155\varpi+31\delta < 1 and 220\varpi + 60\delta < 1 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>

<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>

<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
|-
| Jun 18
|
| 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])

5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])

5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])

5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])

| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)

58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])

[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])

53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
|-
| Jun 19
|
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])

5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])

[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5455_51540_4678.txt 51,540?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24050 xfxie]/[http://math.mit.edu/~drew/admissible_5455_51540.txt Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])

53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])

[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
|-
| Jun 20
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])

<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
|
|
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
|-
| Jun 21
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])

1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records9.txt here] (update: As of June 27 these tables have been merged and uploaded to an [http://math.mit.edu/~primegaps/ online database] of current bounds on <math>H(k)</math> for <math>k</math> up to 5000).
|-
| Jun 22
|
| <strike>1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235740 Harcos]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235759 v08ltu])</strike>
| <strike>[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])</strike>
| Slight improvement in the <math>\tilde \theta</math> parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of <math>k_0</math>
|-
| Jun 23
|
| 1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235891 Paldi]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235905 Harcos])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| An improved monotonicity formula for <math>G_{k_0-1,\tilde \theta}</math> reduces <math>\kappa_3</math> somewhat
|-
| Jun 24
| <math>(134 + \tfrac{2}{3}) \varpi + 28\delta \le 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])

<math>140\varpi + 32 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 Tao])

1/88?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])

1/74?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])
| 1,268? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/k1268_10206.txt 10,206?] ([http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/ Engelsma])
| A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
|-
| Jun 25
| <math>116\varpi+30\delta<1</math>? ([http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation Fouvry-Kowalski-Michel-Nelson]/[http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236237 Tao])
| 1,346? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236123 Hannes])

502?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236162 Trevino])

1,007? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236242 Hannes])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/k1346_10876.txt 10,876]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/ Engelsma])

[http://www.opertech.com/primes/webdata/k2-999/k500-599/k502_3612.txt 3,612]?? ([http://www.opertech.com/primes/webdata/k2-999/k500-599/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1007_7860.txt 7,860]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/ Engelsma])

| Optimistic projections arise from combining the Graham-Ringrose numerology with the announced Fouvry-Kowalski-Michel-Nelson results on d_3 distribution
|-
| Jun 26
| <math>116\varpi + 25.5 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236346 Nielsen])

<math>(112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236387 Tao])
| 962? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236406 Hannes])
| [http://www.opertech.com/primes/webdata/k2-999/k900-999/k962_7470.txt 7,470]? ([http://www.opertech.com/primes/webdata/k2-999/k900-999 Engelsma])
| Beginning to flesh out various "levels" of Type I, Type II, and Type III estimates, see [[Distribution of primes in smooth moduli|this page]], in particular optimising van der Corput in the Type I sums. Integrated tuples page [http://math.mit.edu/~primegaps/ now online].
|-
| Jun 27
| <math>108\varpi + 30 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236502 Tao])
| 902? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236507 Hannes])
| [http://math.mit.edu/~primegaps/tuples/admissible_902_6966.txt 6,966]? ([http://math.mit.edu/~primegaps/ Engelsma])
| Improved the Type III estimates by averaging in <math>\alpha</math>; also some slight improvements to the Type II sums. [http://math.mit.edu/~primegaps/ Tuples page] is now accepting submissions.
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

== Polymath threads ==

* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart], Scott Morrison, 30 May 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang], Terence Tao, 3 June 2013. Inactive
* [http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/ Polymath proposal: bounded gaps between primes], Terence Tao, 4 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ Online reading seminar for Zhang’s “bounded gaps between primes”], Terence Tao, 4 June 2013. Inactive
* [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/ More narrow admissible sets], Scott Morrison, 5 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ The elementary Selberg sieve and bounded prime gaps], Terence Tao, 8 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/ A combinatorial subset sum problem associated with bounded prime gaps], Terence Tao, 10 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Further analysis of the truncated GPY sieve], Terence Tao, 11 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/ Estimation of the Type I and Type II sums], Terence Tao, 12 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/ Estimation of the Type III sums], Terence Tao, 14 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/ A truncated elementary Selberg sieve of Pintz], Terence Tao, 18 June, 2013. Active
* [http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/ Bounding short exponential sums on smooth moduli via Weyl differencing], Terence Tao, 22 June, 2013. Active
* [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/ The distribution of primes in densely divisible moduli], Terence Tao, 23 June, 2013. Active

== Code and data ==

* [https://github.com/semorrison/polymath8 Hensely-Richards sequences], Scott Morrison
** [http://tqft.net/misc/finding%20k_0.nb A mathematica notebook for finding k_0], Scott Morrison
** [https://github.com/avi-levy/dhl python implementation], Avi Levy
* [http://www.opertech.com/primes/k-tuples.html k-tuple pattern data], Thomas J Engelsma
** [https://perswww.kuleuven.be/~u0040935/k0graph.png A graph of this data]
** [http://www.opertech.com/primes/webdata/ Tuples giving this data]
* [https://github.com/vit-tucek/admissible_sets Sifted sequences], Vit Tucek
* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Other sifted sequences], Christian Elsholtz
* [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/S0025-5718-01-01348-5.pdf Size of largest admissible tuples in intervals of length up to 1050], Clark and Jarvis
* [http://math.mit.edu/~drew/admissable_v0.1.tar C implementation of the "greedy-greedy" algorithm], Andrew Sutherland
* [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Dropbox for sequences], pedant
* [https://docs.google.com/spreadsheet/ccc?key=0Ao3urQ79oleSdEZhRS00X1FLQjM3UlJTZFRqd19ySGc&usp=sharing Spreadsheet for admissible sequences], Vit Tucek
* [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.1.zip Java code for optimising a given tuple V1.1], xfxie

== Errata ==

Page numbers refer to the file linked to for the relevant paper.

# Errata for Zhang's "[http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes]"
## Page 5: In the first display, <math>\mathcal{E}</math> should be multiplied by <math>\mathcal{L}^{2k_0+2l_0}</math>, because <math>\lambda(n)^2</math> in (2.2) can be that large, cf. (2.4).
## Page 14: In the final display, the constraint <math>(n,d_1=1</math> should be <math>(n,d_1)=1</math>.
## Page 35: In the display after (10.5), the subscript on <math>{\mathcal J}_i</math> should be deleted.
## Page 36: In the third display, a factor of <math>\tau(q_0r)^{O(1)}</math> may be needed on the right-hand side (but is ultimately harmless).
## Page 38: In the display after (10.14), <math>\xi(r,a;q_1,b_1;q_2,b_2;n,k)</math> should be <math>\xi(r,a;k;q_1,b_1;q_2,b_2;n)</math>.
## Page 42: In (12.3), <math>B</math> should probably be 2.
## Page 47: In the third display after (13.13), the condition <math>l \in {\mathcal I}_i(h)</math> should be <math>l \in {\mathcal I}_i(sh)</math>.
## Page 49: In the top line, a comma in <math>(h_1,h_2;,n_1,n_2)</math> should be deleted.
## Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
## Page 54: Three displays before (14.17), <math>\bar{r_2}(m_1+m_2)q</math> should be <math>\bar{r_2}(m_1+m_2)/q</math>.
# Errata for Motohashi-Pintz's "[http://arxiv.org/pdf/math/0602599v1.pdf A smoothed GPY sieve]"
## Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of <math>(\log \frac{R}{|D|})^{2\ell+1}</math>, (4.15) contains instead a factor of <math>(\log \frac{R/w}{|K|})^{2\ell+1}</math> which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of <math>\exp(-k\omega/3)</math> in (4.15) does not seem to be available for estimating the second sum in (5.14).
# Errata for Pintz's "[http://arxiv.org/pdf/1306.1497v1.pdf A note on bounded gaps between primes]", version 1. Update: the errata below have been corrected in subsequent versions of Pintz's paper.
## Page 7: In (2.39), the exponent of <math>3a/2</math> should instead be <math>-5a/2</math> (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
## Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

== Other relevant blog posts ==

* [http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/ Marker lecture III: “Small gaps between primes”], Terence Tao, 19 Nov 2008.
* [http://blogs.ethz.ch/kowalski/2009/01/22/the-goldston-pintz-yildirim-result-and-how-far-do-we-have-to-walk-to-twin-primes/ The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?], Emmanuel Kowalski, 22 Jan 2009.
* [http://www.math.columbia.edu/~woit/wordpress/?p=5865 Number Theory News], Peter Woit, 12 May 2013.
* [http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html Bounded Gaps Between Primes], Emily Riehl, 14 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ Bounded gaps between primes!], Emmanuel Kowalski, 21 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/04/bounded-gaps-between-primes-some-grittier-details/ Bounded gaps between primes: some grittier details], Emmanuel Kowalski, 4 June 2013.
** [http://www.math.ethz.ch/~kowalski/zhang-notes.pdf The slides from the talk mentioned in that post]
* [http://aperiodical.com/2013/06/bound-on-prime-gaps-bound-decreasing-by-leaps-and-bounds/ Bound on prime gaps bound decreasing by leaps and bounds], Christian Perfect, 8 June 2013.
* [http://blogs.ams.org/blogonmathblogs/2013/06/14/narrowing-the-gap/ Narrowing the Gap], Brie Finegold, AMS Blog on Math Blogs, 14 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/22/bounded-gaps-between-primes-the-dawn-of-some-enlightenment/ Bounded gaps between primes: the dawn of (some) enlightenment], Emmanuel Kowalski, 22 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation/ A ternary divisor variation], Emmanuel Kowalski, 25 June 2013.

== MathOverflow ==

* [http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture], 20 May 2013.
* [http://mathoverflow.net/questions/131825/a-technical-question-related-to-zhangs-result-of-bounded-prime-gaps A technical question related to Zhang’s result of bounded prime gaps], 25 May 2013.
* [http://mathoverflow.net/questions/132452/how-does-yitang-zhang-use-cauchys-inequality-and-theorem-2-to-obtain-the-error How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the <math>S_2</math> sum], 31 May 2013.
* [http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed Tightening Zhang’s bound], 3 June 2013.
** [http://meta.mathoverflow.net/discussion/1605/tightening-zhangs-bound/ Metathread for this post]
* [http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-len Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?], 3 June 2013.

== Wikipedia and other references ==

* [http://en.wikipedia.org/wiki/Bessel_function Bessel function]
* [http://en.wikipedia.org/wiki/Bombieri-Vinogradov_theorem Bombieri-Vinogradov theorem]
* [http://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem Brun-Titchmarsh theorem]
* [http://www.encyclopediaofmath.org/index.php/Dispersion_method Dispersion method]
* [http://en.wikipedia.org/wiki/Prime_gap Prime gap]
* [http://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture Second Hardy-Littlewood conjecture]
* [http://en.wikipedia.org/wiki/Twin_prime_conjecture Twin prime conjecture]

== Recent papers and notes ==

* [http://arxiv.org/abs/1304.3199 On the exponent of distribution of the ternary divisor function], Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
* [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes], Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages. [http://arxiv.org/abs/1306.2133 arXiv]
* [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes], Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
* [http://arxiv.org/abs/1305.6289 Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture], Janos Pintz, 27 May 2013.
* [http://arxiv.org/abs/1305.6369 A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million], T. S. Trudgian, 28 May 2013.
* [http://www.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf The Friedlander-Iwaniec sum], É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
* [http://terrytao.files.wordpress.com/2013/06/bounds.pdf Notes on Zhang's prime gaps paper], Terence Tao, 1 June 2013.
* [http://arxiv.org/abs/1306.0511 Bounded prime gaps in short intervals], Johan Andersson, 3 June 2013.
* [http://arxiv.org/abs/1306.0948 Bounded length intervals containing two primes and an almost-prime II], James Maynard, 5 June 2013.
* [http://arxiv.org/abs/1306.1497 A note on bounded gaps between primes], Janos Pintz, 6 June 2013.
* [https://sites.google.com/site/avishaytal/files/Primes.pdf Lower Bounds for Admissible k-tuples], Avishay Tal, 15 June 2013.
* Notes in a truncated elementary Selberg sieve ([http://terrytao.files.wordpress.com/2013/06/file-1.pdf Section 1], [http://terrytao.files.wordpress.com/2013/06/file-2.pdf Section 2], [http://terrytao.files.wordpress.com/2013/06/file-3.pdf Section 3]), Janos Pintz, 18 June 2013.

== Media ==

* [http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989 First proof that infinitely many prime numbers come in pairs], Maggie McKee, Nature, 14 May 2013.
* [http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html Proof that an infinite number of primes are paired], Lisa Grossman, New Scientist, 14 May 2013.
* [https://www.simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/ Unheralded Mathematician Bridges the Prime Gap], Erica Klarreich, Simons science news, 20 May 2013.
** The article also appeared on Wired as "[http://www.wired.com/wiredscience/2013/05/twin-primes/ Unknown Mathematician Proves Elusive Property of Prime Numbers]".
* [http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html The Beauty of Bounded Gaps], Jordan Ellenberg, Slate, 22 May 2013.
* [http://www.newscientist.com/article/dn23644 Game of proofs boosts prime pair result by millions], Jacob Aron, New Scientist, 4 June 2013.

== Bibliography ==

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

* [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. [http://www.ams.org/mathscinet-getitem?mr=834613 MathSciNet] [http://link.springer.com/article/10.1007%2FBF02399204 Article]
* [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. [http://www.ams.org/mathscinet-getitem?mr=891581 MathSciNet] [https://eudml.org/doc/164255 Article]
* [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. [http://www.ams.org/mathscinet-getitem?mr=976723 MathSciNet] [http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Article]
* [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
* [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 [http://www.ams.org/mathscinet-getitem?mr=1836929 MathSciNet] [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Article]
* [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). [http://www.ams.org/mathscinet-getitem?mr=610700 MathSciNet] [http://www.math.ethz.ch/~kowalski/fouvry-iwaniec-on-a-theorem.pdf Article]
* [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. [http://www.ams.org/mathscinet-getitem?mr=719249 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4226.pdf Article]
* [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. [http://www.ams.org/mathscinet-getitem?mr=786351 MathSciNet] [http://www.jstor.org/stable/1971175 JSTOR] [http://www.jstor.org/stable/1971176 Appendix]
* [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. [http://arxiv.org/abs/math/0508185 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2552109 MathSciNet]
* [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. [http://www.ams.org/mathscinet-getitem?mr=1726073 MathSciNet] [http://www.ccrwest.org/gordon/ants.pdf Article]
* [GR1980] S. W. Graham, C. J. Ringrose, Lower bounds for least quadratic nonresidues. Analytic number theory (Allerton Park, IL, 1989), 269–309, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990. [http://www.ams.org/mathscinet-getitem?mr=1084186 MathSciNet] [http://link.springer.com/content/pdf/10.1007%2F978-1-4612-3464-7_18.pdf Article]
* [HB1978] D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions. Invent. Math. 47 (1978), no. 2, 149–170. [http://www.ams.org/mathscinet-getitem?mr=485727 MathSciNet] [http://link.springer.com/article/10.1007%2FBF01578069 Article]
* [HB1986] D. R. Heath-Brown, The divisor function d3(n) in arithmetic progressions. Acta Arith. 47 (1986), no. 1, 29–56. [http://www.ams.org/mathscinet-getitem?mr=866901 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa47/aa4713.pdf Article]
* [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. [http://www.ams.org/mathscinet-getitem?mr=340194 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/S0002-9904-1974-13434-8.pdf Article]
* [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. [http://www.ams.org/mathscinet-getitem?mr=396440 MathSciNet] [https://eudml.org/doc/205282 Article]
* [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. [http://arxiv.org/abs/math/0602599 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2414788 MathSciNet] [http://blms.oxfordjournals.org/content/40/2/298 Article]
* [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. [http://www.ams.org/mathscinet-getitem?mr=374060 MathSciNet] [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6718308 Article]
* [M1978] Hugh L. Montgomery, The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. [http://www.ams.org/mathscinet-getitem?mr=466048 MathSciNet] [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540922 Article]
* [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. [http://www.ams.org/mathscinet-getitem?mr=337832 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/home.html Article]
* [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. [http://www.ams.org/mathscinet-getitem?mr=130203 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa7/aa711.pdf Article]
* [S2007] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. [http://www.ams.org/mathscinet-getitem?mr=2265008 MathSciNet] [http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/ Article] [http://arxiv.org/abs/math/0605696 arXiv]

Bounded gaps between primes

2013-06-25T17:31:20Z

Hannes: Undo revision 8087 by Andres Caicedo (Talk) (140 varpi + 32 delta < 1 not yet confirmed)

== World records ==

* <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
* <math>k_0</math> is a quantity such that every admissible <math>k_0</math>-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in <math>k_0</math> lead to improvements in <math>H</math>. (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].)
* <math>\varpi</math> is a technical parameter related to a specialized form of the [http://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture Elliott-Halberstam conjecture]. Would like to be as large as possible. Improvements in <math>\varpi</math> lead to improvements in <math>k_0</math>. (The relationship is roughly of the form <math>k_0 \sim \varpi^{-3/2}</math>; there is an active discussion on optimising these improvements [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ here].) In more recent work, the single parameter <math>\varpi</math> is replaced by a pair <math>(\varpi,\delta)</math> (in previous work we had <math>\delta=\varpi</math>). Discussion on improving the values of <math>(\varpi,\delta)</math> is currently being held [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ here]. In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored.

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| 14 May
| 1/1,168 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| All subsequent work is based on Zhang's breakthrough paper.
|-
| 21 May
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| 28 May
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| 30 May
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| 31 May
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| 1 Jun
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| 2 Jun
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| 3 Jun
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| 4 Jun
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| 5 Jun
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| 6 Jun
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| 7 Jun
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>207 \varpi + 43\delta < \frac{1}{4}</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| Jun 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])

[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])

[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
|-
| Jun 12
|
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])

[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
|-
| Jun 13
|
|
|

[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
|
|-
| Jun 14
|
|
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
|
|-
| Jun 15
| <math>87\varpi+17\delta < \frac{1}{4}</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])

6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
|-
| Jun 16
| <math>87\varpi+17\delta < \frac{1}{4}</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])

<strike>155\varpi+31\delta < 1 and 11\varpi + 3\delta < \frac{1}{20} ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>

<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>

<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
|-
| Jun 18
|
| 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])

5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])

5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])

5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])

| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)

58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])

[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])

53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
|-
| Jun 19
|
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])

5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])

[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5455_51540_4678.txt 51,540?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24050 xfxie]/[http://math.mit.edu/~drew/admissible_5455_51540.txt Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])

53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])

[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
|-
| Jun 20
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])

<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
|
|
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
|-
| Jun 21
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])

1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records9.txt here]
|-
| Jun 22
|
| <strike>1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235740 Harcos]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235759 v08ltu])</strike>
| <strike>[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])</strike>
| Slight improvement in the <math>\tilde \theta</math> parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of <math>k_0</math>
|-
| Jun 23
|
| 1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235891 Paldi]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235905 Harcos])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| An improved monotonicity formula for <math>G_{k_0-1,\tilde \theta}</math> reduces <math>\kappa_3</math> somewhat
|-
| Jun 24
| <math>101\varpi + 21\delta \le 3/4</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])

<math>140\varpi + 32 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 Tao])

1/88?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])

1/74?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])
| 1,268? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/k1268_10206.txt 10,206?] ([http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/ Engelsma])
| A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
|-
| Jun 25
|
| 1,346? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236123 Hannes])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/k1346_10876.txt 10,876]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/ Engelsma])
|
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

== Polymath threads ==

* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart], Scott Morrison, 30 May 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang], Terence Tao, 3 June 2013. Inactive
* [http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/ Polymath proposal: bounded gaps between primes], Terence Tao, 4 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ Online reading seminar for Zhang’s “bounded gaps between primes”], Terence Tao, 4 June 2013. Inactive
* [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/ More narrow admissible sets], Scott Morrison, 5 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ The elementary Selberg sieve and bounded prime gaps], Terence Tao, 8 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/ A combinatorial subset sum problem associated with bounded prime gaps], Terence Tao, 10 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Further analysis of the truncated GPY sieve], Terence Tao, 11 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/ Estimation of the Type I and Type II sums], Terence Tao, 12 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/ Estimation of the Type III sums], Terence Tao, 14 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/ A truncated elementary Selberg sieve of Pintz], Terence Tao, 18 June, 2013. Active
* [http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/ Bounding short exponential sums on smooth moduli via Weyl differencing], Terence Tao, 22 June, 2013. Active
* [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/ The distribution of primes in densely divisible moduli], Terence Tao, 23 June, 2013. Active

== Code and data ==

* [https://github.com/semorrison/polymath8 Hensely-Richards sequences], Scott Morrison
** [http://tqft.net/misc/finding%20k_0.nb A mathematica notebook for finding k_0], Scott Morrison
** [https://github.com/avi-levy/dhl python implementation], Avi Levy
* [http://www.opertech.com/primes/k-tuples.html k-tuple pattern data], Thomas J Engelsma
** [https://perswww.kuleuven.be/~u0040935/k0graph.png A graph of this data]
** [http://www.opertech.com/primes/webdata/ Tuples giving this data]
* [https://github.com/vit-tucek/admissible_sets Sifted sequences], Vit Tucek
* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Other sifted sequences], Christian Elsholtz
* [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/S0025-5718-01-01348-5.pdf Size of largest admissible tuples in intervals of length up to 1050], Clark and Jarvis
* [http://math.mit.edu/~drew/admissable_v0.1.tar C implementation of the "greedy-greedy" algorithm], Andrew Sutherland
* [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Dropbox for sequences], pedant
* [https://docs.google.com/spreadsheet/ccc?key=0Ao3urQ79oleSdEZhRS00X1FLQjM3UlJTZFRqd19ySGc&usp=sharing Spreadsheet for admissible sequences], Vit Tucek
* [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.1.zip Java code for optimising a given tuple V1.1], xfxie

== Errata ==

Page numbers refer to the file linked to for the relevant paper.

# Errata for Zhang's "[http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes]"
## Page 5: In the first display, <math>\mathcal{E}</math> should be multiplied by <math>\mathcal{L}^{2k_0+2l_0}</math>, because <math>\lambda(n)^2</math> in (2.2) can be that large, cf. (2.4).
## Page 14: In the final display, the constraint <math>(n,d_1=1</math> should be <math>(n,d_1)=1</math>.
## Page 35: In the display after (10.5), the subscript on <math>{\mathcal J}_i</math> should be deleted.
## Page 36: In the third display, a factor of <math>\tau(q_0r)^{O(1)}</math> may be needed on the right-hand side (but is ultimately harmless).
## Page 38: In the display after (10.14), <math>\xi(r,a;q_1,b_1;q_2,b_2;n,k)</math> should be <math>\xi(r,a;k;q_1,b_1;q_2,b_2;n)</math>.
## Page 42: In (12.3), <math>B</math> should probably be 2.
## Page 47: In the third display after (13.13), the condition <math>l \in {\mathcal I}_i(h)</math> should be <math>l \in {\mathcal I}_i(sh)</math>.
## Page 49: In the top line, a comma in <math>(h_1,h_2;,n_1,n_2)</math> should be deleted.
## Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
## Page 54: Three displays before (14.17), <math>\bar{r_2}(m_1+m_2)q</math> should be <math>\bar{r_2}(m_1+m_2)/q</math>.
# Errata for Motohashi-Pintz's "[http://arxiv.org/pdf/math/0602599v1.pdf A smoothed GPY sieve]"
## Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of <math>(\log \frac{R}{|D|})^{2\ell+1}</math>, (4.15) contains instead a factor of <math>(\log \frac{R/w}{|K|})^{2\ell+1}</math> which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of <math>\exp(-k\omega/3)</math> in (4.15) does not seem to be available for estimating the second sum in (5.14).
# Errata for Pintz's "[http://arxiv.org/pdf/1306.1497v1.pdf A note on bounded gaps between primes]", version 1. Update: the errata below have been corrected in subsequent versions of Pintz's paper.
## Page 7: In (2.39), the exponent of <math>3a/2</math> should instead be <math>-5a/2</math> (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
## Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

== Other relevant blog posts ==

* [http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/ Marker lecture III: “Small gaps between primes”], Terence Tao, 19 Nov 2008.
* [http://blogs.ethz.ch/kowalski/2009/01/22/the-goldston-pintz-yildirim-result-and-how-far-do-we-have-to-walk-to-twin-primes/ The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?], Emmanuel Kowalski, 22 Jan 2009.
* [http://www.math.columbia.edu/~woit/wordpress/?p=5865 Number Theory News], Peter Woit, 12 May 2013.
* [http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html Bounded Gaps Between Primes], Emily Riehl, 14 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ Bounded gaps between primes!], Emmanuel Kowalski, 21 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/04/bounded-gaps-between-primes-some-grittier-details/ Bounded gaps between primes: some grittier details], Emmanuel Kowalski, 4 June 2013.
** [http://www.math.ethz.ch/~kowalski/zhang-notes.pdf The slides from the talk mentioned in that post]
* [http://aperiodical.com/2013/06/bound-on-prime-gaps-bound-decreasing-by-leaps-and-bounds/ Bound on prime gaps bound decreasing by leaps and bounds], Christian Perfect, 8 June 2013.
* [http://blogs.ams.org/blogonmathblogs/2013/06/14/narrowing-the-gap/ Narrowing the Gap], Brie Finegold, AMS Blog on Math Blogs, 14 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/22/bounded-gaps-between-primes-the-dawn-of-some-enlightenment/ Bounded gaps between primes: the dawn of (some) enlightenment], Emmanuel Kowalski, 22 June 2013.

== MathOverflow ==

* [http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture], 20 May 2013.
* [http://mathoverflow.net/questions/131825/a-technical-question-related-to-zhangs-result-of-bounded-prime-gaps A technical question related to Zhang’s result of bounded prime gaps], 25 May 2013.
* [http://mathoverflow.net/questions/132452/how-does-yitang-zhang-use-cauchys-inequality-and-theorem-2-to-obtain-the-error How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the <math>S_2</math> sum], 31 May 2013.
* [http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed Tightening Zhang’s bound], 3 June 2013.
** [http://meta.mathoverflow.net/discussion/1605/tightening-zhangs-bound/ Metathread for this post]
* [http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-len Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?], 3 June 2013.

== Wikipedia and other references ==

* [http://en.wikipedia.org/wiki/Bessel_function Bessel function]
* [http://en.wikipedia.org/wiki/Bombieri-Vinogradov_theorem Bombieri-Vinogradov theorem]
* [http://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem Brun-Titchmarsh theorem]
* [http://www.encyclopediaofmath.org/index.php/Dispersion_method Dispersion method]
* [http://en.wikipedia.org/wiki/Prime_gap Prime gap]
* [http://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture Second Hardy-Littlewood conjecture]
* [http://en.wikipedia.org/wiki/Twin_prime_conjecture Twin prime conjecture]

== Recent papers and notes ==

* [http://arxiv.org/abs/1304.3199 On the exponent of distribution of the ternary divisor function], Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
* [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes], Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages. [http://arxiv.org/abs/1306.2133 arXiv]
* [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes], Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
* [http://arxiv.org/abs/1305.6289 Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture], Janos Pintz, 27 May 2013.
* [http://arxiv.org/abs/1305.6369 A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million], T. S. Trudgian, 28 May 2013.
* [http://www.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf The Friedlander-Iwaniec sum], É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
* [http://terrytao.files.wordpress.com/2013/06/bounds.pdf Notes on Zhang's prime gaps paper], Terence Tao, 1 June 2013.
* [http://arxiv.org/abs/1306.0511 Bounded prime gaps in short intervals], Johan Andersson, 3 June 2013.
* [http://arxiv.org/abs/1306.0948 Bounded length intervals containing two primes and an almost-prime II], James Maynard, 5 June 2013.
* [http://arxiv.org/abs/1306.1497 A note on bounded gaps between primes], Janos Pintz, 6 June 2013.
* [https://sites.google.com/site/avishaytal/files/Primes.pdf Lower Bounds for Admissible k-tuples], Avishay Tal, 15 June 2013.
* Notes in a truncated elementary Selberg sieve ([http://terrytao.files.wordpress.com/2013/06/file-1.pdf Section 1], [http://terrytao.files.wordpress.com/2013/06/file-2.pdf Section 2], [http://terrytao.files.wordpress.com/2013/06/file-3.pdf Section 3]), Janos Pintz, 18 June 2013.

== Media ==

* [http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989 First proof that infinitely many prime numbers come in pairs], Maggie McKee, Nature, 14 May 2013.
* [http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html Proof that an infinite number of primes are paired], Lisa Grossman, New Scientist, 14 May 2013.
* [https://www.simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/ Unheralded Mathematician Bridges the Prime Gap], Erica Klarreich, Simons science news, 20 May 2013.
** The article also appeared on Wired as "[http://www.wired.com/wiredscience/2013/05/twin-primes/ Unknown Mathematician Proves Elusive Property of Prime Numbers]".
* [http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html The Beauty of Bounded Gaps], Jordan Ellenberg, Slate, 22 May 2013.
* [http://www.newscientist.com/article/dn23644 Game of proofs boosts prime pair result by millions], Jacob Aron, New Scientist, 4 June 2013.

== Bibliography ==

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

* [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. [http://www.ams.org/mathscinet-getitem?mr=834613 MathSciNet] [http://link.springer.com/article/10.1007%2FBF02399204 Article]
* [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. [http://www.ams.org/mathscinet-getitem?mr=891581 MathSciNet] [https://eudml.org/doc/164255 Article]
* [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. [http://www.ams.org/mathscinet-getitem?mr=976723 MathSciNet] [http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Article]
* [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
* [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 [http://www.ams.org/mathscinet-getitem?mr=1836929 MathSciNet] [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Article]
* [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). [http://www.ams.org/mathscinet-getitem?mr=610700 MathSciNet] [http://www.math.ethz.ch/~kowalski/fouvry-iwaniec-on-a-theorem.pdf Article]
* [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. [http://www.ams.org/mathscinet-getitem?mr=719249 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4226.pdf Article]
* [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. [http://www.ams.org/mathscinet-getitem?mr=786351 MathSciNet] [http://www.jstor.org/stable/1971175 JSTOR] [http://www.jstor.org/stable/1971176 Appendix]
* [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. [http://arxiv.org/abs/math/0508185 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2552109 MathSciNet]
* [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. [http://www.ams.org/mathscinet-getitem?mr=1726073 MathSciNet] [http://www.ccrwest.org/gordon/ants.pdf Article]
* [GR1980] S. W. Graham, C. J. Ringrose, Lower bounds for least quadratic nonresidues. Analytic number theory (Allerton Park, IL, 1989), 269–309, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990. [http://www.ams.org/mathscinet-getitem?mr=1084186 MathSciNet] [http://link.springer.com/content/pdf/10.1007%2F978-1-4612-3464-7_18.pdf Article]
* [HB1978] D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions. Invent. Math. 47 (1978), no. 2, 149–170. [http://www.ams.org/mathscinet-getitem?mr=485727 MathSciNet] [http://link.springer.com/article/10.1007%2FBF01578069 Article]
* [HB1986] D. R. Heath-Brown, The divisor function d3(n) in arithmetic progressions. Acta Arith. 47 (1986), no. 1, 29–56. [http://www.ams.org/mathscinet-getitem?mr=866901 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa47/aa4713.pdf Article]
* [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. [http://www.ams.org/mathscinet-getitem?mr=340194 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/S0002-9904-1974-13434-8.pdf Article]
* [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. [http://www.ams.org/mathscinet-getitem?mr=396440 MathSciNet] [https://eudml.org/doc/205282 Article]
* [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. [http://arxiv.org/abs/math/0602599 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2414788 MathSciNet] [http://blms.oxfordjournals.org/content/40/2/298 Article]
* [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. [http://www.ams.org/mathscinet-getitem?mr=374060 MathSciNet] [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6718308 Article]
* [M1978] Hugh L. Montgomery, The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. [http://www.ams.org/mathscinet-getitem?mr=466048 MathSciNet] [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540922 Article]
* [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. [http://www.ams.org/mathscinet-getitem?mr=337832 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/home.html Article]
* [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. [http://www.ams.org/mathscinet-getitem?mr=130203 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa7/aa711.pdf Article]
* [S2007] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. [http://www.ams.org/mathscinet-getitem?mr=2265008 MathSciNet] [http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/ Article] [http://arxiv.org/abs/math/0605696 arXiv]

Bounded gaps between primes

2013-06-25T01:49:40Z

Hannes: /* Polymath threads */ typo

== World records ==

* <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
* <math>k_0</math> is a quantity such that every admissible <math>k_0</math>-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in <math>k_0</math> lead to improvements in <math>H</math>. (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].)
* <math>\varpi</math> is a technical parameter related to a specialized form of the [http://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture Elliott-Halberstam conjecture]. Would like to be as large as possible. Improvements in <math>\varpi</math> lead to improvements in <math>k_0</math>. (The relationship is roughly of the form <math>k_0 \sim \varpi^{-3/2}</math>; there is an active discussion on optimising these improvements [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ here].) In more recent work, the single parameter <math>\varpi</math> is replaced by a pair <math>(\varpi,\delta)</math> (in previous work we had <math>\delta=\varpi</math>). Discussion on improving the values of <math>(\varpi,\delta)</math> is currently being held [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ here]. In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored.

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| 14 May
| 1/1,168 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| All subsequent work is based on Zhang's breakthrough paper.
|-
| 21 May
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| 28 May
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| 30 May
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| 31 May
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| 1 Jun
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| 2 Jun
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| 3 Jun
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| 4 Jun
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| 5 Jun
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| 6 Jun
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| 7 Jun
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>207 \varpi + 43\delta < \frac{1}{4}</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| Jun 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])

[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])

[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
|-
| Jun 12
|
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])

[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
|-
| Jun 13
|
|
|

[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
|
|-
| Jun 14
|
|
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
|
|-
| Jun 15
| <math>87\varpi+17\delta < \frac{1}{4}</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])

6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
|-
| Jun 16
| <math>87\varpi+17\delta < \frac{1}{4}</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])

<strike>155\varpi+31\delta < 1 and 11\varpi + 3\delta < \frac{1}{20} ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>

<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>

<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
|-
| Jun 18
|
| 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])

5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])

5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])

5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])

| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)

58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])

[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])

53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
|-
| Jun 19
|
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])

5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])

[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5455_51540_4678.txt 51,540?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24050 xfxie]/[http://math.mit.edu/~drew/admissible_5455_51540.txt Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])

53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])

[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
|-
| Jun 20
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])

<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
|
|
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
|-
| Jun 21
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])

1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records7.txt here]
|-
| Jun 22
|
| <strike>1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235740 Harcos]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235759 v08ltu])</strike>
| <strike>[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])</strike>
| Slight improvement in the <math>\tilde \theta</math> parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of <math>k_0</math>
|-
| Jun 23
|
| 1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235891 Paldi]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235905 Harcos])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| An improved monotonicity formula for <math>G_{k_0-1,\tilde \theta}</math> reduces <math>\kappa_3</math> somewhat
|-
| Jun 24
| <math>101\varpi + (21+O(1))\delta < 3/4</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])

<math>140\varpi + 32 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 Tao])

1/88?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])

1/74?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])
| 1,268? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/k1268_10206.txt 10,206?] ([http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/ Engelsma])
| A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

== Polymath threads ==

* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart], Scott Morrison, 30 May 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang], Terence Tao, 3 June 2013. Inactive
* [http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/ Polymath proposal: bounded gaps between primes], Terence Tao, 4 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ Online reading seminar for Zhang’s “bounded gaps between primes”], Terence Tao, 4 June 2013. Inactive
* [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/ More narrow admissible sets], Scott Morrison, 5 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ The elementary Selberg sieve and bounded prime gaps], Terence Tao, 8 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/ A combinatorial subset sum problem associated with bounded prime gaps], Terence Tao, 10 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Further analysis of the truncated GPY sieve], Terence Tao, 11 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/ Estimation of the Type I and Type II sums], Terence Tao, 12 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/ Estimation of the Type III sums], Terence Tao, 14 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/ A truncated elementary Selberg sieve of Pintz], Terence Tao, 18 June, 2013. Active
* [http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/ Bounding short exponential sums on smooth moduli via Weyl differencing], Terence Tao, 22 June, 2013. Active
* [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/ The distribution of primes in densely divisible moduli], Terence Tao, 23 June, 2013. Active

== Code and data ==

* [https://github.com/semorrison/polymath8 Hensely-Richards sequences], Scott Morrison
** [http://tqft.net/misc/finding%20k_0.nb A mathematica notebook for finding k_0], Scott Morrison
** [https://github.com/avi-levy/dhl python implementation], Avi Levy
* [http://www.opertech.com/primes/k-tuples.html k-tuple pattern data], Thomas J Engelsma
** [https://perswww.kuleuven.be/~u0040935/k0graph.png A graph of this data]
** [http://www.opertech.com/primes/webdata/ Tuples giving this data]
* [https://github.com/vit-tucek/admissible_sets Sifted sequences], Vit Tucek
* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Other sifted sequences], Christian Elsholtz
* [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/S0025-5718-01-01348-5.pdf Size of largest admissible tuples in intervals of length up to 1050], Clark and Jarvis
* [http://math.mit.edu/~drew/admissable_v0.1.tar C implementation of the "greedy-greedy" algorithm], Andrew Sutherland
* [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Dropbox for sequences], pedant
* [https://docs.google.com/spreadsheet/ccc?key=0Ao3urQ79oleSdEZhRS00X1FLQjM3UlJTZFRqd19ySGc&usp=sharing Spreadsheet for admissible sequences], Vit Tucek
* [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.1.zip Java code for optimising a given tuple V1.1], xfxie

== Errata ==

Page numbers refer to the file linked to for the relevant paper.

# Errata for Zhang's "[http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes]"
## Page 5: In the first display, <math>\mathcal{E}</math> should be multiplied by <math>\mathcal{L}^{2k_0+2l_0}</math>, because <math>\lambda(n)^2</math> in (2.2) can be that large, cf. (2.4).
## Page 14: In the final display, the constraint <math>(n,d_1=1</math> should be <math>(n,d_1)=1</math>.
## Page 35: In the display after (10.5), the subscript on <math>{\mathcal J}_i</math> should be deleted.
## Page 36: In the third display, a factor of <math>\tau(q_0r)^{O(1)}</math> may be needed on the right-hand side (but is ultimately harmless).
## Page 38: In the display after (10.14), <math>\xi(r,a;q_1,b_1;q_2,b_2;n,k)</math> should be <math>\xi(r,a;k;q_1,b_1;q_2,b_2;n)</math>.
## Page 42: In (12.3), <math>B</math> should probably be 2.
## Page 47: In the third display after (13.13), the condition <math>l \in {\mathcal I}_i(h)</math> should be <math>l \in {\mathcal I}_i(sh)</math>.
## Page 49: In the top line, a comma in <math>(h_1,h_2;,n_1,n_2)</math> should be deleted.
## Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
## Page 54: Three displays before (14.17), <math>\bar{r_2}(m_1+m_2)q</math> should be <math>\bar{r_2}(m_1+m_2)/q</math>.
# Errata for Motohashi-Pintz's "[http://arxiv.org/pdf/math/0602599v1.pdf A smoothed GPY sieve]"
## Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of <math>(\log \frac{R}{|D|})^{2\ell+1}</math>, (4.15) contains instead a factor of <math>(\log \frac{R/w}{|K|})^{2\ell+1}</math> which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of <math>\exp(-k\omega/3)</math> in (4.15) does not seem to be available for estimating the second sum in (5.14).
# Errata for Pintz's "[http://arxiv.org/pdf/1306.1497v1.pdf A note on bounded gaps between primes]", version 1. Update: the errata below have been corrected in subsequent versions of Pintz's paper.
## Page 7: In (2.39), the exponent of <math>3a/2</math> should instead be <math>-5a/2</math> (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
## Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

== Other relevant blog posts ==

* [http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/ Marker lecture III: “Small gaps between primes”], Terence Tao, 19 Nov 2008.
* [http://blogs.ethz.ch/kowalski/2009/01/22/the-goldston-pintz-yildirim-result-and-how-far-do-we-have-to-walk-to-twin-primes/ The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?], Emmanuel Kowalski, 22 Jan 2009.
* [http://www.math.columbia.edu/~woit/wordpress/?p=5865 Number Theory News], Peter Woit, 12 May 2013.
* [http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html Bounded Gaps Between Primes], Emily Riehl, 14 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ Bounded gaps between primes!], Emmanuel Kowalski, 21 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/04/bounded-gaps-between-primes-some-grittier-details/ Bounded gaps between primes: some grittier details], Emmanuel Kowalski, 4 June 2013.
** [http://www.math.ethz.ch/~kowalski/zhang-notes.pdf The slides from the talk mentioned in that post]
* [http://aperiodical.com/2013/06/bound-on-prime-gaps-bound-decreasing-by-leaps-and-bounds/ Bound on prime gaps bound decreasing by leaps and bounds], Christian Perfect, 8 June 2013.
* [http://blogs.ams.org/blogonmathblogs/2013/06/14/narrowing-the-gap/ Narrowing the Gap], Brie Finegold, AMS Blog on Math Blogs, 14 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/22/bounded-gaps-between-primes-the-dawn-of-some-enlightenment/ Bounded gaps between primes: the dawn of (some) enlightenment], Emmanuel Kowalski, 22 June 2013.

== MathOverflow ==

* [http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture], 20 May 2013.
* [http://mathoverflow.net/questions/131825/a-technical-question-related-to-zhangs-result-of-bounded-prime-gaps A technical question related to Zhang’s result of bounded prime gaps], 25 May 2013.
* [http://mathoverflow.net/questions/132452/how-does-yitang-zhang-use-cauchys-inequality-and-theorem-2-to-obtain-the-error How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the <math>S_2</math> sum], 31 May 2013.
* [http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed Tightening Zhang’s bound], 3 June 2013.
** [http://meta.mathoverflow.net/discussion/1605/tightening-zhangs-bound/ Metathread for this post]
* [http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-len Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?], 3 June 2013.

== Wikipedia and other references ==

* [http://en.wikipedia.org/wiki/Bessel_function Bessel function]
* [http://en.wikipedia.org/wiki/Bombieri-Vinogradov_theorem Bombieri-Vinogradov theorem]
* [http://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem Brun-Titchmarsh theorem]
* [http://www.encyclopediaofmath.org/index.php/Dispersion_method Dispersion method]
* [http://en.wikipedia.org/wiki/Prime_gap Prime gap]
* [http://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture Second Hardy-Littlewood conjecture]
* [http://en.wikipedia.org/wiki/Twin_prime_conjecture Twin prime conjecture]

== Recent papers and notes ==

* [http://arxiv.org/abs/1304.3199 On the exponent of distribution of the ternary divisor function], Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
* [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes], Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages. [http://arxiv.org/abs/1306.2133 arXiv]
* [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes], Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
* [http://arxiv.org/abs/1305.6289 Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture], Janos Pintz, 27 May 2013.
* [http://arxiv.org/abs/1305.6369 A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million], T. S. Trudgian, 28 May 2013.
* [http://www.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf The Friedlander-Iwaniec sum], É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
* [http://terrytao.files.wordpress.com/2013/06/bounds.pdf Notes on Zhang's prime gaps paper], Terence Tao, 1 June 2013.
* [http://arxiv.org/abs/1306.0511 Bounded prime gaps in short intervals], Johan Andersson, 3 June 2013.
* [http://arxiv.org/abs/1306.0948 Bounded length intervals containing two primes and an almost-prime II], James Maynard, 5 June 2013.
* [http://arxiv.org/abs/1306.1497 A note on bounded gaps between primes], Janos Pintz, 6 June 2013.
* [https://sites.google.com/site/avishaytal/files/Primes.pdf Lower Bounds for Admissible k-tuples], Avishay Tal, 15 June 2013.
* Notes in a truncated elementary Selberg sieve ([http://terrytao.files.wordpress.com/2013/06/file-1.pdf Section 1], [http://terrytao.files.wordpress.com/2013/06/file-2.pdf Section 2], [http://terrytao.files.wordpress.com/2013/06/file-3.pdf Section 3]), Janos Pintz, 18 June 2013.

== Media ==

* [http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989 First proof that infinitely many prime numbers come in pairs], Maggie McKee, Nature, 14 May 2013.
* [http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html Proof that an infinite number of primes are paired], Lisa Grossman, New Scientist, 14 May 2013.
* [https://www.simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/ Unheralded Mathematician Bridges the Prime Gap], Erica Klarreich, Simons science news, 20 May 2013.
** The article also appeared on Wired as "[http://www.wired.com/wiredscience/2013/05/twin-primes/ Unknown Mathematician Proves Elusive Property of Prime Numbers]".
* [http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html The Beauty of Bounded Gaps], Jordan Ellenberg, Slate, 22 May 2013.
* [http://www.newscientist.com/article/dn23644 Game of proofs boosts prime pair result by millions], Jacob Aron, New Scientist, 4 June 2013.

== Bibliography ==

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

* [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. [http://www.ams.org/mathscinet-getitem?mr=834613 MathSciNet] [http://link.springer.com/article/10.1007%2FBF02399204 Article]
* [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. [http://www.ams.org/mathscinet-getitem?mr=891581 MathSciNet] [https://eudml.org/doc/164255 Article]
* [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. [http://www.ams.org/mathscinet-getitem?mr=976723 MathSciNet] [http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Article]
* [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
* [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 [http://www.ams.org/mathscinet-getitem?mr=1836929 MathSciNet] [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Article]
* [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). [http://www.ams.org/mathscinet-getitem?mr=610700 MathSciNet] [http://www.math.ethz.ch/~kowalski/fouvry-iwaniec-on-a-theorem.pdf Article]
* [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. [http://www.ams.org/mathscinet-getitem?mr=719249 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4226.pdf Article]
* [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. [http://www.ams.org/mathscinet-getitem?mr=786351 MathSciNet] [http://www.jstor.org/stable/1971175 JSTOR] [http://www.jstor.org/stable/1971176 Appendix]
* [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. [http://arxiv.org/abs/math/0508185 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2552109 MathSciNet]
* [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. [http://www.ams.org/mathscinet-getitem?mr=1726073 MathSciNet] [http://www.ccrwest.org/gordon/ants.pdf Article]
* [GR1980] S. W. Graham, C. J. Ringrose, Lower bounds for least quadratic nonresidues. Analytic number theory (Allerton Park, IL, 1989), 269–309, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990. [http://www.ams.org/mathscinet-getitem?mr=1084186 MathSciNet] [http://link.springer.com/content/pdf/10.1007%2F978-1-4612-3464-7_18.pdf Article]
* [HB1978] D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions. Invent. Math. 47 (1978), no. 2, 149–170. [http://www.ams.org/mathscinet-getitem?mr=485727 MathSciNet] [http://link.springer.com/article/10.1007%2FBF01578069 Article]
* [HB1986] D. R. Heath-Brown, The divisor function d3(n) in arithmetic progressions. Acta Arith. 47 (1986), no. 1, 29–56. [http://www.ams.org/mathscinet-getitem?mr=866901 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa47/aa4713.pdf Article]
* [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. [http://www.ams.org/mathscinet-getitem?mr=340194 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/S0002-9904-1974-13434-8.pdf Article]
* [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. [http://www.ams.org/mathscinet-getitem?mr=396440 MathSciNet] [https://eudml.org/doc/205282 Article]
* [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. [http://arxiv.org/abs/math/0602599 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2414788 MathSciNet] [http://blms.oxfordjournals.org/content/40/2/298 Article]
* [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. [http://www.ams.org/mathscinet-getitem?mr=374060 MathSciNet] [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6718308 Article]
* [M1978] Hugh L. Montgomery, The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. [http://www.ams.org/mathscinet-getitem?mr=466048 MathSciNet] [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540922 Article]
* [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. [http://www.ams.org/mathscinet-getitem?mr=337832 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/home.html Article]
* [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. [http://www.ams.org/mathscinet-getitem?mr=130203 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa7/aa711.pdf Article]
* [S2007] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. [http://www.ams.org/mathscinet-getitem?mr=2265008 MathSciNet] [http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/ Article] [http://arxiv.org/abs/math/0605696 arXiv]

Bounded gaps between primes

2013-06-25T01:46:29Z

Hannes: https -> http on a few places

== World records ==

* <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
* <math>k_0</math> is a quantity such that every admissible <math>k_0</math>-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in <math>k_0</math> lead to improvements in <math>H</math>. (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].)
* <math>\varpi</math> is a technical parameter related to a specialized form of the [http://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture Elliott-Halberstam conjecture]. Would like to be as large as possible. Improvements in <math>\varpi</math> lead to improvements in <math>k_0</math>. (The relationship is roughly of the form <math>k_0 \sim \varpi^{-3/2}</math>; there is an active discussion on optimising these improvements [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ here].) In more recent work, the single parameter <math>\varpi</math> is replaced by a pair <math>(\varpi,\delta)</math> (in previous work we had <math>\delta=\varpi</math>). Discussion on improving the values of <math>(\varpi,\delta)</math> is currently being held [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ here]. In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored.

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| 14 May
| 1/1,168 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| All subsequent work is based on Zhang's breakthrough paper.
|-
| 21 May
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| 28 May
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| 30 May
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| 31 May
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| 1 Jun
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| 2 Jun
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| 3 Jun
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| 4 Jun
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| 5 Jun
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| 6 Jun
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| 7 Jun
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>207 \varpi + 43\delta < \frac{1}{4}</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| Jun 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])

[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])

[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
|-
| Jun 12
|
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])

[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
|-
| Jun 13
|
|
|

[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
|
|-
| Jun 14
|
|
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
|
|-
| Jun 15
| <math>87\varpi+17\delta < \frac{1}{4}</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])

6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
|-
| Jun 16
| <math>87\varpi+17\delta < \frac{1}{4}</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])

<strike>155\varpi+31\delta < 1 and 11\varpi + 3\delta < \frac{1}{20} ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>

<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>

<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
|-
| Jun 18
|
| 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])

5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])

5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])

5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])

| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)

58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])

[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])

53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
|-
| Jun 19
|
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])

5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])

[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5455_51540_4678.txt 51,540?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24050 xfxie]/[http://math.mit.edu/~drew/admissible_5455_51540.txt Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])

53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])

[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
|-
| Jun 20
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])

<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
|
|
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
|-
| Jun 21
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])

1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records7.txt here]
|-
| Jun 22
|
| <strike>1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235740 Harcos]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235759 v08ltu])</strike>
| <strike>[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])</strike>
| Slight improvement in the <math>\tilde \theta</math> parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of <math>k_0</math>
|-
| Jun 23
|
| 1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235891 Paldi]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235905 Harcos])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| An improved monotonicity formula for <math>G_{k_0-1,\tilde \theta}</math> reduces <math>\kappa_3</math> somewhat
|-
| Jun 24
| <math>101\varpi + (21+O(1))\delta < 3/4</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])

<math>140\varpi + 32 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 Tao])

1/88?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])

1/74?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])
| 1,268? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/k1268_10206.txt 10,206?] ([http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/ Engelsma])
| A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

== Polymath threads ==

* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart], Scott Morrison, 30 May 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang], Terence Tao, 3 June 2013. Inactive
* [http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/ Polymath proposal: bounded gaps between primes], Terence Tao, 4 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ Online reading seminar for Zhang’s “bounded gaps between primes], Terence Tao, 4 June 2013. Inactive
* [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/ More narrow admissible sets], Scott Morrison, 5 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ The elementary Selberg sieve and bounded prime gaps], Terence Tao, 8 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/ A combinatorial subset sum problem associated with bounded prime gaps], Terence Tao, 10 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Further analysis of the truncated GPY sieve], Terence Tao, 11 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/ Estimation of the Type I and Type II sums], Terence Tao, 12 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/ Estimation of the Type III sums], Terence Tao, 14 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/ A truncated elementary Selberg sieve of Pintz], Terence Tao, 18 June, 2013. Active
* [http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/ Bounding short exponential sums on smooth moduli via Weyl differencing], Terence Tao, 22 June, 2013. Active
* [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/ The distribution of primes in densely divisible moduli], Terence Tao, 23 June, 2013. Active

== Code and data ==

* [https://github.com/semorrison/polymath8 Hensely-Richards sequences], Scott Morrison
** [http://tqft.net/misc/finding%20k_0.nb A mathematica notebook for finding k_0], Scott Morrison
** [https://github.com/avi-levy/dhl python implementation], Avi Levy
* [http://www.opertech.com/primes/k-tuples.html k-tuple pattern data], Thomas J Engelsma
** [https://perswww.kuleuven.be/~u0040935/k0graph.png A graph of this data]
** [http://www.opertech.com/primes/webdata/ Tuples giving this data]
* [https://github.com/vit-tucek/admissible_sets Sifted sequences], Vit Tucek
* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Other sifted sequences], Christian Elsholtz
* [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/S0025-5718-01-01348-5.pdf Size of largest admissible tuples in intervals of length up to 1050], Clark and Jarvis
* [http://math.mit.edu/~drew/admissable_v0.1.tar C implementation of the "greedy-greedy" algorithm], Andrew Sutherland
* [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Dropbox for sequences], pedant
* [https://docs.google.com/spreadsheet/ccc?key=0Ao3urQ79oleSdEZhRS00X1FLQjM3UlJTZFRqd19ySGc&usp=sharing Spreadsheet for admissible sequences], Vit Tucek
* [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.1.zip Java code for optimising a given tuple V1.1], xfxie

== Errata ==

Page numbers refer to the file linked to for the relevant paper.

# Errata for Zhang's "[http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes]"
## Page 5: In the first display, <math>\mathcal{E}</math> should be multiplied by <math>\mathcal{L}^{2k_0+2l_0}</math>, because <math>\lambda(n)^2</math> in (2.2) can be that large, cf. (2.4).
## Page 14: In the final display, the constraint <math>(n,d_1=1</math> should be <math>(n,d_1)=1</math>.
## Page 35: In the display after (10.5), the subscript on <math>{\mathcal J}_i</math> should be deleted.
## Page 36: In the third display, a factor of <math>\tau(q_0r)^{O(1)}</math> may be needed on the right-hand side (but is ultimately harmless).
## Page 38: In the display after (10.14), <math>\xi(r,a;q_1,b_1;q_2,b_2;n,k)</math> should be <math>\xi(r,a;k;q_1,b_1;q_2,b_2;n)</math>.
## Page 42: In (12.3), <math>B</math> should probably be 2.
## Page 47: In the third display after (13.13), the condition <math>l \in {\mathcal I}_i(h)</math> should be <math>l \in {\mathcal I}_i(sh)</math>.
## Page 49: In the top line, a comma in <math>(h_1,h_2;,n_1,n_2)</math> should be deleted.
## Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
## Page 54: Three displays before (14.17), <math>\bar{r_2}(m_1+m_2)q</math> should be <math>\bar{r_2}(m_1+m_2)/q</math>.
# Errata for Motohashi-Pintz's "[http://arxiv.org/pdf/math/0602599v1.pdf A smoothed GPY sieve]"
## Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of <math>(\log \frac{R}{|D|})^{2\ell+1}</math>, (4.15) contains instead a factor of <math>(\log \frac{R/w}{|K|})^{2\ell+1}</math> which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of <math>\exp(-k\omega/3)</math> in (4.15) does not seem to be available for estimating the second sum in (5.14).
# Errata for Pintz's "[http://arxiv.org/pdf/1306.1497v1.pdf A note on bounded gaps between primes]", version 1. Update: the errata below have been corrected in subsequent versions of Pintz's paper.
## Page 7: In (2.39), the exponent of <math>3a/2</math> should instead be <math>-5a/2</math> (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
## Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

== Other relevant blog posts ==

* [http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/ Marker lecture III: “Small gaps between primes”], Terence Tao, 19 Nov 2008.
* [http://blogs.ethz.ch/kowalski/2009/01/22/the-goldston-pintz-yildirim-result-and-how-far-do-we-have-to-walk-to-twin-primes/ The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?], Emmanuel Kowalski, 22 Jan 2009.
* [http://www.math.columbia.edu/~woit/wordpress/?p=5865 Number Theory News], Peter Woit, 12 May 2013.
* [http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html Bounded Gaps Between Primes], Emily Riehl, 14 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ Bounded gaps between primes!], Emmanuel Kowalski, 21 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/04/bounded-gaps-between-primes-some-grittier-details/ Bounded gaps between primes: some grittier details], Emmanuel Kowalski, 4 June 2013.
** [http://www.math.ethz.ch/~kowalski/zhang-notes.pdf The slides from the talk mentioned in that post]
* [http://aperiodical.com/2013/06/bound-on-prime-gaps-bound-decreasing-by-leaps-and-bounds/ Bound on prime gaps bound decreasing by leaps and bounds], Christian Perfect, 8 June 2013.
* [http://blogs.ams.org/blogonmathblogs/2013/06/14/narrowing-the-gap/ Narrowing the Gap], Brie Finegold, AMS Blog on Math Blogs, 14 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/22/bounded-gaps-between-primes-the-dawn-of-some-enlightenment/ Bounded gaps between primes: the dawn of (some) enlightenment], Emmanuel Kowalski, 22 June 2013.

== MathOverflow ==

* [http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture], 20 May 2013.
* [http://mathoverflow.net/questions/131825/a-technical-question-related-to-zhangs-result-of-bounded-prime-gaps A technical question related to Zhang’s result of bounded prime gaps], 25 May 2013.
* [http://mathoverflow.net/questions/132452/how-does-yitang-zhang-use-cauchys-inequality-and-theorem-2-to-obtain-the-error How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the <math>S_2</math> sum], 31 May 2013.
* [http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed Tightening Zhang’s bound], 3 June 2013.
** [http://meta.mathoverflow.net/discussion/1605/tightening-zhangs-bound/ Metathread for this post]
* [http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-len Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?], 3 June 2013.

== Wikipedia and other references ==

* [http://en.wikipedia.org/wiki/Bessel_function Bessel function]
* [http://en.wikipedia.org/wiki/Bombieri-Vinogradov_theorem Bombieri-Vinogradov theorem]
* [http://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem Brun-Titchmarsh theorem]
* [http://www.encyclopediaofmath.org/index.php/Dispersion_method Dispersion method]
* [http://en.wikipedia.org/wiki/Prime_gap Prime gap]
* [http://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture Second Hardy-Littlewood conjecture]
* [http://en.wikipedia.org/wiki/Twin_prime_conjecture Twin prime conjecture]

== Recent papers and notes ==

* [http://arxiv.org/abs/1304.3199 On the exponent of distribution of the ternary divisor function], Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
* [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes], Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages. [http://arxiv.org/abs/1306.2133 arXiv]
* [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes], Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
* [http://arxiv.org/abs/1305.6289 Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture], Janos Pintz, 27 May 2013.
* [http://arxiv.org/abs/1305.6369 A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million], T. S. Trudgian, 28 May 2013.
* [http://www.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf The Friedlander-Iwaniec sum], É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
* [http://terrytao.files.wordpress.com/2013/06/bounds.pdf Notes on Zhang's prime gaps paper], Terence Tao, 1 June 2013.
* [http://arxiv.org/abs/1306.0511 Bounded prime gaps in short intervals], Johan Andersson, 3 June 2013.
* [http://arxiv.org/abs/1306.0948 Bounded length intervals containing two primes and an almost-prime II], James Maynard, 5 June 2013.
* [http://arxiv.org/abs/1306.1497 A note on bounded gaps between primes], Janos Pintz, 6 June 2013.
* [https://sites.google.com/site/avishaytal/files/Primes.pdf Lower Bounds for Admissible k-tuples], Avishay Tal, 15 June 2013.
* Notes in a truncated elementary Selberg sieve ([http://terrytao.files.wordpress.com/2013/06/file-1.pdf Section 1], [http://terrytao.files.wordpress.com/2013/06/file-2.pdf Section 2], [http://terrytao.files.wordpress.com/2013/06/file-3.pdf Section 3]), Janos Pintz, 18 June 2013.

== Media ==

* [http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989 First proof that infinitely many prime numbers come in pairs], Maggie McKee, Nature, 14 May 2013.
* [http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html Proof that an infinite number of primes are paired], Lisa Grossman, New Scientist, 14 May 2013.
* [https://www.simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/ Unheralded Mathematician Bridges the Prime Gap], Erica Klarreich, Simons science news, 20 May 2013.
** The article also appeared on Wired as "[http://www.wired.com/wiredscience/2013/05/twin-primes/ Unknown Mathematician Proves Elusive Property of Prime Numbers]".
* [http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html The Beauty of Bounded Gaps], Jordan Ellenberg, Slate, 22 May 2013.
* [http://www.newscientist.com/article/dn23644 Game of proofs boosts prime pair result by millions], Jacob Aron, New Scientist, 4 June 2013.

== Bibliography ==

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

* [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. [http://www.ams.org/mathscinet-getitem?mr=834613 MathSciNet] [http://link.springer.com/article/10.1007%2FBF02399204 Article]
* [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. [http://www.ams.org/mathscinet-getitem?mr=891581 MathSciNet] [https://eudml.org/doc/164255 Article]
* [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. [http://www.ams.org/mathscinet-getitem?mr=976723 MathSciNet] [http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Article]
* [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
* [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 [http://www.ams.org/mathscinet-getitem?mr=1836929 MathSciNet] [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Article]
* [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). [http://www.ams.org/mathscinet-getitem?mr=610700 MathSciNet] [http://www.math.ethz.ch/~kowalski/fouvry-iwaniec-on-a-theorem.pdf Article]
* [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. [http://www.ams.org/mathscinet-getitem?mr=719249 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4226.pdf Article]
* [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. [http://www.ams.org/mathscinet-getitem?mr=786351 MathSciNet] [http://www.jstor.org/stable/1971175 JSTOR] [http://www.jstor.org/stable/1971176 Appendix]
* [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. [http://arxiv.org/abs/math/0508185 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2552109 MathSciNet]
* [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. [http://www.ams.org/mathscinet-getitem?mr=1726073 MathSciNet] [http://www.ccrwest.org/gordon/ants.pdf Article]
* [GR1980] S. W. Graham, C. J. Ringrose, Lower bounds for least quadratic nonresidues. Analytic number theory (Allerton Park, IL, 1989), 269–309, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990. [http://www.ams.org/mathscinet-getitem?mr=1084186 MathSciNet] [http://link.springer.com/content/pdf/10.1007%2F978-1-4612-3464-7_18.pdf Article]
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* [HB1986] D. R. Heath-Brown, The divisor function d3(n) in arithmetic progressions. Acta Arith. 47 (1986), no. 1, 29–56. [http://www.ams.org/mathscinet-getitem?mr=866901 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa47/aa4713.pdf Article]
* [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. [http://www.ams.org/mathscinet-getitem?mr=340194 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/S0002-9904-1974-13434-8.pdf Article]
* [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. [http://www.ams.org/mathscinet-getitem?mr=396440 MathSciNet] [https://eudml.org/doc/205282 Article]
* [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. [http://arxiv.org/abs/math/0602599 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2414788 MathSciNet] [http://blms.oxfordjournals.org/content/40/2/298 Article]
* [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. [http://www.ams.org/mathscinet-getitem?mr=374060 MathSciNet] [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6718308 Article]
* [M1978] Hugh L. Montgomery, The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. [http://www.ams.org/mathscinet-getitem?mr=466048 MathSciNet] [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540922 Article]
* [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. [http://www.ams.org/mathscinet-getitem?mr=337832 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/home.html Article]
* [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. [http://www.ams.org/mathscinet-getitem?mr=130203 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa7/aa711.pdf Article]
* [S2007] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. [http://www.ams.org/mathscinet-getitem?mr=2265008 MathSciNet] [http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/ Article] [http://arxiv.org/abs/math/0605696 arXiv]

Bounded gaps between primes

2013-06-22T06:12:54Z

Hannes: /* World records */

== World records ==

* <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
* <math>k_0</math> is a quantity such that every admissible <math>k_0</math>-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in <math>k_0</math> lead to improvements in <math>H</math>. (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].)
* <math>\varpi</math> is a technical parameter related to a specialized form of the [https://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture Elliott-Halberstam conjecture]. Would like to be as large as possible. Improvements in <math>\varpi</math> lead to improvements in <math>k_0</math>. (The relationship is roughly of the form <math>k_0 \sim \varpi^{-3/2}</math>; there is an active discussion on optimising these improvements [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ here].) In more recent work, the single parameter <math>\varpi</math> is replaced by a pair <math>(\varpi,\delta)</math> (in previous work we had <math>\delta=\varpi</math>). Discussion on improving the values of <math>(\varpi,\delta)</math> is currently being held [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ here]. In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored.

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| 14 May
| 1/1,168 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| All subsequent work is based on Zhang's breakthrough paper.
|-
| 21 May
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| 28 May
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| 30 May
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| 31 May
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| 1 Jun
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| 2 Jun
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| 3 Jun
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| 4 Jun
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| 5 Jun
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| 6 Jun
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| 7 Jun
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>207 \varpi + 43\delta < \frac{1}{4}</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| Jun 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])

[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])

[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
|-
| Jun 12
|
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])

[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
|-
| Jun 13
|
|
|

[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
|
|-
| Jun 14
|
|
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
|
|-
| Jun 15
| <math>87\varpi+17\delta < \frac{1}{4}</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])

6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
|-
| Jun 16
| <math>87\varpi+17\delta < \frac{1}{4}</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])

<strike>155\varpi+31\delta < 1 and 11\varpi + 3\delta < \frac{1}{20} ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>

<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>

<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
|-
| Jun 18
|
| 5,937? (Pintz/[https://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[https://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])

5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])

5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])

5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])

| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([https://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)

58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])

[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])

53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
|-
| Jun 19
|
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])

5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])

[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])

53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])

[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
|-
| Jun 20
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])

<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
|
|
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
|-
| Jun 21
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])

1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records5.txt here]
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

== Polymath threads ==

* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart], Scott Morrison, 30 May 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang], Terence Tao, 3 June 2013. Inactive
* [http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/ Polymath proposal: bounded gaps between primes], Terence Tao, 4 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ Online reading seminar for Zhang’s “bounded gaps between primes], Terence Tao, 4 June 2013. Inactive
* [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/ More narrow admissible sets], Scott Morrison, 5 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ The elementary Selberg sieve and bounded prime gaps], Terence Tao, 8 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/ A combinatorial subset sum problem associated with bounded prime gaps], Terence Tao, 10 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Further analysis of the truncated GPY sieve], Terence Tao, 11 June 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/ Estimation of the Type I and Type II sums], Terence Tao, 12 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/ Estimation of the Type III sums], Terence Tao, 14 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/ A truncated elementary Selberg sieve of Pintz], Terence Tao, 18 June, 2013. Active

== Code and data ==

* [https://github.com/semorrison/polymath8 Hensely-Richards sequences], Scott Morrison
** [http://tqft.net/misc/finding%20k_0.nb A mathematica notebook for finding k_0], Scott Morrison
** [https://github.com/avi-levy/dhl python implementation], Avi Levy
* [http://www.opertech.com/primes/k-tuples.html k-tuple pattern data], Thomas J Engelsma
** [https://perswww.kuleuven.be/~u0040935/k0graph.png A graph of this data]
** [http://www.opertech.com/primes/webdata/ Tuples giving this data]
* [https://github.com/vit-tucek/admissible_sets Sifted sequences], Vit Tucek
* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Other sifted sequences], Christian Elsholtz
* [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/S0025-5718-01-01348-5.pdf Size of largest admissible tuples in intervals of length up to 1050], Clark and Jarvis
* [http://math.mit.edu/~drew/admissable_v0.1.tar C implementation of the "greedy-greedy" algorithm], Andrew Sutherland
* [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Dropbox for sequences], pedant
* [https://docs.google.com/spreadsheet/ccc?key=0Ao3urQ79oleSdEZhRS00X1FLQjM3UlJTZFRqd19ySGc&usp=sharing Spreadsheet for admissible sequences], Vit Tucek
* [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.1.zip Java code for optimising a given tuple V1.1], xfxie

== Errata ==

Page numbers refer to the file linked to for the relevant paper.

# Errata for Zhang's "[http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes]"
## Page 5: In the first display, <math>\mathcal{E}</math> should be multiplied by <math>\mathcal{L}^{2k_0+2l_0}</math>, because <math>\lambda(n)^2</math> in (2.2) can be that large, cf. (2.4).
## Page 14: In the final display, the constraint <math>(n,d_1=1</math> should be <math>(n,d_1)=1</math>.
## Page 35: In the display after (10.5), the subscript on <math>{\mathcal J}_i</math> should be deleted.
## Page 36: In the third display, a factor of <math>\tau(q_0r)^{O(1)}</math> may be needed on the right-hand side (but is ultimately harmless).
## Page 38: In the display after (10.14), <math>\xi(r,a;q_1,b_1;q_2,b_2;n,k)</math> should be <math>\xi(r,a;k;q_1,b_1;q_2,b_2;n)</math>.
## Page 42: In (12.3), <math>B</math> should probably be 2.
## Page 47: In the third display after (13.13), the condition <math>l \in {\mathcal I}_i(h)</math> should be <math>l \in {\mathcal I}_i(sh)</math>.
## Page 49: In the top line, a comma in <math>(h_1,h_2;,n_1,n_2)</math> should be deleted.
## Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
## Page 54: Three displays before (14.17), <math>\bar{r_2}(m_1+m_2)q</math> should be <math>\bar{r_2}(m_1+m_2)/q</math>.
# Errata for Motohashi-Pintz's "[http://arxiv.org/pdf/math/0602599v1.pdf A smoothed GPY sieve]"
## Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of <math>(\log \frac{R}{|D|})^{2\ell+1}</math>, (4.15) contains instead a factor of <math>(\log \frac{R/w}{|K|})^{2\ell+1}</math> which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of <math>\exp(-k\omega/3)</math> in (4.15) does not seem to be available for estimating the second sum in (5.14).
# Errata for Pintz's "[http://arxiv.org/pdf/1306.1497v1.pdf A note on bounded gaps between primes]", version 1. Update: the errata below have been corrected in subsequent versions of Pintz's paper.
## Page 7: In (2.39), the exponent of <math>3a/2</math> should instead be <math>-5a/2</math> (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
## Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

== Other relevant blog posts ==

* [http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/ Marker lecture III: “Small gaps between primes”], Terence Tao, 19 Nov 2008.
* [http://blogs.ethz.ch/kowalski/2009/01/22/the-goldston-pintz-yildirim-result-and-how-far-do-we-have-to-walk-to-twin-primes/ The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?], Emmanuel Kowalski, 22 Jan 2009.
* [http://www.math.columbia.edu/~woit/wordpress/?p=5865 Number Theory News], Peter Woit, 12 May 2013.
* [http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html Bounded Gaps Between Primes], Emily Riehl, 14 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ Bounded gaps between primes!], Emmanuel Kowalski, 21 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/04/bounded-gaps-between-primes-some-grittier-details/ Bounded gaps between primes: some grittier details], Emmanuel Kowalski, 4 June 2013.
** [http://www.math.ethz.ch/~kowalski/zhang-notes.pdf The slides from the talk mentioned in that post]
* [http://aperiodical.com/2013/06/bound-on-prime-gaps-bound-decreasing-by-leaps-and-bounds/ Bound on prime gaps bound decreasing by leaps and bounds], Christian Perfect, 8 June 2013.
* [http://blogs.ams.org/blogonmathblogs/2013/06/14/narrowing-the-gap/ Narrowing the Gap], Brie Finegold, AMS Blog on Math Blogs, 14 June 2013.

== MathOverflow ==

* [http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture], 20 May 2013.
* [http://mathoverflow.net/questions/131825/a-technical-question-related-to-zhangs-result-of-bounded-prime-gaps A technical question related to Zhang’s result of bounded prime gaps], 25 May 2013.
* [http://mathoverflow.net/questions/132452/how-does-yitang-zhang-use-cauchys-inequality-and-theorem-2-to-obtain-the-error How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the <math>S_2</math> sum], 31 May 2013.
* [http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed Tightening Zhang’s bound], 3 June 2013.
** [http://meta.mathoverflow.net/discussion/1605/tightening-zhangs-bound/ Metathread for this post]
* [http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-len Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?], 3 June 2013.

== Wikipedia and other references ==

* [http://en.wikipedia.org/wiki/Bessel_function Bessel function]
* [http://en.wikipedia.org/wiki/Bombieri-Vinogradov_theorem Bombieri-Vinogradov theorem]
* [http://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem Brun-Titchmarsh theorem]
* [http://www.encyclopediaofmath.org/index.php/Dispersion_method Dispersion method]
* [http://en.wikipedia.org/wiki/Prime_gap Prime gap]
* [http://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture Second Hardy-Littlewood conjecture]
* [http://en.wikipedia.org/wiki/Twin_prime_conjecture Twin prime conjecture]

== Recent papers and notes ==

* [http://arxiv.org/abs/1304.3199 On the exponent of distribution of the ternary divisor function], Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
* [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes], Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages. [http://arxiv.org/abs/1306.2133 arXiv]
* [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes], Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
* [http://arxiv.org/abs/1305.6289 Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture], Janos Pintz, 27 May 2013.
* [http://arxiv.org/abs/1305.6369 A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million], T. S. Trudgian, 28 May 2013.
* [http://www.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf The Friedlander-Iwaniec sum], É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
* [http://terrytao.files.wordpress.com/2013/06/bounds.pdf Notes on Zhang's prime gaps paper], Terence Tao, 1 June 2013.
* [http://arxiv.org/abs/1306.0511 Bounded prime gaps in short intervals], Johan Andersson, 3 June 2013.
* [http://arxiv.org/abs/1306.0948 Bounded length intervals containing two primes and an almost-prime II], James Maynard, 5 June 2013.
* [http://arxiv.org/abs/1306.1497 A note on bounded gaps between primes], Janos Pintz, 6 June 2013.
* Notes in a truncated elementary Selberg sieve ([http://terrytao.files.wordpress.com/2013/06/file-1.pdf Section 1], [http://terrytao.files.wordpress.com/2013/06/file-2.pdf Section 2], [http://terrytao.files.wordpress.com/2013/06/file-3.pdf Section 3]), Janos Pintz, 18 June 2013.

== Media ==

* [http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989 First proof that infinitely many prime numbers come in pairs], Maggie McKee, Nature, 14 May 2013.
* [http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html Proof that an infinite number of primes are paired], Lisa Grossman, New Scientist, 14 May 2013.
* [https://www.simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/ Unheralded Mathematician Bridges the Prime Gap], Erica Klarreich, Simons science news, 20 May 2013.
** The article also appeared on Wired as "[http://www.wired.com/wiredscience/2013/05/twin-primes/ Unknown Mathematician Proves Elusive Property of Prime Numbers]".
* [http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html The Beauty of Bounded Gaps], Jordan Ellenberg, Slate, 22 May 2013.
* [http://www.newscientist.com/article/dn23644 Game of proofs boosts prime pair result by millions], Jacob Aron, New Scientist, 4 June 2013.

== Bibliography ==

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

* [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. [http://www.ams.org/mathscinet-getitem?mr=834613 MathSciNet] [http://link.springer.com/article/10.1007%2FBF02399204 Article]
* [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. [http://www.ams.org/mathscinet-getitem?mr=891581 MathSciNet] [https://eudml.org/doc/164255 Article]
* [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. [http://www.ams.org/mathscinet-getitem?mr=976723 MathSciNet] [http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Article]
* [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
* [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 [http://www.ams.org/mathscinet-getitem?mr=1836929 MathSciNet] [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Article]
* [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). [http://www.ams.org/mathscinet-getitem?mr=610700 MathSciNet] [http://www.math.ethz.ch/~kowalski/fouvry-iwaniec-on-a-theorem.pdf Article]
* [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. [http://www.ams.org/mathscinet-getitem?mr=719249 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4226.pdf Article]
* [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. [http://www.ams.org/mathscinet-getitem?mr=786351 MathSciNet] [http://www.jstor.org/stable/1971175 JSTOR] [http://www.jstor.org/stable/1971176 Appendix]
* [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. [http://arxiv.org/abs/math/0508185 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2552109 MathSciNet]
* [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. [http://www.ams.org/mathscinet-getitem?mr=1726073 MathSciNet] [http://www.ccrwest.org/gordon/ants.pdf Article]
* [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. [http://www.ams.org/mathscinet-getitem?mr=340194 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/S0002-9904-1974-13434-8.pdf Article]
* [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. [http://www.ams.org/mathscinet-getitem?mr=396440 MathSciNet] [https://eudml.org/doc/205282 Article]
* [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. [http://arxiv.org/abs/math/0602599 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2414788 MathSciNet] [http://blms.oxfordjournals.org/content/40/2/298 Article]
* [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. [http://www.ams.org/mathscinet-getitem?mr=374060 MathSciNet] [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6718308 Article]
* [M1978] Hugh L. Montgomery, The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. [http://www.ams.org/mathscinet-getitem?mr=466048 MathSciNet] [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540922 Article]
* [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. [http://www.ams.org/mathscinet-getitem?mr=337832 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/home.html Article]
* [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. [http://www.ams.org/mathscinet-getitem?mr=130203 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa7/aa711.pdf Article]
* [S2007] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. [http://www.ams.org/mathscinet-getitem?mr=2265008 MathSciNet] [http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/ Article] [http://arxiv.org/abs/math/0605696 arXiv]

Finding narrow admissible tuples

2013-06-22T02:12:29Z

Hannes: /* Benchmarks */ Made 51,526 non-bold

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719_114806.txt 114,806]
| [http://math.mit.edu/~drew/admissible_6329_64176.txt 64,176]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
|
|
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
|-
| Greedy-greedy sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386382.txt 386,382]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285208_-147296.txt 285,208]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_10719_108450_-116422.txt 108,450]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5000_46810_3946.txt 46,810]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 61,727
| 52,371
| 47,586
|-
! Lower bounds
|-
|Inclusion-exclusion
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 29,508,018]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 1,513,556]
|
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23953 193,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 85,878]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24008 49,464]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24037 41,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 38,048]
|-
|Partitioning
|
|
|
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 156,614]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 73,094]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 43,130]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24037 37,224]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 34,068]
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719

[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| 55,149
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| 24,012
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,000 !! 672 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 8,424
| 5,406
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_342_2414.txt 2,414]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
|
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_342_2352.txt 2,352]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_4000_36612_2554.txt 36,612]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 31,098
| 27,019
| 17,202
| 7,907
| 5,046
| 2,338
|-
! Lower bounds
|-
|Inclusion-exclusion
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 29,746]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 21,884]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 14,082]
|
|
|
|-
|Partitioning
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 27,248]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 20,434]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 13,620]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 6,802]
| [https://sites.google.com/site/avishaytal/files/Primes.pdf 4,574]
| [http://www.opertech.com/primes/k-tuples.html 342]
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 4,167
| 2,648
| 1,214
|-
|First Montgomery-Vaughan
| 18,768
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| 8,448
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 3,959
| 2,558
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples (due to Engelsma). For <math>k_0</math>=342 and below the exact values of <math>H(k_0)</math> have been determined by Engelsma (each is one less than the corresponding value of <math>w</math> listed in his [http://www.opertech.com/primes/k-tuples.html tables]).

For the Zhang tuples the optimal <math>m < \pi(10^{10})</math> that produced an admissible <math>k_0</math>-tuple was used. This is not always the least <math>m</math> that produces an admissible <math>k_0</math>-tuple; for <math>k_0=</math>22,949, for example, the minimal <math>m=</math>586 yields an admissible <math>k_0</math>-tuple of diameter of 264,460, but <math>m=</math>599 yields a narrower admissible <math>k_0</math>-tuple with the listed diameter of 264,414. A list of table entries for which this occurs can be found [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here] (and also for <math>k_0</math>=6,329).

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed under in the inclusion-exclusion and partitioning rows [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 due to Avishay] and computed as described in this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition).

Finding narrow admissible tuples

2013-06-22T02:10:10Z

Hannes: /* Benchmarks */ Flipped order of Engelsma and Best known

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719_114806.txt 114,806]
| [http://math.mit.edu/~drew/admissible_6329_64176.txt 64,176]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
|
|
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
|-
| Greedy-greedy sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386382.txt 386,382]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285208_-147296.txt 285,208]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_10719_108450_-116422.txt 108,450]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt '''51,526''']
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5000_46810_3946.txt 46,810]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 61,727
| 52,371
| 47,586
|-
! Lower bounds
|-
|Inclusion-exclusion
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 29,508,018]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 1,513,556]
|
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23953 193,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 85,878]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24008 49,464]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24037 41,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 38,048]
|-
|Partitioning
|
|
|
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 156,614]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 73,094]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 43,130]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24037 37,224]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 34,068]
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719

[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| 55,149
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| 24,012
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,000 !! 672 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 8,424
| 5,406
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_342_2414.txt 2,414]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
|
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_342_2352.txt 2,352]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_4000_36612_2554.txt 36,612]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 31,098
| 27,019
| 17,202
| 7,907
| 5,046
| 2,338
|-
! Lower bounds
|-
|Inclusion-exclusion
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 29,746]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 21,884]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 14,082]
|
|
|
|-
|Partitioning
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 27,248]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 20,434]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 13,620]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 6,802]
| [https://sites.google.com/site/avishaytal/files/Primes.pdf 4,574]
| [http://www.opertech.com/primes/k-tuples.html 342]
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 4,167
| 2,648
| 1,214
|-
|First Montgomery-Vaughan
| 18,768
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| 8,448
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 3,959
| 2,558
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples (due to Engelsma). For <math>k_0</math>=342 and below the exact values of <math>H(k_0)</math> have been determined by Engelsma (each is one less than the corresponding value of <math>w</math> listed in his [http://www.opertech.com/primes/k-tuples.html tables]).

For the Zhang tuples the optimal <math>m < \pi(10^{10})</math> that produced an admissible <math>k_0</math>-tuple was used. This is not always the least <math>m</math> that produces an admissible <math>k_0</math>-tuple; for <math>k_0=</math>22,949, for example, the minimal <math>m=</math>586 yields an admissible <math>k_0</math>-tuple of diameter of 264,460, but <math>m=</math>599 yields a narrower admissible <math>k_0</math>-tuple with the listed diameter of 264,414. A list of table entries for which this occurs can be found [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here] (and also for <math>k_0</math>=6,329).

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed under in the inclusion-exclusion and partitioning rows [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 due to Avishay] and computed as described in this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition).

Finding narrow admissible tuples

2013-06-22T02:08:33Z

Hannes: /* Benchmarks */ Added links to Engelsma's data

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719_114806.txt 114,806]
| [http://math.mit.edu/~drew/admissible_6329_64176.txt 64,176]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
|
|
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
|-
| Greedy-greedy sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386382.txt 386,382]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285208_-147296.txt 285,208]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_10719_108450_-116422.txt 108,450]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt '''51,526''']
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5000_46810_3946.txt 46,810]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 61,727
| 52,371
| 47,586
|-
! Lower bounds
|-
|Inclusion-exclusion
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 29,508,018]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 1,513,556]
|
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23953 193,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 85,878]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24008 49,464]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24037 41,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 38,048]
|-
|Partitioning
|
|
|
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 156,614]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 73,094]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 43,130]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24037 37,224]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 34,068]
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719

[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| 55,149
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| 24,012
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,000 !! 672 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 8,424
| 5,406
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_342_2414.txt 2,414]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
|
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_342_2352.txt 2,352]
|-
|Best known tuple
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_4000_36612_2554.txt 36,612]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 31,098
| 27,019
| 17,202
| 7,907
| 5,046
| 2,338
|-
! Lower bounds
|-
|Inclusion-exclusion
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 29,746]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 21,884]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 14,082]
|
|
|
|-
|Partitioning
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 27,248]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 20,434]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 13,620]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 6,802]
| [https://sites.google.com/site/avishaytal/files/Primes.pdf 4,574]
| [http://www.opertech.com/primes/k-tuples.html 342]
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 4,167
| 2,648
| 1,214
|-
|First Montgomery-Vaughan
| 18,768
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| 8,448
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 3,959
| 2,558
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples (due to Engelsma). For <math>k_0</math>=342 and below the exact values of <math>H(k_0)</math> have been determined by Engelsma (each is one less than the corresponding value of <math>w</math> listed in his [http://www.opertech.com/primes/k-tuples.html tables]).

For the Zhang tuples the optimal <math>m < \pi(10^{10})</math> that produced an admissible <math>k_0</math>-tuple was used. This is not always the least <math>m</math> that produces an admissible <math>k_0</math>-tuple; for <math>k_0=</math>22,949, for example, the minimal <math>m=</math>586 yields an admissible <math>k_0</math>-tuple of diameter of 264,460, but <math>m=</math>599 yields a narrower admissible <math>k_0</math>-tuple with the listed diameter of 264,414. A list of table entries for which this occurs can be found [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here] (and also for <math>k_0</math>=6,329).

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed under in the inclusion-exclusion and partitioning rows [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 due to Avishay] and computed as described in this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition).

Finding narrow admissible tuples

2013-06-19T16:39:07Z

Hannes: /* Benchmarks */ added some lower bounds

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719_114806.txt 114,806]
| [http://math.mit.edu/~drew/admissible_6329_64176.txt 64,176]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
|
|
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
|-
| Greedy-greedy sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
|
|
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386382.txt 386,382]
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_10719_108450_-116422.txt 108,450]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt '''51,526''']
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5000_46810_3946.txt 46,810]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 61,727
| 52,371
| 47,586
|-
! Lower bounds
|-
|Inclusion-exclusion
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 29,508,018]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 1,513,556]
|
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23953 193,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 85,878]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24008 49,464]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 38,048]
|-
|Partitioning
|
|
|
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 156,614]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 73,094]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 43,130]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 34,068]
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719

[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| 55,149
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| 24,012
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,000 !! 672 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 8,424
| 5,406
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_342_2414.txt 2,414]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
|
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
|
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
|
|-
|Best known tuple
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_4000_36612_2554.txt 36,612]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_5010.txt 5,010*]
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
| Engelsma data
| 36,622
| 30,606
| 26,622
| 16,978
| 7,802
| 4,998
| 2,328
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 31,098
| 27,019
| 17,202
| 7,907
| 5,046
| 2,338
|-
! Lower bounds
|-
|Inclusion-exclusion
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 29,746]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 21,884]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 14,082]
|
|
|
|-
|Partitioning
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 27,248]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 20,434]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 13,620]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24015 6,802]
| [https://sites.google.com/site/avishaytal/files/Primes.pdf 4,574]
| [http://www.opertech.com/primes/k-tuples.html 342]
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 4,167
| 2,648
| 1,214
|-
|First Montgomery-Vaughan
| 18,768
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| 8,448
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 3,959
| 2,558
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples (due to Engelsma). For <math>k_0</math>=342 and below the exact values of <math>H(k_0)</math> have been determined by Engelsma (each is one less than the corresponding value of <math>w</math> listed in his [http://www.opertech.com/primes/k-tuples.html tables]).

For the Zhang tuples the optimal <math>m < \pi(10^{10})</math> that produced an admissible <math>k_0</math>-tuple was used. This is not always the least <math>m</math> that produces an admissible <math>k_0</math>-tuple; for <math>k_0=</math>22,949, for example, the minimal <math>m=</math>586 yields an admissible <math>k_0</math>-tuple of diameter of 264,460, but <math>m=</math>599 yields a narrower admissible <math>k_0</math>-tuple with the listed diameter of 264,414. A list of table entries for which this occurs can be found [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here] (and also for <math>k_0</math>=6,329).

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed under in the inclusion-exclusion and partitioning rows [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23925 due to Avishay] and computed as described in this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition).

Finding narrow admissible tuples

2013-06-17T00:17:12Z

Hannes: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 || 10,719 !! 6,329 || 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
|
| 50,840
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719_114806.txt 114,806]
| [http://math.mit.edu/~drew/admissible_6329_64176.txt 64,176]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
|
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
|
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
|
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
|-
| Greedy-greedy sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
|
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386382.txt 386,382]
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_10719_108450_-116422.txt 108,450]
| [http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5000_46810_3946.txt 46,810]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 61,727
| 47,586
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 32,916
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719

[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| 55,149
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 24,012
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,000 !! 672
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
|
| 28,972
| 18,386
| 8,424
| 5,406
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
|
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
|
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
|
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
|
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
|
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36636.txt 36,636*]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://math.mit.edu/~drew/admissible_2000_16984.txt 16,984*]
| [http://math.mit.edu/~drew/admissible_1000_7806.txt 7,806*]
| [http://math.mit.edu/~drew/admissible_672_5010.txt 5,010*]
|-
| Engelsma data
| 36,622
| 30,606
| 26,622
| 16,978
| 7,802
| 4,998
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
|
| 27,019
| 17,202
| 7,907
| 5,046
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 4,167
| 2,648
|-
|First Montgomery-Vaughan
| 18,768
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| 8,448
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 3,959
| 2,558
|}

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples ([http://www.opertech.com/primes/k-tuples.html due to Engelsma]).

For the Zhang tuples the optimal <math>m < \pi(10^{10})</math> that produced an admissible <math>k_0</math>-tuple was used. This is not always the least <math>m</math> that produces an admissible <math>k_0</math>-tuple; for <math>k_0=</math>22,949, for example, the minimal <math>m=</math>586 yields an admissible <math>k_0</math>-tuple of diameter of 264,460, but <math>m=</math>599 yields a narrower admissible <math>k_0</math>-tuple with the listed diameter of 264,414. A list of table entries for which this occurs can be found [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here] (and also for <math>k_0</math>=6,329).

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

Finding narrow admissible tuples

2013-06-16T23:05:43Z

Hannes: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 || 10,719 !! 6,329 || 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
|
| 50,840
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719_114806.txt 114,806]
| [http://math.mit.edu/~drew/admissible_6329_64176.txt 64,176]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
|
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
|
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
|
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
|-
| Greedy-greedy sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
|
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386382.txt 386,382]
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_10719_108450_-116422.txt 108,450]
| [http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5000_46810_3946.txt 46,810]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 61,727
| 47,586
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 32,916
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719

[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| 55,149
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 24,012
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,000 !! 672
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
|
| 28,972
| 18,386
| 8,424
| 5,406
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
|
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
|
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
|
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
|
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
|
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36636.txt 36,636*]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://math.mit.edu/~drew/admissible_2000_16984.txt 16,984*]
| [http://math.mit.edu/~drew/admissible_1000_7806.txt 7,806*]
| [http://math.mit.edu/~drew/admissible_672_5010.txt 5,010*]
|-
| Engelsma data
| 36,622
| 30,606
| 26,622
| 16,978
| 7,802
| 4,998
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
|
| 27,019
| 17,202
| 7,907
| 5,046
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
|-
|Brun-Titchmarsh
| 19,785
|
| 14,358
| 9,118
| 4,167
| 2,648
|-
|First Montgomery-Vaughan
| 18,768
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
|
| 13,696
| 8,448
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 3,959
| 2,558
|}

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples ([http://www.opertech.com/primes/k-tuples.html due to Engelsma]).

For the Zhang tuples the optimal <math>m < \pi(10^{10})</math> that produced an admissible <math>k_0</math>-tuple was used. This is not always the least <math>m</math> that produces an admissible <math>k_0</math>-tuple; for <math>k_0=</math>22,949, for example, the minimal <math>m=</math>586 yields an admissible <math>k_0</math>-tuple of diameter of 264,460, but <math>m=</math>599 yields a narrower admissible <math>k_0</math>-tuple with the listed diameter of 264,414. A list of table entries for which this occurs can be found [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here] (and also for <math>k_0</math>=6,329).

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

Finding narrow admissible tuples

2013-06-16T08:34:59Z

Hannes: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 || 10,719 !! 6,329 || 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
|
| 50,840
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719.txt 114,806]
|
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
|
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
|
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
|
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
|-
| Greedy-greedy sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
|
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386432.txt 386,432]
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898]
| [http://math.mit.edu/~drew/admissible_10719_108462.txt 108,462]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760]
| [http://math.mit.edu/~drew/admissible_5000_46824.txt 46,824]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
|
| 47,586
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 32,916
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719

[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| 55,149
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 24,012
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,000 !! 2,000 !! 1,000 !! 672
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 28,972
| 18,386
| 8,424
| 5,406
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36636.txt 36,636*]
| [http://math.mit.edu/~drew/admissible_3000_26610.txt 26,610]
| [http://math.mit.edu/~drew/admissible_2000_16984.txt 16,984*]
| [http://math.mit.edu/~drew/admissible_1000_7806.txt 7,806*]
| [http://math.mit.edu/~drew/admissible_672_5010.txt 5,010*]
|-
| Engelsma data
| 36,622
| 26,622
| 16,978
| 7,802
| 4,998
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 27,019
| 17,202
| 7,907
| 5,046
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
|-
|Brun-Titchmarsh
| 19,785
| 14,358
| 9,118
| 4,167
| 2,648
|-
|First Montgomery-Vaughan
| 18,768
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 13,696
| 8,448
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 3,959
| 2,558
|}

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples ([http://www.opertech.com/primes/k-tuples.html due to Engelsma]).

For the Zhang tuples the optimal <math>m < \pi(10^{10})</math> that produced an admissible <math>k_0</math>-tuple was used. This is not always the least <math>m</math> that produces an admissible <math>k_0</math>-tuple; for <math>k_0=</math>22949, for example, the minimal <math>m=</math>586 yields an admissible <math>k_0</math>-tuple of diameter of 264,460, but <math>m=</math>599 yields a narrower admissible <math>k_0</math>-tuple with the listed diameter of 264,414. A complete list of the table entries for which this occurs can be found [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

Finding narrow admissible tuples

2013-06-16T08:31:34Z

Hannes: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 || 10,719 !! 6,329 || 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
|
| 50,840
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719.txt 114,806]
|
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
|
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
|
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
|
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
|-
| Greedy-greedy sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
|
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386432.txt 386,432]
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898]
| [http://math.mit.edu/~drew/admissible_10719_108462.txt 108,462]
|
| [http://math.mit.edu/~drew/admissible_5000_46824.txt 46,824]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
|
| 47,586
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 32,916
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719

[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| 55,149
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 24,012
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,000 !! 2,000 !! 1,000 !! 672
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 28,972
| 18,386
| 8,424
| 5,406
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36636.txt 36,636*]
| [http://math.mit.edu/~drew/admissible_3000_26610.txt 26,610]
| [http://math.mit.edu/~drew/admissible_2000_16984.txt 16,984*]
| [http://math.mit.edu/~drew/admissible_1000_7806.txt 7,806*]
| [http://math.mit.edu/~drew/admissible_672_5010.txt 5,010*]
|-
| Engelsma data
| 36,622
| 26,622
| 16,978
| 7,802
| 4,998
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 27,019
| 17,202
| 7,907
| 5,046
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
|-
|Brun-Titchmarsh
| 19,785
| 14,358
| 9,118
| 4,167
| 2,648
|-
|First Montgomery-Vaughan
| 18,768
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 13,696
| 8,448
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 3,959
| 2,558
|}

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples ([http://www.opertech.com/primes/k-tuples.html due to Engelsma]).

For the Zhang tuples the optimal <math>m < \pi(10^{10})</math> that produced an admissible <math>k_0</math>-tuple was used. This is not always the least <math>m</math> that produces an admissible <math>k_0</math>-tuple; for <math>k_0=</math>22949, for example, the minimal <math>m=</math>586 yields an admissible <math>k_0</math>-tuple of diameter of 264,460, but <math>m=</math>599 yields a narrower admissible <math>k_0</math>-tuple with the listed diameter of 264,414. A complete list of the table entries for which this occurs can be found [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

Finding narrow admissible tuples

2013-06-16T08:28:04Z

Hannes: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 || 10,719 !! 6,329 || 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
|
| 50,840
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719.txt 114,806]
|
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
|
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
|
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
|
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
|-
| Greedy-greedy sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
|
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386432.txt 386,432]
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898]
| [http://math.mit.edu/~drew/admissible_10719_108462.txt 108,462]
|
| [http://math.mit.edu/~drew/admissible_5000_46824.txt 46,824]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
|
| 47,586
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
|
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719

[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| 55,149
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
|
| 24,012
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,000 !! 2,000 !! 1,000 !! 672
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 28,972
| 18,386
| 8,424
| 5,406
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36636.txt 36,636*]
| [http://math.mit.edu/~drew/admissible_3000_26610.txt 26,610]
| [http://math.mit.edu/~drew/admissible_2000_16984.txt 16,984*]
| [http://math.mit.edu/~drew/admissible_1000_7806.txt 7,806*]
| [http://math.mit.edu/~drew/admissible_672_5010.txt 5,010*]
|-
| Engelsma data
| 36,622
| 26,622
| 16,978
| 7,802
| 4,998
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 27,019
| 17,202
| 7,907
| 5,046
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
|-
|Brun-Titchmarsh
| 19,785
| 14,358
| 9,118
| 4,167
| 2,648
|-
|First Montgomery-Vaughan
| 18,768
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 13,696
| 8,448
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 3,959
| 2,558
|}

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples ([http://www.opertech.com/primes/k-tuples.html due to Engelsma]).

For the Zhang tuples the optimal <math>m < \pi(10^{10})</math> that produced an admissible <math>k_0</math>-tuple was used. This is not always the least <math>m</math> that produces an admissible <math>k_0</math>-tuple; for <math>k_0=</math>22949, for example, the minimal <math>m=</math>586 yields an admissible <math>k_0</math>-tuple of diameter of 264,460, but <math>m=</math>599 yields a narrower admissible <math>k_0</math>-tuple with the listed diameter of 264,414. A complete list of the table entries for which this occurs can be found [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

Finding narrow admissible tuples

2013-06-15T12:32:55Z

Hannes: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 || 10,719 !! 5,000 !! 4,000 !! 3,000 !! 2,000 !! 1,000 !! 672
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 50,840
| 39,660
| 28,972
| 18,386
| 8,424
| 5,406
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719.txt 114,806]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
|-
|Shifted Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|-
| Greedy-greedy sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386432.txt 386,432]
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910]
| [http://math.mit.edu/~drew/admissible_10719_108462.txt 108,462]
| [http://math.mit.edu/~drew/admissible_5000_46824.txt 46,824]
| [http://math.mit.edu/~drew/admissible_4000_36636.txt 36,636*]
| [http://math.mit.edu/~drew/admissible_3000_26610.txt 26,610]
| [http://math.mit.edu/~drew/admissible_2000_16984.txt 16,984*]
| [http://math.mit.edu/~drew/admissible_1000_7806.txt 7,806*]
| [http://math.mit.edu/~drew/admissible_672_5010.txt 5,010*]
|-
| Engelsma data
| -
| -
| -
| -
| -
| -
| -
| -
| 36,622
| 26,622
| 16,978
| 7,802
| 4,998
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 47,586
| 37,176
| 27,019
| 17,202
| 7,907
| 5,046
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
|-
|Second Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 25,351
| 19,785
| 14,358
| 9,118
| 4,167
| 2,648
|-
|First Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719

[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| 55,149
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| 24,012
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
| 18,768
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 13,696
| 8,448
[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 3,959
| 2,558
|}

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples ([http://www.opertech.com/primes/k-tuples.html due to Engelsma]).

For the Zhang tuples the optimal <math>m < \pi(10^{10})</math> that produced an admissible <math>k_0</math>-tuple was used. This is not always the least <math>m</math> that produces an admissible <math>k_0</math>-tuple; for <math>k_0=</math>22949, for example, the minimal <math>m=</math>586 yields an admissible <math>k_0</math>-tuple of diameter of 264,460, but <math>m=</math>599 yields a narrower admissible <math>k_0</math>-tuple with the listed diameter of 264,414. A complete list of the table entries for which this occurs can be found [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

Finding narrow admissible tuples

2013-06-15T12:25:29Z

Hannes: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 || 10,719 !! 5,000 !! 4,000 !! 3,000 !! 2,000 !! 1,000 !! 672
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 50,840
| 39,660
| 28,972
| 18,386
| 8,424
| 5,406
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719.txt 114,806]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
|-
|Shifted Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|-
| Greedy-greedy sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386432.txt 386,432]
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910]
| [http://math.mit.edu/~drew/admissible_10719_108462.txt 108,462]
| [http://math.mit.edu/~drew/admissible_5000_46824.txt 46,824]
| [http://math.mit.edu/~drew/admissible_4000_36636.txt 36,636*]
| [http://math.mit.edu/~drew/admissible_3000_26610.txt 26,610]
| [http://math.mit.edu/~drew/admissible_2000_16984.txt 16,984*]
| [http://math.mit.edu/~drew/admissible_1000_7806.txt 7,806*]
| [http://math.mit.edu/~drew/admissible_672_5010.txt 5,010*]
|-
| Engelsma data
| -
| -
| -
| -
| -
| -
| -
| -
| 36,622
| 26,622
| 16,978
| 7,802
| 4,998
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 47,586
| 37,176
| 27,019
| 17,202
| 7,907
| 5,046
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
|-
|Second Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 25,351
| 19,785
| 14,358
| 9,118
| 4,167
| 2,648
|-
|First Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
|
| 55,149
| 24,012
| 18,768
| 13,696
| 8,448
| 3,959
| 2,558
|}

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples ([http://www.opertech.com/primes/k-tuples.html due to Engelsma]).

For the Zhang tuples the optimal <math>m < \pi(10^{10})</math> that produced an admissible <math>k_0</math>-tuple was used. This is not always the least <math>m</math> that produces an admissible <math>k_0</math>-tuple; for <math>k_0=</math>22949, for example, the minimal <math>m=</math>586 yields an admissible <math>k_0</math>-tuple of diameter of 264,460, but <math>m=</math>599 yields a narrower admissible <math>k_0</math>-tuple with the listed diameter of 264,414. A complete list of the table entries for which this occurs can be found [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

Finding narrow admissible tuples

2013-06-14T22:11:14Z

Hannes: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 || 10,719 !! 5,000 !! 4,000 !! 3,000 !! 2,000 !! 1,000 !! 672
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 50,840
| 39,660
| 28,972
| 18,386
| 8,424
| 5,406
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719.txt 114,806]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
|-
|Shifted Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|-
| Greedy-greedy sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532]
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910]
| [http://math.mit.edu/~drew/admissible_10719_108462.txt 108,462]
| [http://math.mit.edu/~drew/admissible_5000_46824.txt 46,824]
| [http://math.mit.edu/~drew/admissible_4000_36636.txt 36,636*]
| [http://math.mit.edu/~drew/admissible_3000_26610.txt 26,610]
| [http://math.mit.edu/~drew/admissible_2000_16984.txt 16,984*]
| [http://math.mit.edu/~drew/admissible_1000_7806.txt 7,806*]
| [http://math.mit.edu/~drew/admissible_672_5010.txt 5,010*]
|-
| Engelsma data
| -
| -
| -
| -
| -
| -
| -
| -
| 36,622
| 26,622
| 16,978
| 7,802
| 4,998
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 47,586
| 37,176
| 27,019
| 17,202
| 7,907
| 5,046
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
|-
|Second Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 25,351
| 19,785
| 14,358
| 9,118
| 4,167
| 2,648
|-
|First Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
|
| 55,149
| 24,012
| 18,768
| 13,696
| 8,448
| 3,959
| 2,558
|}

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples ([http://www.opertech.com/primes/k-tuples.html due to Engelsma]).

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

Finding narrow admissible tuples

2013-06-14T01:21:25Z

Hannes: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works (in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z</math>, the Schinzel sieve (discussed in [HR1973], [CJ2001] first sieves by <math>1\bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . For a given choice of <math>y</math>, the parameter <math>z</math> is minimized subject to ensuring that the first <math>k_0</math> survivors (after the first) form an admissible sequence <math>\mathcal{H}</math>, so the only free parameter is <math>y</math>, which is chosen to minimize the diameter of <math>\mathcal{H}</math>. The case <math>y=1</math> corresponds to a sieve of Eratosthenes, which will typically yield the same sequence as Zhang with the minimal (but not necessarily optimal) value of <math>m</math> that yields an admissible <math>k_0</math>-tuple. As originally proposed, the Schinzel sieve works over the positive integers, but one can apply the sieve to any given interval, and as with the Hensley-Richards sieve, it is generally better to use an asymmetric interval (which need not contain the origin).

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-Schinzel sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a Schinzel sieve with <math>y=2</math> and <math>z=\sqrt{x_1-x_0}</math> and then continuing in a greedy fashion
This method was proposed by Sutherland and originally referred to as a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 "greedy-greedy"] approach. This nomenclature arose from the fact that one optimization that can be applied to the standard Schinzel sieve on a given interval is to "greedily" avoid sieving modulo primes where the set of survivors is already admissible (this may occur for primes less than the minimal value of <math>z</math> that yields <math>k_0</math>-survivors), while a second optimization is to use a value of <math>z</math> that is intentionally smaller than necessary and switch to greedy sieving for primes greater than <math>z</math>. With the choice <math>z=\sqrt{x_1-x_0}</math>, unless the initial interval is much larger than necessary, all primes up to <math>z</math> will require a residue class to be sieved and the first "greedy" seldom applies.

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 || 10,719 !! 5,000 !! 4,000 !! 3,000 !! 2,000 !! 1,000 !! 672
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 50,840
| 39,660
| 28,972
| 18,386
| 8,424
| 5,406
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719.txt 114,806]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
|-
|Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|-
|greedy-Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532]
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910]
| [http://math.mit.edu/~drew/admissible_10719_108462.txt 108,462]
| [http://math.mit.edu/~drew/admissible_5000_46824.txt 46,824]
| [http://math.mit.edu/~drew/admissible_4000_36636.txt 36,636*]
| [http://math.mit.edu/~drew/admissible_3000_26622.txt 26,622]
| [http://math.mit.edu/~drew/admissible_2000_16984.txt 16,984*]
| [http://math.mit.edu/~drew/admissible_1000_7808.txt 7,808*]
| [http://math.mit.edu/~drew/admissible_672_5010.txt 5,010*]
|-
| Engelsma data
| -
| -
| -
| -
| -
| -
| -
| -
| 36,622
| 26,622
| 16,978
| 7,802
| 4,998
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 47,586
| 37,176
| 27,019
| 17,202
| 7,907
| 5,046
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
|-
|MV with <math>c=3.2/\pi</math>
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
|-
|Second Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 25,351
| 19,785
| 14,358
| 9,118
| 4,167
| 2,648
|-
|First Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
|
| 55,149
| 24,012
| 18,768
| 13,696
| 8,448
| 3,959
| 2,558
|}

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples ([http://www.opertech.com/primes/k-tuples.html due to Engelsma]).

The Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-Schinzel tuples were generated by breaking ties downward in every case, as in Sutherland's original greedy-greedy algorithm (and the optimal interval was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.
As with the Schinzel tuples, the chosen intervals are not guaranteed to be optimal but are believed to be so.

Finding narrow admissible tuples

2013-06-14T01:17:50Z

Hannes: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works (in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z</math>, the Schinzel sieve (discussed in [HR1973], [CJ2001] first sieves by <math>1\bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . For a given choice of <math>y</math>, the parameter <math>z</math> is minimized subject to ensuring that the first <math>k_0</math> survivors (after the first) form an admissible sequence <math>\mathcal{H}</math>, so the only free parameter is <math>y</math>, which is chosen to minimize the diameter of <math>\mathcal{H}</math>. The case <math>y=1</math> corresponds to a sieve of Eratosthenes, which will typically yield the same sequence as Zhang with the minimal (but not necessarily optimal) value of <math>m</math> that yields an admissible <math>k_0</math>-tuple. As originally proposed, the Schinzel sieve works over the positive integers, but one can apply the sieve to any given interval, and as with the Hensley-Richards sieve, it is generally better to use an asymmetric interval (which need not contain the origin).

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-Schinzel sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a Schinzel sieve with <math>y=2</math> and <math>z=\sqrt{x_1-x_0}</math> and then continuing in a greedy fashion
This method was proposed by Sutherland and originally referred to as a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 "greedy-greedy"] approach. This nomenclature arose from the fact that one optimization that can be applied to the standard Schinzel sieve on a given interval is to "greedily" avoid sieving modulo primes where the set of survivors is already admissible (this may occur for primes less than the minimal value of <math>z</math> that yields <math>k_0</math>-survivors), while a second optimization is to use a value of <math>z</math> that is intentionally smaller than necessary and switch to greedy sieving for primes greater than <math>z</math>. With the choice <math>z=\sqrt{x_1-x_0}</math>, unless the initial interval is much larger than necessary, all primes up to <math>z</math> will require a residue class to be sieved and the first "greedy" seldom applies.

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 || 10,719 !! 5,000 !! 4,000 !! 3,000 !! 2,000 !! 1,000 !! 672
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 50,840
| 39,660
| 28,972
| 18,386
| 8,424
| 5,406
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719.txt 114,806]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
|-
|Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|-
|greedy-Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532]
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910]
| [http://math.mit.edu/~drew/admissible_10719_108462.txt 108,462]
| [http://math.mit.edu/~drew/admissible_5000_46824.txt 46,824]
| [http://math.mit.edu/~drew/admissible_4000_36636.txt 36,636*]
| [http://math.mit.edu/~drew/admissible_3000_26622.txt 26,622]
| [http://math.mit.edu/~drew/admissible_2000_16984.txt 16,984*]
| [http://math.mit.edu/~drew/admissible_1000_7808.txt 7,808*]
| [http://math.mit.edu/~drew/admissible_672_5010.txt 5,010*]
|-
| Engelsma data
| -
| -
| -
| -
| -
| -
| -
| -
| 36,622
| 26,622
| 16,978
| 7,802
| 4,998
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 47,586
| 37,176
| 27,019
| 17,202
| 7,907
| 5,046
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28781]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22564]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10500]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3124]
|-
|MV with <math>c=3.2/\pi</math>
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28737]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22523]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10480]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3118]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27708]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21701]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10061]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2979]
|-
|Second Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27696]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21690]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2977]
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
|
| 58,863
| 25,351
| 19,785
| 14,358
| 9,118
| 4,167
| 2,648
|-
|First Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
|
| 55,149
| 24,012
| 18,768
| 13,696
| 8,448
| 3,959
| 2,558
|}

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples ([http://www.opertech.com/primes/k-tuples.html due to Engelsma]).

The Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-Schinzel tuples were generated by breaking ties downward in every case, as in Sutherland's original greedy-greedy algorithm (and the optimal interval was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.
As with the Schinzel tuples, the chosen intervals are not guaranteed to be optimal but are believed to be so.

Finding narrow admissible tuples

2013-06-14T01:09:48Z

Hannes: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works (in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z</math>, the Schinzel sieve (discussed in [HR1973], [CJ2001] first sieves by <math>1\bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . For a given choice of <math>y</math>, the parameter <math>z</math> is minimized subject to ensuring that the first <math>k_0</math> survivors (after the first) form an admissible sequence <math>\mathcal{H}</math>, so the only free parameter is <math>y</math>, which is chosen to minimize the diameter of <math>\mathcal{H}</math>. The case <math>y=1</math> corresponds to a sieve of Eratosthenes, which will typically yield the same sequence as Zhang with the minimal (but not necessarily optimal) value of <math>m</math> that yields an admissible <math>k_0</math>-tuple. As originally proposed, the Schinzel sieve works over the positive integers, but one can apply the sieve to any given interval, and as with the Hensley-Richards sieve, it is generally better to use an asymmetric interval (which need not contain the origin).

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-Schinzel sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a Schinzel sieve with <math>y=2</math> and <math>z=\sqrt{x_1-x_0}</math> and then continuing in a greedy fashion
This method was proposed by Sutherland and originally referred to as a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 "greedy-greedy"] approach. This nomenclature arose from the fact that one optimization that can be applied to the standard Schinzel sieve on a given interval is to "greedily" avoid sieving modulo primes where the set of survivors is already admissible (this may occur for primes less than the minimal value of <math>z</math> that yields <math>k_0</math>-survivors), while a second optimization is to use a value of <math>z</math> that is intentionally smaller than necessary and switch to greedy sieving for primes greater than <math>z</math>. With the choice <math>z=\sqrt{x_1-x_0}</math>, unless the initial interval is much larger than necessary, all primes up to <math>z</math> will require a residue class to be sieved and the first "greedy" seldom applies.

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 || 10,719 !! 5,000 !! 4,000 !! 3,000 !! 2,000 !! 1,000 !! 672
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 50,840
| 39,660
| 28,972
| 18,386
| 8,424
| 5,406
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719.txt 114,806]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
|-
|Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|-
|greedy-Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532]
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910]
| [http://math.mit.edu/~drew/admissible_10719_108462.txt 108,462]
| [http://math.mit.edu/~drew/admissible_5000_46824.txt 46,824]
| [http://math.mit.edu/~drew/admissible_4000_36636.txt 36,636*]
| [http://math.mit.edu/~drew/admissible_3000_26622.txt 26,622]
| [http://math.mit.edu/~drew/admissible_2000_16984.txt 16,984*]
| [http://math.mit.edu/~drew/admissible_1000_7808.txt 7,808*]
| [http://math.mit.edu/~drew/admissible_672_5010.txt 5,010*]
|-
| Engelsma data
| -
| -
| -
| -
| -
| -
| -
| -
| 36,622
| 26,622
| 16,978
| 7,802
| 4,998
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 47,586
| 37,176
| 27,019
| 17,202
| 7,907
| 5,046
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
|
|
|
|
|
|
|-
|MV with <math>c=3.2/\pi</math>
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
|
|
|
|
|
|
|-
|MV with <math>c=\sqrt{22}/\pi</math>
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
|
|
|
|
|
|
|-
|Second Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
|
|
|
|
|
|
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
|
| 58,863
| 25,351
| 19,785
| 14,358
| 9,118
| 4,167
| 2,648
|-
|First Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
|
| 55,149
| 24,012
| 18,768
| 13,696
| 8,448
| 3,959
| 2,558
|}

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples ([http://www.opertech.com/primes/k-tuples.html due to Engelsma]).

The Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-Schinzel tuples were generated by breaking ties downward in every case, as in Sutherland's original greedy-greedy algorithm (and the optimal interval was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.
As with the Schinzel tuples, the chosen intervals are not guaranteed to be optimal but are believed to be so.

Finding narrow admissible tuples

2013-06-13T15:42:53Z

Hannes: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is taken to optimize the diameter <math>p_{m+k_0}-p_{m+1}</math> while staying admissible (in practice, this basically means making <math>m</math> as small as possible). Certainly any <math>m</math> with <math>p_{m+1} > k_0</math> works (in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is again optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z</math>, the Schinzel sieve (discussed in [HR1973], [CJ2001] first sieves by <math>1\bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . For a given choice of <math>y</math>, the parameter <math>z</math> is minimized subject to ensuring that the first <math>k_0</math> survivors (after the first) form an admissible sequence <math>\mathcal{H}</math>, so the only free parameter is <math>y</math>, which is chosen to minimize the diameter of <math>\mathcal{H}</math>. The case <math>y=1</math> corresponds to a sieve of Eratosthenes, which will typically yield the same sequence as Zhang with the minimal (but not necessarily optimal) value of <math>m</math> that yields an admissible <math>k_0</math>-tuple. As originally proposed, the Schinzel sieve works over the positive integers, but one can apply the sieve to any given interval, and as with the Hensley-Richards sieve, it is generally better to use an asymmetric interval (which need not contain the origin).

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-Schinzel sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a Schinzel sieve with <math>y=2</math> and <math>z=\sqrt{x_1-x_0}</math> and then continuing in a greedy fashion
This method was proposed by Sutherland and originally referred to as a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 "greedy-greedy"] approach. This nomenclature arose from the fact that one optimization that can be applied to the standard Schinzel sieve on a given interval is to "greedily" avoid sieving modulo primes where the set of survivors is already admissible (this may occur for primes less than the minimal value of <math>z</math> that yields <math>k_0</math>-survivors), while a second optimization is to use a value of <math>z</math> that is intentionally smaller than necessary and switch to greedy sieving for primes greater than <math>z</math>. With the choice <math>z=\sqrt{x_1-x_0}</math>, unless the initial interval is much larger than necessary, all primes up to <math>z</math> will require a residue class to be sieved and the first "greedy" seldom applies.

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!!3,500,000!! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 || 10,719 !! 5,000 !! 4,000 !! 3,000 !! 2,000 !! 1,000 !! 672
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 50,840
| 39,660
| 28,972
| 18,386
| 8,424
| 5,406
|-
|Zhang sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 59,093,364]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411932.txt 411,932]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23282_268536.txt 268,536]
| [http://math.mit.edu/~drew/admissible_22949_264414.txt 264,414]
| [http://math.mit.edu/~drew/admissible_10719.txt 114,806]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1000_8212.txt 8,212]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
|-
|Hensley-Richards sieve
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
|-
|Asymmetric Hensley-Richards
|
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
|-
|Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2413228.txt 2,413,228]
| [http://math.mit.edu/~drew/admissible_34429_400512.txt 400,512]
| [http://math.mit.edu/~drew/admissible_26024_295162.txt 295,162]
| [http://math.mit.edu/~drew/admissible_23283_262206.txt 262,206]
| [http://math.mit.edu/~drew/admissible_22949_258000.txt 258,000]
| [http://math.mit.edu/~drew/admissible_10719_112440.txt 112,440]
| [http://math.mit.edu/~drew/admissible_5000_48726.txt 48,726]
| [http://math.mit.edu/~drew/admissible_4000_38168.txt 38,168]
| [http://math.mit.edu/~drew/admissible_3000_27632.txt 27,632]
| [http://math.mit.edu/~drew/admissible_2000_17616.txt 17,616]
| [http://math.mit.edu/~drew/admissible_1000_8160.txt 8,160]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
|-
|greedy-Schinzel sieve
|
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
|-
|Best known tuple
| [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 57,554,086]
| [http://math.mit.edu/~drew/admissible_181000_2326476.txt 2,326,476]
| [http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532]
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210]
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804]
| [http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970]
| [http://math.mit.edu/~drew/admissible_10719_108462.txt 108,462]
| [http://math.mit.edu/~drew/admissible_5000_46824.txt 46,824]
| [http://math.mit.edu/~drew/admissible_4000_36636.txt 36,636*]
| [http://math.mit.edu/~drew/admissible_3000_26622.txt 26,622]
| [http://math.mit.edu/~drew/admissible_2000_16984.txt 16,984*]
| [http://math.mit.edu/~drew/admissible_1000_7808.txt 7,808*]
| [http://math.mit.edu/~drew/admissible_672_5010.txt 5,010*]
|-
| Engelsma data
| -
| -
| -
| -
| -
| -
| -
| -
| 36,622
| 26,622
| 16,978
| 7,802
| 4,998
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 47,586
| 37,176
| 27,019
| 17,202
| 7,907
| 5,046
|-
! Lower bounds
|-
|MV with <math>c=1</math> (conjectural)
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23648 234,642]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 172,924]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
|
|
|
|
|
|
|-
|MV with <math>c=3.2/\pi</math>
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23648 234,322]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 172,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
|
|
|
|
|
|
|-
|MV with <math>c=\sqrt{22}/\pi</math>
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
|
|
|
|
|
|
|-
|Second Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
|
|
|
|
|
|
|-
|Brun-Titchmarsh
| 30,137,225
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
|
| 58,863
| 25,351
| 19,785
| 14,358
| 9,118
| 4,167
| 2,648
|-
|First Montgomery-Vaughan
|
|
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23621 196,729]
196,719
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
145,461
| 128,971
|
| 55,149
| 24,012
| 18,768
| 13,696
| 8,448
| 3,959
| 2,558
|}

<nowiki>*</nowiki> indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of <math>k_0</math>, but are not as narrow as the literally best known tuples ([http://www.opertech.com/primes/k-tuples.html due to Engelsma]).

The Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-Schinzel tuples were generated by breaking ties downward in every case, as in Sutherland's original greedy-greedy algorithm (and the optimal interval was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.
As with the Schinzel tuples, the chosen intervals are not guaranteed to be optimal but are believed to be so.

Finding narrow admissible tuples

2013-06-11T17:34:09Z

Hannes: /* Benchmarks */

Bounded gaps between primes

2013-06-11T13:13:28Z

Hannes: /* World records */

== World records ==

* <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
* <math>k_0</math> is a quantity such that every admissible <math>k_0</math>-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in <math>k_0</math> lead to improvements in <math>H</math>. (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].)
* <math>\varpi</math> is a technical parameter related to a specialized form of the [https://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture Elliott-Halberstam conjecture]. Would like to be as large as possible. Improvements in <math>\varpi</math> lead to improvements in <math>k_0</math>. (The relationship is roughly of the form <math>k_0 \sim \varpi^{-3/2}</math>; there is an active discussion on optimising these improvements [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ here].) In more recent work, the single parameter <math>\varpi</math> is replaced by a pair <math>(\varpi,\delta)</math> (in previous work we had <math>\delta=\varpi</math>). Discussion on improving the values of <math>(\varpi,\delta)</math> is currently being held [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ here]. In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored.

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| 14 May
| 1/1,168 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| All subsequent work is based on Zhang's breakthrough paper.
|-
| 21 May
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| 28 May
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| 30 May
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| 31 May
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| 1 Jun
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| 2 Jun
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| 3 Jun
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| 4 Jun
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| 5 Jun
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429 ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| 6 Jun
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| 7 Jun
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>207 \varpi + 43\delta < \frac{1}{4}</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024 ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| June 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000* ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338* ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])
[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283 ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])
|
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

== Polymath threads ==

* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart], Scott Morrison, 30 May 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang], Terence Tao, 3 June 2013. Inactive
* [http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/ Polymath proposal: bounded gaps between primes], Terence Tao, 4 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ Online reading seminar for Zhang’s “bounded gaps between primes], Terence Tao, 4 June 2013. Active
* [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/ More narrow admissible sets], Scott Morrison, 5 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ The elementary Selberg sieve and bounded prime gaps], Terence Tao, 8 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/ A combinatorial subset sum problem associated with bounded prime gaps], Terence Tao, 10 June 2013. Active

== Code and data ==

* [https://github.com/semorrison/polymath8 Hensely-Richards sequences], Scott Morrison
** [http://tqft.net/misc/finding%20k_0.nb A mathematica notebook for finding k_0], Scott Morrison
** [https://github.com/avi-levy/dhl python implementation], Avi Levy
* [http://www.opertech.com/primes/k-tuples.html k-tuple pattern data], Thomas J Engelsma
** [https://perswww.kuleuven.be/~u0040935/k0graph.png A graph of this data]
* [https://github.com/vit-tucek/admissible_sets Sifted sequences], Vit Tucek
* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Other sifted sequences], Christian Elsholtz
* [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/S0025-5718-01-01348-5.pdf Size of largest admissible tuples in intervals of length up to 1050], Clark and Jarvis
* [http://math.mit.edu/~drew/admissable_v0.1.tar C implementation of the "greedy-greedy" algorithm], Andrew Sutherland
* [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Dropbox for sequences], pedant
* [https://docs.google.com/spreadsheet/ccc?key=0Ao3urQ79oleSdEZhRS00X1FLQjM3UlJTZFRqd19ySGc&usp=sharing Spreadsheet for admissible sequences], Vit Tucek
* [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.0.zip Java code for optimising a given tuple V1.0], [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.1.zip Java code for optimising a given tuple V1.1], xfxie

== Errata ==

Page numbers refer to the file linked to for the relevant paper.

# Errata for Zhang's "[http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes]"
## Page 5: In the first display, <math>\mathcal{E}</math> should be multiplied by <math>\mathcal{L}^{2k_0+2l_0}</math>, because <math>\lambda(n)^2</math> in (2.2) can be that large, cf. (2.4).
## Page 14: In the final display, the constraint <math>(n,d_1=1</math> should be <math>(n,d_1)=1</math>.
## Page 35: In the display after (10.5), the subscript on <math>{\mathcal J}_i</math> should be deleted.
## Page 36: In the third display, a factor of <math>\tau(q_0r)^{O(1)}</math> may be needed on the right-hand side (but is ultimately harmless).
## Page 38: In the display after (10.14), <math>\xi(r,a;q_1,b_1;q_2,b_2;n,k)</math> should be <math>\xi(r,a;k;q_1,b_1;q_2,b_2;n)</math>.
## Page 42: In (12.3), <math>B</math> should probably be 2.
## Page 47: In the third display after (13.13), the condition <math>l \in {\mathcal I}_i(h)</math> should be <math>l \in {\mathcal I}_i(sh)</math>.
## Page 49: In the top line, a comma in <math>(h_1,h_2;,n_1,n_2)</math> should be deleted.
## Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
## Page 54: Three displays before (14.17), <math>\bar{r_2}(m_1+m_2)q</math> should be <math>\bar{r_2}(m_1+m_2)/q</math>.
# Errata for Motohashi-Pintz's "[http://arxiv.org/pdf/math/0602599v1.pdf A smoothed GPY sieve]"
## Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of <math>(\log \frac{R}{|D|})^{2\ell+1}</math>, (4.15) contains instead a factor of <math>(\log \frac{R/w}{|K|})^{2\ell+1}</math> which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of <math>\exp(-k\omega/3)</math> in (4.15) does not seem to be available for estimating the second sum in (5.14).
# Errata for Pintz's "[http://arxiv.org/pdf/1306.1497v1.pdf A note on bounded gaps between primes]"
## Page 7: In (2.39), the exponent of <math>3a/2</math> should instead be <math>-5a/2</math> (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
## Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

== Other relevant blog posts ==

* [http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/ Marker lecture III: “Small gaps between primes”], Terence Tao, 19 Nov 2008.
* [http://blogs.ethz.ch/kowalski/2009/01/22/the-goldston-pintz-yildirim-result-and-how-far-do-we-have-to-walk-to-twin-primes/ The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?], Emmanuel Kowalski, 22 Jan 2009.
* [http://www.math.columbia.edu/~woit/wordpress/?p=5865 Number Theory News], Peter Woit, 12 May 2013.
* [http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html Bounded Gaps Between Primes], Emily Riehl, 14 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ Bounded gaps between primes!], Emmanuel Kowalski, 21 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/04/bounded-gaps-between-primes-some-grittier-details/ Bounded gaps between primes: some grittier details], Emmanuel Kowalski, 4 June 2013.
** [http://www.math.ethz.ch/~kowalski/zhang-notes.pdf The slides from the talk mentioned in that post]
* [http://aperiodical.com/2013/06/bound-on-prime-gaps-bound-decreasing-by-leaps-and-bounds/ Bound on prime gaps bound decreasing by leaps and bounds], Christian Perfect, 8 June 2013.

== MathOverflow ==

* [http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture], 20 May 2013.
* [http://mathoverflow.net/questions/131825/a-technical-question-related-to-zhangs-result-of-bounded-prime-gaps A technical question related to Zhang’s result of bounded prime gaps], 25 May 2013.
* [http://mathoverflow.net/questions/132452/how-does-yitang-zhang-use-cauchys-inequality-and-theorem-2-to-obtain-the-error How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the <math>S_2</math> sum], 31 May 2013.
* [http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed Tightening Zhang’s bound], 3 June 2013.
** [http://meta.mathoverflow.net/discussion/1605/tightening-zhangs-bound/ Metathread for this post]
* [http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-len Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?], 3 June 2013.

== Wikipedia and other references ==

* [http://en.wikipedia.org/wiki/Bessel_function Bessel function]
* [http://en.wikipedia.org/wiki/Bombieri-Vinogradov_theorem Bombieri-Vinogradov theorem]
* [http://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem Brun-Titchmarsh theorem]
* [http://www.encyclopediaofmath.org/index.php/Dispersion_method Dispersion method]
* [http://en.wikipedia.org/wiki/Prime_gap Prime gap]
* [http://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture Second Hardy-Littlewood conjecture]
* [http://en.wikipedia.org/wiki/Twin_prime_conjecture Twin prime conjecture]

== Recent papers and notes ==

* [http://arxiv.org/abs/1304.3199 On the exponent of distribution of the ternary divisor function], Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
* [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes], Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages.
* [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes], Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
* [http://arxiv.org/abs/1305.6289 Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture], Janos Pintz, 27 May 2013.
* [http://arxiv.org/abs/1305.6369 A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million], T. S. Trudgian, 28 May 2013.
* [http://www.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf The Friedlander-Iwaniec sum], É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
* [http://terrytao.files.wordpress.com/2013/06/bounds.pdf Notes on Zhang's prime gaps paper], Terence Tao, 1 June 2013.
* [http://arxiv.org/abs/1306.0511 Bounded prime gaps in short intervals], Johan Andersson, 3 June 2013.
* [http://arxiv.org/abs/1306.0948 Bounded length intervals containing two primes and an almost-prime II], James Maynard, 5 June 2013.
* [http://arxiv.org/abs/1306.1497 A note on bounded gaps between primes], Janos Pintz, 6 June 2013.

== Media ==

* [http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989 First proof that infinitely many prime numbers come in pairs], Maggie McKee, Nature, 14 May 2013.
* [http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html Proof that an infinite number of primes are paired], Lisa Grossman, New Scientist, 14 May 2013.
* [https://www.simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/ Unheralded Mathematician Bridges the Prime Gap], Erica Klarreich, Simons science news, 20 May 2013.
** The article also appeared on Wired as "[http://www.wired.com/wiredscience/2013/05/twin-primes/ Unknown Mathematician Proves Elusive Property of Prime Numbers]".
* [http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html The Beauty of Bounded Gaps], Jordan Ellenberg, Slate, 22 May 2013.
* [http://www.newscientist.com/article/dn23644 Game of proofs boosts prime pair result by millions], Jacob Aron, New Scientist, 4 June 2013.

== Bibliography ==

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

* [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. [http://www.ams.org/mathscinet-getitem?mr=834613 MathSciNet] [http://link.springer.com/article/10.1007%2FBF02399204 Article]
* [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. [http://www.ams.org/mathscinet-getitem?mr=891581 MathSciNet] [https://eudml.org/doc/164255 Article]
* [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. [http://www.ams.org/mathscinet-getitem?mr=976723 MathSciNet] [http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Article]
* [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
* [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 [http://www.ams.org/mathscinet-getitem?mr=1836929 MathSciNet] [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Article]
* [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). [http://www.ams.org/mathscinet-getitem?mr=610700 MathSciNet] [http://www.math.ethz.ch/~kowalski/fouvry-iwaniec-on-a-theorem.pdf Article]
* [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. [http://www.ams.org/mathscinet-getitem?mr=719249 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4226.pdf Article]
* [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. [http://www.ams.org/mathscinet-getitem?mr=786351 MathSciNet] [http://www.jstor.org/stable/1971175 JSTOR]
* [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. [http://arxiv.org/abs/math/0508185 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2552109 MathSciNet]
* [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. [http://www.ams.org/mathscinet-getitem?mr=1726073 MathSciNet] [http://www.ccrwest.org/gordon/ants.pdf Article]
* [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. [http://www.ams.org/mathscinet-getitem?mr=340194 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/S0002-9904-1974-13434-8.pdf Article]
* [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. [http://www.ams.org/mathscinet-getitem?mr=396440 MathSciNet] [https://eudml.org/doc/205282 Article]
* [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. [http://arxiv.org/abs/math/0602599 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2414788 MathSciNet] [http://blms.oxfordjournals.org/content/40/2/298 Article]
* [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. [http://www.ams.org/mathscinet-getitem?mr=374060 MathSciNet] [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6718308 Article]
* [M1978] Hugh L. Montgomery, The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. [http://www.ams.org/mathscinet-getitem?mr=466048 MathSciNet] [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540922 Article]
* [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. [http://www.ams.org/mathscinet-getitem?mr=337832 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/home.html Article]
* [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. [http://www.ams.org/mathscinet-getitem?mr=130203 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa7/aa711.pdf Article]
* [S2007] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. [http://www.ams.org/mathscinet-getitem?mr=2265008 MathSciNet] [http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/ Article] [http://arxiv.org/abs/math/0605696 arXiv]

Bounded gaps between primes

2013-06-11T13:12:09Z

Hannes: /* World records */

== World records ==

* <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
* <math>k_0</math> is a quantity such that every admissible <math>k_0</math>-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in <math>k_0</math> lead to improvements in <math>H</math>. (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].)
* <math>\varpi</math> is a technical parameter related to a specialized form of the [https://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture Elliott-Halberstam conjecture]. Would like to be as large as possible. Improvements in <math>\varpi</math> lead to improvements in <math>k_0</math>. (The relationship is roughly of the form <math>k_0 \sim \varpi^{-3/2}</math>; there is an active discussion on optimising these improvements [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ here].) In more recent work, the single parameter <math>\varpi</math> is replaced by a pair <math>(\varpi,\delta)</math> (in previous work we had <math>\delta=\varpi</math>). Discussion on improving the values of <math>(\varpi,\delta)</math> is currently being held [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ here]. In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored.

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| 14 May
| 1/1,168 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| All subsequent work is based on Zhang's breakthrough paper.
|-
| 21 May
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| 28 May
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| 30 May
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| 31 May
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| 1 Jun
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| 2 Jun
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| 3 Jun
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| 4 Jun
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| 5 Jun
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429 ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| 6 Jun
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| 7 Jun
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>207 \varpi + 43\delta < \frac{1}{4}</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024 ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| June 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000* ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338* ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])
[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283 ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

== Polymath threads ==

* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart], Scott Morrison, 30 May 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang], Terence Tao, 3 June 2013. Inactive
* [http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/ Polymath proposal: bounded gaps between primes], Terence Tao, 4 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ Online reading seminar for Zhang’s “bounded gaps between primes], Terence Tao, 4 June 2013. Active
* [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/ More narrow admissible sets], Scott Morrison, 5 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ The elementary Selberg sieve and bounded prime gaps], Terence Tao, 8 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/ A combinatorial subset sum problem associated with bounded prime gaps], Terence Tao, 10 June 2013. Active

== Code and data ==

* [https://github.com/semorrison/polymath8 Hensely-Richards sequences], Scott Morrison
** [http://tqft.net/misc/finding%20k_0.nb A mathematica notebook for finding k_0], Scott Morrison
** [https://github.com/avi-levy/dhl python implementation], Avi Levy
* [http://www.opertech.com/primes/k-tuples.html k-tuple pattern data], Thomas J Engelsma
** [https://perswww.kuleuven.be/~u0040935/k0graph.png A graph of this data]
* [https://github.com/vit-tucek/admissible_sets Sifted sequences], Vit Tucek
* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Other sifted sequences], Christian Elsholtz
* [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/S0025-5718-01-01348-5.pdf Size of largest admissible tuples in intervals of length up to 1050], Clark and Jarvis
* [http://math.mit.edu/~drew/admissable_v0.1.tar C implementation of the "greedy-greedy" algorithm], Andrew Sutherland
* [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Dropbox for sequences], pedant
* [https://docs.google.com/spreadsheet/ccc?key=0Ao3urQ79oleSdEZhRS00X1FLQjM3UlJTZFRqd19ySGc&usp=sharing Spreadsheet for admissible sequences], Vit Tucek
* [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.0.zip Java code for optimising a given tuple V1.0], [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.1.zip Java code for optimising a given tuple V1.1], xfxie

== Errata ==

Page numbers refer to the file linked to for the relevant paper.

# Errata for Zhang's "[http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes]"
## Page 5: In the first display, <math>\mathcal{E}</math> should be multiplied by <math>\mathcal{L}^{2k_0+2l_0}</math>, because <math>\lambda(n)^2</math> in (2.2) can be that large, cf. (2.4).
## Page 14: In the final display, the constraint <math>(n,d_1=1</math> should be <math>(n,d_1)=1</math>.
## Page 35: In the display after (10.5), the subscript on <math>{\mathcal J}_i</math> should be deleted.
## Page 36: In the third display, a factor of <math>\tau(q_0r)^{O(1)}</math> may be needed on the right-hand side (but is ultimately harmless).
## Page 38: In the display after (10.14), <math>\xi(r,a;q_1,b_1;q_2,b_2;n,k)</math> should be <math>\xi(r,a;k;q_1,b_1;q_2,b_2;n)</math>.
## Page 42: In (12.3), <math>B</math> should probably be 2.
## Page 47: In the third display after (13.13), the condition <math>l \in {\mathcal I}_i(h)</math> should be <math>l \in {\mathcal I}_i(sh)</math>.
## Page 49: In the top line, a comma in <math>(h_1,h_2;,n_1,n_2)</math> should be deleted.
## Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
## Page 54: Three displays before (14.17), <math>\bar{r_2}(m_1+m_2)q</math> should be <math>\bar{r_2}(m_1+m_2)/q</math>.
# Errata for Motohashi-Pintz's "[http://arxiv.org/pdf/math/0602599v1.pdf A smoothed GPY sieve]"
## Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of <math>(\log \frac{R}{|D|})^{2\ell+1}</math>, (4.15) contains instead a factor of <math>(\log \frac{R/w}{|K|})^{2\ell+1}</math> which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of <math>\exp(-k\omega/3)</math> in (4.15) does not seem to be available for estimating the second sum in (5.14).
# Errata for Pintz's "[http://arxiv.org/pdf/1306.1497v1.pdf A note on bounded gaps between primes]"
## Page 7: In (2.39), the exponent of <math>3a/2</math> should instead be <math>-5a/2</math> (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
## Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

== Other relevant blog posts ==

* [http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/ Marker lecture III: “Small gaps between primes”], Terence Tao, 19 Nov 2008.
* [http://blogs.ethz.ch/kowalski/2009/01/22/the-goldston-pintz-yildirim-result-and-how-far-do-we-have-to-walk-to-twin-primes/ The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?], Emmanuel Kowalski, 22 Jan 2009.
* [http://www.math.columbia.edu/~woit/wordpress/?p=5865 Number Theory News], Peter Woit, 12 May 2013.
* [http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html Bounded Gaps Between Primes], Emily Riehl, 14 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ Bounded gaps between primes!], Emmanuel Kowalski, 21 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/04/bounded-gaps-between-primes-some-grittier-details/ Bounded gaps between primes: some grittier details], Emmanuel Kowalski, 4 June 2013.
** [http://www.math.ethz.ch/~kowalski/zhang-notes.pdf The slides from the talk mentioned in that post]
* [http://aperiodical.com/2013/06/bound-on-prime-gaps-bound-decreasing-by-leaps-and-bounds/ Bound on prime gaps bound decreasing by leaps and bounds], Christian Perfect, 8 June 2013.

== MathOverflow ==

* [http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture], 20 May 2013.
* [http://mathoverflow.net/questions/131825/a-technical-question-related-to-zhangs-result-of-bounded-prime-gaps A technical question related to Zhang’s result of bounded prime gaps], 25 May 2013.
* [http://mathoverflow.net/questions/132452/how-does-yitang-zhang-use-cauchys-inequality-and-theorem-2-to-obtain-the-error How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the <math>S_2</math> sum], 31 May 2013.
* [http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed Tightening Zhang’s bound], 3 June 2013.
** [http://meta.mathoverflow.net/discussion/1605/tightening-zhangs-bound/ Metathread for this post]
* [http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-len Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?], 3 June 2013.

== Wikipedia and other references ==

* [http://en.wikipedia.org/wiki/Bessel_function Bessel function]
* [http://en.wikipedia.org/wiki/Bombieri-Vinogradov_theorem Bombieri-Vinogradov theorem]
* [http://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem Brun-Titchmarsh theorem]
* [http://www.encyclopediaofmath.org/index.php/Dispersion_method Dispersion method]
* [http://en.wikipedia.org/wiki/Prime_gap Prime gap]
* [http://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture Second Hardy-Littlewood conjecture]
* [http://en.wikipedia.org/wiki/Twin_prime_conjecture Twin prime conjecture]

== Recent papers and notes ==

* [http://arxiv.org/abs/1304.3199 On the exponent of distribution of the ternary divisor function], Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
* [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes], Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages.
* [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes], Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
* [http://arxiv.org/abs/1305.6289 Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture], Janos Pintz, 27 May 2013.
* [http://arxiv.org/abs/1305.6369 A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million], T. S. Trudgian, 28 May 2013.
* [http://www.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf The Friedlander-Iwaniec sum], É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
* [http://terrytao.files.wordpress.com/2013/06/bounds.pdf Notes on Zhang's prime gaps paper], Terence Tao, 1 June 2013.
* [http://arxiv.org/abs/1306.0511 Bounded prime gaps in short intervals], Johan Andersson, 3 June 2013.
* [http://arxiv.org/abs/1306.0948 Bounded length intervals containing two primes and an almost-prime II], James Maynard, 5 June 2013.
* [http://arxiv.org/abs/1306.1497 A note on bounded gaps between primes], Janos Pintz, 6 June 2013.

== Media ==

* [http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989 First proof that infinitely many prime numbers come in pairs], Maggie McKee, Nature, 14 May 2013.
* [http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html Proof that an infinite number of primes are paired], Lisa Grossman, New Scientist, 14 May 2013.
* [https://www.simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/ Unheralded Mathematician Bridges the Prime Gap], Erica Klarreich, Simons science news, 20 May 2013.
** The article also appeared on Wired as "[http://www.wired.com/wiredscience/2013/05/twin-primes/ Unknown Mathematician Proves Elusive Property of Prime Numbers]".
* [http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html The Beauty of Bounded Gaps], Jordan Ellenberg, Slate, 22 May 2013.
* [http://www.newscientist.com/article/dn23644 Game of proofs boosts prime pair result by millions], Jacob Aron, New Scientist, 4 June 2013.

== Bibliography ==

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

* [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. [http://www.ams.org/mathscinet-getitem?mr=834613 MathSciNet] [http://link.springer.com/article/10.1007%2FBF02399204 Article]
* [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. [http://www.ams.org/mathscinet-getitem?mr=891581 MathSciNet] [https://eudml.org/doc/164255 Article]
* [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. [http://www.ams.org/mathscinet-getitem?mr=976723 MathSciNet] [http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Article]
* [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
* [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 [http://www.ams.org/mathscinet-getitem?mr=1836929 MathSciNet] [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Article]
* [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). [http://www.ams.org/mathscinet-getitem?mr=610700 MathSciNet] [http://www.math.ethz.ch/~kowalski/fouvry-iwaniec-on-a-theorem.pdf Article]
* [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. [http://www.ams.org/mathscinet-getitem?mr=719249 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4226.pdf Article]
* [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. [http://www.ams.org/mathscinet-getitem?mr=786351 MathSciNet] [http://www.jstor.org/stable/1971175 JSTOR]
* [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. [http://arxiv.org/abs/math/0508185 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2552109 MathSciNet]
* [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. [http://www.ams.org/mathscinet-getitem?mr=1726073 MathSciNet] [http://www.ccrwest.org/gordon/ants.pdf Article]
* [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. [http://www.ams.org/mathscinet-getitem?mr=340194 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/S0002-9904-1974-13434-8.pdf Article]
* [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. [http://www.ams.org/mathscinet-getitem?mr=396440 MathSciNet] [https://eudml.org/doc/205282 Article]
* [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. [http://arxiv.org/abs/math/0602599 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2414788 MathSciNet] [http://blms.oxfordjournals.org/content/40/2/298 Article]
* [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. [http://www.ams.org/mathscinet-getitem?mr=374060 MathSciNet] [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6718308 Article]
* [M1978] Hugh L. Montgomery, The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. [http://www.ams.org/mathscinet-getitem?mr=466048 MathSciNet] [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540922 Article]
* [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. [http://www.ams.org/mathscinet-getitem?mr=337832 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/home.html Article]
* [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. [http://www.ams.org/mathscinet-getitem?mr=130203 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa7/aa711.pdf Article]
* [S2007] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. [http://www.ams.org/mathscinet-getitem?mr=2265008 MathSciNet] [http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/ Article] [http://arxiv.org/abs/math/0605696 arXiv]

Finding narrow admissible tuples

2013-06-11T13:08:35Z

Hannes: /* Benchmarks */

Finding narrow admissible tuples

2013-06-11T12:52:46Z

Hannes: /* Benchmarks */

Finding narrow admissible tuples

2013-06-10T21:35:30Z

Hannes: /* Benchmarks */

Bounded gaps between primes

2013-06-10T21:32:41Z

Hannes: /* World records */

== World records ==

* <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
* <math>k_0</math> is a quantity such that every admissible <math>k_0</math>-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in <math>k_0</math> lead to improvements in <math>H</math>. (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].)
* <math>\varpi</math> is a technical parameter related to a specialized form of the [https://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture Elliott-Halberstam conjecture]. Would like to be as large as possible. Improvements in <math>\varpi</math> lead to improvements in <math>k_0</math>. (The relationship is roughly of the form <math>k_0 \sim \varpi^{-3/2}</math>; there is an active discussion on optimising these improvements [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ here].) In more recent work, the single parameter <math>\varpi</math> is replaced by a pair <math>(\varpi,\delta)</math> (in previous work we had <math>\delta=\varpi</math>). Discussion on improving the values of <math>(\varpi,\delta)</math> is currently being held [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ here]. In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored.

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| 14 May
| 1/1,168 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| All subsequent work is based on Zhang's breakthrough paper.
|-
| 21 May
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| 28 May
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| 30 May
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| 31 May
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| 1 Jun
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| 2 Jun
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| 3 Jun
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| 4 Jun
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| 5 Jun
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429 ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| 6 Jun
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| 7 Jun
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>207 \varpi + 43\delta < \frac{1}{4}</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024 ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| June 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
|
| [http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])
[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283 ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])
|
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

== Polymath threads ==

* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart], Scott Morrison, 30 May 2013. Inactive
* [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang], Terence Tao, 3 June 2013. Inactive
* [http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/ Polymath proposal: bounded gaps between primes], Terence Tao, 4 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ Online reading seminar for Zhang’s “bounded gaps between primes], Terence Tao, 4 June 2013. Active
* [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/ More narrow admissible sets], Scott Morrison, 5 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ The elementary Selberg sieve and bounded prime gaps], Terence Tao, 8 June 2013. Active
* [http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/ A combinatorial subset sum problem associated with bounded prime gaps], Terence Tao, 10 June 2013. Active

== Code and data ==

* [https://github.com/semorrison/polymath8 Hensely-Richards sequences], Scott Morrison
** [http://tqft.net/misc/finding%20k_0.nb A mathematica notebook for finding k_0], Scott Morrison
** [https://github.com/avi-levy/dhl python implementation], Avi Levy
* [http://www.opertech.com/primes/k-tuples.html k-tuple pattern data], Thomas J Engelsma
** [https://perswww.kuleuven.be/~u0040935/k0graph.png A graph of this data]
* [https://github.com/vit-tucek/admissible_sets Sifted sequences], Vit Tucek
* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Other sifted sequences], Christian Elsholtz
* [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/S0025-5718-01-01348-5.pdf Size of largest admissible tuples in intervals of length up to 1050], Clark and Jarvis
* [http://math.mit.edu/~drew/admissable_v0.1.tar C implementation of the "greedy-greedy" algorithm], Andrew Sutherland
* [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Dropbox for sequences], pedant
* [https://docs.google.com/spreadsheet/ccc?key=0Ao3urQ79oleSdEZhRS00X1FLQjM3UlJTZFRqd19ySGc&usp=sharing Spreadsheet for admissible sequences], Vit Tucek
* [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.0.zip Java code for optimising a given tuple V1.0], [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.1.zip Java code for optimising a given tuple V1.1], xfxie

== Errata ==

Page numbers refer to the file linked to for the relevant paper.

# Errata for Zhang's "[http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes]"
## Page 5: In the first display, <math>\mathcal{E}</math> should be multiplied by <math>\mathcal{L}^{2k_0+2l_0}</math>, because <math>\lambda(n)^2</math> in (2.2) can be that large, cf. (2.4).
## Page 14: In the final display, the constraint <math>(n,d_1=1</math> should be <math>(n,d_1)=1</math>.
## Page 35: In the display after (10.5), the subscript on <math>{\mathcal J}_i</math> should be deleted.
## Page 36: In the third display, a factor of <math>\tau(q_0r)^{O(1)}</math> may be needed on the right-hand side (but is ultimately harmless).
## Page 38: In the display after (10.14), <math>\xi(r,a;q_1,b_1;q_2,b_2;n,k)</math> should be <math>\xi(r,a;k;q_1,b_1;q_2,b_2;n)</math>.
## Page 42: In (12.3), <math>B</math> should probably be 2.
## Page 47: In the third display after (13.13), the condition <math>l \in {\mathcal I}_i(h)</math> should be <math>l \in {\mathcal I}_i(sh)</math>.
## Page 49: In the top line, a comma in <math>(h_1,h_2;,n_1,n_2)</math> should be deleted.
## Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
## Page 54: Three displays before (14.17), <math>\bar{r_2}(m_1+m_2)q</math> should be <math>\bar{r_2}(m_1+m_2)/q</math>.
# Errata for Motohashi-Pintz's "[http://arxiv.org/pdf/math/0602599v1.pdf A smoothed GPY sieve]"
## Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of <math>(\log \frac{R}{|D|})^{2\ell+1}</math>, (4.15) contains instead a factor of <math>(\log \frac{R/w}{|K|})^{2\ell+1}</math> which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of <math>\exp(-k\omega/3)</math> in (4.15) does not seem to be available for estimating the second sum in (5.14).
# Errata for Pintz's "[http://arxiv.org/pdf/1306.1497v1.pdf A note on bounded gaps between primes]"
## Page 7: In (2.39), the exponent of <math>3a/2</math> should instead be <math>-5a/2</math> (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
## Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

== Other relevant blog posts ==

* [http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/ Marker lecture III: “Small gaps between primes”], Terence Tao, 19 Nov 2008.
* [http://blogs.ethz.ch/kowalski/2009/01/22/the-goldston-pintz-yildirim-result-and-how-far-do-we-have-to-walk-to-twin-primes/ The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?], Emmanuel Kowalski, 22 Jan 2009.
* [http://www.math.columbia.edu/~woit/wordpress/?p=5865 Number Theory News], Peter Woit, 12 May 2013.
* [http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html Bounded Gaps Between Primes], Emily Riehl, 14 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ Bounded gaps between primes!], Emmanuel Kowalski, 21 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/04/bounded-gaps-between-primes-some-grittier-details/ Bounded gaps between primes: some grittier details], Emmanuel Kowalski, 4 June 2013.
** [http://www.math.ethz.ch/~kowalski/zhang-notes.pdf The slides from the talk mentioned in that post]
* [http://aperiodical.com/2013/06/bound-on-prime-gaps-bound-decreasing-by-leaps-and-bounds/ Bound on prime gaps bound decreasing by leaps and bounds], Christian Perfect, 8 June 2013.

== MathOverflow ==

* [http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture], 20 May 2013.
* [http://mathoverflow.net/questions/131825/a-technical-question-related-to-zhangs-result-of-bounded-prime-gaps A technical question related to Zhang’s result of bounded prime gaps], 25 May 2013.
* [http://mathoverflow.net/questions/132452/how-does-yitang-zhang-use-cauchys-inequality-and-theorem-2-to-obtain-the-error How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the <math>S_2</math> sum], 31 May 2013.
* [http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed Tightening Zhang’s bound], 3 June 2013.
** [http://meta.mathoverflow.net/discussion/1605/tightening-zhangs-bound/ Metathread for this post]
* [http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-len Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?], 3 June 2013.

== Wikipedia and other references ==

* [http://en.wikipedia.org/wiki/Bessel_function Bessel function]
* [http://en.wikipedia.org/wiki/Bombieri-Vinogradov_theorem Bombieri-Vinogradov theorem]
* [http://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem Brun-Titchmarsh theorem]
* [http://www.encyclopediaofmath.org/index.php/Dispersion_method Dispersion method]
* [http://en.wikipedia.org/wiki/Prime_gap Prime gap]
* [http://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture Second Hardy-Littlewood conjecture]
* [http://en.wikipedia.org/wiki/Twin_prime_conjecture Twin prime conjecture]

== Recent papers and notes ==

* [http://arxiv.org/abs/1304.3199 On the exponent of distribution of the ternary divisor function], Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
* [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes], Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages.
* [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes], Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
* [http://arxiv.org/abs/1305.6289 Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture], Janos Pintz, 27 May 2013.
* [http://arxiv.org/abs/1305.6369 A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million], T. S. Trudgian, 28 May 2013.
* [http://www.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf The Friedlander-Iwaniec sum], É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
* [http://terrytao.files.wordpress.com/2013/06/bounds.pdf Notes on Zhang's prime gaps paper], Terence Tao, 1 June 2013.
* [http://arxiv.org/abs/1306.0511 Bounded prime gaps in short intervals], Johan Andersson, 3 June 2013.
* [http://arxiv.org/abs/1306.0948 Bounded length intervals containing two primes and an almost-prime II], James Maynard, 5 June 2013.
* [http://arxiv.org/abs/1306.1497 A note on bounded gaps between primes], Janos Pintz, 6 June 2013.

== Media ==

* [http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989 First proof that infinitely many prime numbers come in pairs], Maggie McKee, Nature, 14 May 2013.
* [http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html Proof that an infinite number of primes are paired], Lisa Grossman, New Scientist, 14 May 2013.
* [https://www.simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/ Unheralded Mathematician Bridges the Prime Gap], Erica Klarreich, Simons science news, 20 May 2013.
** The article also appeared on Wired as "[http://www.wired.com/wiredscience/2013/05/twin-primes/ Unknown Mathematician Proves Elusive Property of Prime Numbers]".
* [http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html The Beauty of Bounded Gaps], Jordan Ellenberg, Slate, 22 May 2013.
* [http://www.newscientist.com/article/dn23644 Game of proofs boosts prime pair result by millions], Jacob Aron, New Scientist, 4 June 2013.

== Bibliography ==

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

* [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. [http://www.ams.org/mathscinet-getitem?mr=834613 MathSciNet] [http://link.springer.com/article/10.1007%2FBF02399204 Article]
* [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. [http://www.ams.org/mathscinet-getitem?mr=891581 MathSciNet] [https://eudml.org/doc/164255 Article]
* [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. [http://www.ams.org/mathscinet-getitem?mr=976723 MathSciNet] [http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Article]
* [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
* [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 [http://www.ams.org/mathscinet-getitem?mr=1836929 MathSciNet] [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Article]
* [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). [http://www.ams.org/mathscinet-getitem?mr=610700 MathSciNet] [http://www.math.ethz.ch/~kowalski/fouvry-iwaniec-on-a-theorem.pdf Article]
* [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. [http://www.ams.org/mathscinet-getitem?mr=719249 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4226.pdf Article]
* [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. [http://www.ams.org/mathscinet-getitem?mr=786351 MathSciNet] [http://www.jstor.org/stable/1971175 JSTOR]
* [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. [http://arxiv.org/abs/math/0508185 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2552109 MathSciNet]
* [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. [http://www.ams.org/mathscinet-getitem?mr=1726073 MathSciNet] [http://www.ccrwest.org/gordon/ants.pdf Article]
* [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. [http://www.ams.org/mathscinet-getitem?mr=340194 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/S0002-9904-1974-13434-8.pdf Article]
* [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. [http://www.ams.org/mathscinet-getitem?mr=396440 MathSciNet] [https://eudml.org/doc/205282 Article]
* [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. [http://arxiv.org/abs/math/0602599 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2414788 MathSciNet] [http://blms.oxfordjournals.org/content/40/2/298 Article]
* [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. [http://www.ams.org/mathscinet-getitem?mr=374060 MathSciNet] [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6718308 Article]
* [M1978] Hugh L. Montgomery, The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. [http://www.ams.org/mathscinet-getitem?mr=466048 MathSciNet] [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540922 Article]
* [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. [http://www.ams.org/mathscinet-getitem?mr=337832 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/home.html Article]
* [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. [http://www.ams.org/mathscinet-getitem?mr=130203 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa7/aa711.pdf Article]
* [S2007] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. [http://www.ams.org/mathscinet-getitem?mr=2265008 MathSciNet] [http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/ Article] [http://arxiv.org/abs/math/0605696 arXiv]

Finding narrow admissible tuples

2013-06-10T21:14:47Z

Hannes: /* Benchmarks */

Finding narrow admissible tuples

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Hannes: /* Benchmarks */

Bounded gaps between primes

2013-06-10T16:33:10Z

Hannes: added some <math>-tags

Bounded gaps between primes

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Hannes: /* Code and data */

Finding narrow admissible tuples

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Hannes: /* Greedy-Schinzel sieve */

Finding narrow admissible tuples

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Hannes:

Finding narrow admissible tuples

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Hannes: /* Benchmarks */

Finding narrow admissible tuples

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Hannes: